Global Existence of Small Data Solutions to Weakly Coupled Systems of Semi-Linear Fractional σ–Evolution Equations with Mass and Different Nonlinear Memory terms

Abstract

We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the permissible range of exponents in a specific equation. Using $L^p-L^q$ estimates for solutions to the corresponding linear fractional $\sigma$–evolution equations with vanishing right-hand sides, and applying a fixed-point argument, the existence of small data solutions is established for some admissible range of powers $(p_1,p_2,\dots ,p_k)$.

1. Introduction

This paper is devoted to the weakly coupled system of k semi-linear fractional $\sigma$–evolution equations. The system incorporates mass terms and different memory terms and our focus is on small data solutions to the corresponding Cauchy problem.

$$\left\{\begin{array}{c}{\partial }_t^{1+{\alpha }_1}{u}_1+{(-\Delta )}^{{\sigma }_1}{u}_{\ell }+M_1^2{u}_1=F_{{\mu }_1,p_1}\left(u_k\right),\hfill \\ {\partial }_t^{1+{\alpha }_2}{u}_2+{(-\Delta )}^{{\sigma }_2}{u}_2+M_2^2{u}_2=F_{{\mu }_2,p_2}\left(u_1\right),\hfill \\ .\hfill \\ .\hfill \\ {\partial }_t^{1+{\alpha }_k}{u}_k+{(-\Delta )}^{{\sigma }_k}{u}_k+M_k^2{u}_k=F_{{\mu }_k,p_k}\left(u_{k-1}\right),\hfill \\ u_{\ell }(0,x)=u_{0l}\left(x\right),{\partial }_t{u}_{\ell }(0,x)=0,\ell =1,2,\dots ,k,\hfill \end{array}\right.$$

(1)

where $k\ge 2$, for $l=1,\cdots ,k$, ${\alpha }_l\in (0,1)$, ${\mu }_l\in (0,1)$, $p_l>1$, $M_l>0$, ${\sigma }_l\ge 1$, $(t,x)\in [0,\infty )\times {\mathbb{R}}^d$, with

$$F_{\mu ,p}\left(u\right)(t,x):=\underset{0}{\overset{t}{\int }}{(t-s)}^{-\mu }{\left|u(s,x)\right|}^{p}ds.$$

(2)

The fractional derivative is defined as follows: ${\partial }_t^{1+{\alpha }_l}u=D_t^{{\alpha }_l}\left(u_t\right)$, where

$$D_t^{{\alpha }_l}\left(f\right)={\partial }_t\left(I_t^{1-{\alpha }_l}f\right)\mathrm{and}{I}_t^{\beta }f=\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}f\left(s\right)ds\mathrm{for}\beta >0.$$

Here, $D_t^{\alpha }\left(f\right)$ and $I_t^{\beta }f$ denote the fractional Riemann–Liouville derivative and the fractional Riemann–Liouville integral, respectively, of f in $[0,t]$, and $\Gamma$ is the Euler Gamma function.

In this discussion, we will illustrate two distinct Cauchy problems: the semi-linear heat equation and the semi-linear wave equation.

Firstly, let us consider the semi-linear heat equation:

$$u_t-\Delta u={\left|u\right|}^p,u(0,x)=u_0\left(x\right).$$

According to Fujita’s results in [1], the critical exponent for this equation is defined as $p_{\mathrm{Fuj}}:=1+\frac{2}{n}$. It is noteworthy that for $p>p_{\mathrm{Fuj}}$, small initial data solutions exist globally (in time), while for $1<p<p_{\mathrm{Fuj}}$, a blow-up phenomenon occurs. The critical case $p=p_{\mathrm{Fuj}}$ was further studied in [2,3], where it was shown that blow-up does indeed occur.

Moving on, let us shift our focus to the semi-linear wave equation:

$$u_{tt}-\Delta u={\left|u\right|}^p,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right).$$

For the specific case when $n=3$, ref. [4] proved that the critical exponent can be determined as the positive root of the quadratic equation $(n-1)p^2-(n+1)p-2=0$. The exponent obtained from the quadratic equation is known as the Strauss exponent, denoted as $p_S.$ Based on the Strauss exponent $p_S$, we can conclude that there is the global (in time) existence of small data weak solutions when $p\ge p_S$. However, for $p>1$ and large data, we can only expect the local (in time) existence of solutions. The optimality of the Strauss exponent $p_S$ in ${\mathbb{R}}^2$ was demonstrated in [5,6]. After that, the global existence of solutions for $n=2,3$ was treated in [7], while for $n\ge 4$, it was addressed in [8,9]. The nonexistence of solutions with compactly supported data was studied in [10] for the range $1<p<\frac{n+1}{n-1}$. For the specific case of $n=3$, optimal results were proven in [11] for $p=1+\sqrt{2}$. Moreover, in [12], it was proved that for $n>3$ and $1<p<p_S$, there is a nonexistence result for small data.

In 2017, D’Abbicco et al. [13] studied the semi-linear fractional wave equation, which can be expressed as follows:

$${\partial }_t^{1+\lambda }u-\Delta u={\left|u\right|}^p,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),$$

(3)

where $\lambda \in (0,1)$, which represents the fractional Riemann–Liouville derivative. The authors successfully proved the critical power for the existence of solutions with small initial data in spatial dimensions that are relatively low. The case of non-null Cauchy data and the use of the Caputo fractional order were studied in [13].

In [14], they proved the global (in time) existence of small data solutions for semi-linear fractional $\sigma$-evolution equations. These equations incorporated either mass or power nonlinearity. Furthermore, a related problem was addressed in [15], where instead of the power nonlinearity, a memory term was considered.

For the weakly coupled system consisting of semi-linear heat equations, we have the following equations:

$$\begin{array}{c}{u}_t-\Delta u={\left|v\right|}^p,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),\hfill \\ v_t-\Delta v={\left|u\right|}^q,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),\hfill \end{array}$$

where $t\in [0,\infty ),x\in {\mathbb{R}}^d$, and $p,q>1$ with $pq>1$. In [16], it was shown that the exponents p and q satisfying

$$\frac{d}{2}=\frac{\max \{p,q\}+1}{pq-1}$$

are critical. This means that solutions exist globally if $\frac{d}{2}>\frac{\max \{p,q\}+1}{pq-1}$, while blow-up occurs for the opposite case. For more details on the system of semi-linear heat equations, please refer to [17,18,19,20].

Considerations are made in several papers regarding weakly coupled systems of semi-linear classical damped wave equations with power nonlinearities. The specific problem of interest is:

$$\begin{array}{c}{u}_{tt}-\Delta u+u_t={\left|v\right|}^p,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),\hfill \\ v_{tt}-\Delta v+v_t={\left|u\right|}^q,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),\hfill \end{array}$$

(4)

where $t\in [0,\infty ),x\in {\mathbb{R}}^d$. In 2007, Sun and Wang proved in [21] that

$$\lambda :=\frac{\max \{p;q\}+1}{pq-1}<\frac{d}{2},$$

(5)

For the case of $d=1$ or $d=3$, it has been proven that the solution exists globally in time for small initial data in weakly coupled systems of semi-linear classical damped wave equations with power nonlinearities. However, if $\lambda \ge \frac{d}{2}$, it has been shown that every solution with a positive average value does not exist globally.

In the paper [22], these results were generalized to the case where $d=1,2,3$. Additionally, improved time-decay estimates have been provided specifically for the case of $d=2$. In 2014, Nishihara and Wakasugi used the weighted energy method to prove the critical exponent for any space dimension in [23]. Furthermore, considering time-dependent dissipation terms, the authors in [24,25,26] demonstrated the global (in time) existence of small data solutions under certain conditions that illustrate the interplay between the exponents of the power nonlinearities.

During the last years, many authors have studied the Cauchy problem for weakly coupled systems, see, e.g., [24,27,28], where the derivative introduced in their work is the classical derivative. In [29], the authors studied a weakly coupled system where the fractional derivative involves in the equations with special Cauchy data.

The paper is organized into several sections. First, we provide an overview of the study and present the main results (Section 2). Following that, Section 3 introduces the necessary background information and definitions for the foundation used to prove the results. Then, the proofs of the theorems are presented, utilizing previous estimates of linear equations (Section 5). Finally, Section 6 summarizes the study, highlights its contributions, and suggests potential directions for future research.

In a recent paper [30], the author investigated the following Cauchy problem for weakly coupled systems of semi-linear fractional $\sigma$-evolution equations. The system involves mass terms and different power nonlinearities.

$$\left\{\begin{array}{c}{\partial }_t^{1+{\alpha }_1}u+{(-\Delta )}^{{\sigma }_1}u+M_1^{2}u={\left|v\right|}^{p_1},\hfill \\ {\partial }_t^{1+{\alpha }_2}v+{(-\Delta )}^{{\sigma }_2}v+M_2^{2}v={\left|u\right|}^{p_2},\hfill \\ u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right),u_t(0,x)=v_t(0,x)=0,\hfill \end{array}\right.$$

(6)

where ${\alpha }_k\in (0,1)$, ${\sigma }_k\ge 1$, $M_k>0$ for $k=1,2$, $(t,x)\in [0,\infty )\times {\mathbb{R}}^d$, ${\partial }_t^{1+{\alpha }_k}u=D_t^{{\alpha }_k}\left(u_t\right)$ with

$$D_t^{{\alpha }_k}\left(f\right)={\partial }_t\left(I_t^{1-{\alpha }_k}f\right)\mathrm{and}{I}_t^{\beta }f=\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}f\left(s\right)ds\mathrm{for}\beta >0.$$

$D_t^{\alpha }\left(f\right)$ and $I_t^{\beta }f$ are defined as above.

The author proved the following results.

Proposition 1.

Let us assume $0<{\alpha }_1,{\alpha }_2<1$, ${\sigma }_1,{\sigma }_2\ge 1$, $M_1,M_2>0$ and $m_1,m_2\ge 1$. Assume that for all $\delta >0$

$$\begin{array}{c}\hfill p_1>\max \left\{\frac{m_2}{m_1}-\delta ,\frac{1}{1-{\alpha }_2}\right\},\end{array}$$

and

$$\begin{array}{c}\hfill p_2>\max \left\{\frac{m_1}{m_2}-\delta ,\frac{1}{1-{\alpha }_1}\right\}.\end{array}$$

Then, there exists a positive constant ε, such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}:=\left(L^{m_1}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)\times \left(L^{m_2}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right),$$

with $\parallel (u_0,v_0){\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$, we have a uniquely determined global (in time) Sobolev solution

$$(u,v)\in \mathcal{C}\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)\times \mathcal{C}\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)$$

to the Cauchy problem (6). Moreover, for all $s\ge 0$, the solution satisfies the following decay estimates:

$${\parallel u(s,·)\parallel }_{L^r}\lesssim {(1+s)}^{{\alpha }_1-1}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}forallr\in [m_1,\infty ],$$

$${\parallel v(s,·)\parallel }_{L^r}\lesssim {(1+s)}^{{\alpha }_2-1}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}forallr\in [m_2,\infty ].$$

Proposition 2

(Loss of decay). Let us assume $0<{\alpha }_1,{\alpha }_2<1$, ${\sigma }_1,{\sigma }_2\ge 1$, $M_1,M_2>0$ and $m_1,m_2\ge 1$. Assume that for all $\delta >0$

$$\begin{array}{c}\hfill \max \left\{1,\frac{{\alpha }_1}{1-{\alpha }_2},\frac{m_2}{m_1}-\delta \right\}<p_1<\frac{1}{1-{\alpha }_2},\end{array}$$

$$\begin{array}{c}\hfill p_2>\max \left\{\frac{m_1}{m_2}-\delta ,\frac{1}{p_1(1-{\alpha }_2)-{\alpha }_1}\right\}.\end{array}$$

Then, there exists a positive constant ε, such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_{m_1}^{m_2}with{\parallel (u_0,v_0)\parallel }_{{\mathcal{A}}_{m_1}^{m_2}}\le \epsilon$$

we have a uniquely determined global (in time) Sobolev solution

$$(u,v)\in \mathcal{C}\left([0,\infty ),L^{m_1}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)\times \mathcal{C}\left([0,\infty ),L^{m_2}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)$$

to the Cauchy problem (6). Moreover, for all $s\ge 0$, the solution satisfies the following decay estimates:

$$\begin{array}{cc}\hfill & {\parallel u(s,·)\parallel }_{L^r}\lesssim {(1+t)}^{{\alpha }_1-p_1(1-{\alpha }_2)}{\parallel u_0\parallel }_{L^{m_1}\cap L^{\infty }}forallr\in [m_1,\infty ],\hfill \\ & {\parallel v(s,·)\parallel }_{L^r}\lesssim {(1+s)}^{{\alpha }_2-1}{\parallel v_0\parallel }_{L^{m_2}\cap L^{\infty }}forallr\in [m_2,\infty ]\hfill \end{array}$$

Proposition 3

(Loss of decay). Let us assume $0<{\alpha }_1,{\alpha }_2<1$, ${\sigma }_1,{\sigma }_2\ge 1$ and $M_1,M_2>0$. Assume that $\delta >0$ is small enough for all

$$\begin{array}{c}\hfill p_1=\frac{1}{1-{\alpha }_2}and{p}_2>\frac{1}{1-{\alpha }_1-\delta }.\end{array}$$

Then, there exists a positive constant ε, such that for any data

$$(u_0,v_0)\in {\mathcal{A}}_1^{1}with{\parallel (u_0,v_0)\parallel }_{{\mathcal{A}}_1^1}\le \epsilon$$

we have a uniquely determined global (in time) Sobolev solution

$$(u,v)\in \mathcal{C}\left([0,\infty ),L^1\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)\times \mathcal{C}\left([0,\infty ),L^1\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)$$

to the Cauchy problem (6). Moreover, for all $s\ge 0$, the solution satisfies the following decay estimates:

$$\begin{array}{cc}\hfill & {\parallel u(s,·)\parallel }_{L^r}\lesssim ln(2+s){(1+s)}^{{\alpha }_1-1}{\parallel u_0\parallel }_{L^1\cap L^{\infty }}forallr\in [1,\infty ],\hfill \\ & {\parallel v(s,·)\parallel }_{L^r}\lesssim {(1+s)}^{{\alpha }_2-1}{\parallel v_0\parallel }_{L^1\cap L^{\infty }}forallr\in [1,\infty ].\hfill \end{array}$$

In the subsequent sections, we will utilize the notation $f\lesssim g$, indicating the existence of a non-negative constant C, such that $f\le Cg$. Our main findings concerning the global (in time) existence of small data Sobolev solutions will be presented in the following section.

2. Main Results

Theorem 1.

Let us assume $0<{\alpha }_{\ell }<1$, ${\alpha }_{\ell }<{\mu }_{\ell }<1$, ${\sigma }_{\ell }\ge 1$, $m_l\ge 1$, and $M_{\ell }>0$ for all $\ell =1,\dots ,k$. Assume that for all $\delta >0$

$$\begin{array}{cc}\hfill p_1& >\max \left\{\frac{m_k}{m_1}-\delta ,\frac{1}{{\mu }_k-{\alpha }_k}\right\},\hfill \\ \hfill p_{\ell }& >\max \left\{\frac{m_{l-1}}{m_l}-\delta ,\frac{1}{{\mu }_{\ell -1}-{\alpha }_{\ell -1}}\right\},forall\ell =2,\dots ,k.\hfill \end{array}$$

Then, there exists a positive constant ε, such that for any data $(u_{01},\dots ,u_{0k})\in {\mathcal{A}}_k:={\prod }_{\ell =1}^k\left(L^{m_l}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)$ with $\parallel (u_{01},\dots ,u_{0k}){\parallel }_{{\mathcal{A}}_k}\le \epsilon$, we have a uniquely determined global (in time) Sobolev solution

$$u\in \prod _{\ell =1}^k\mathcal{C}\left([0,\infty ),L^{m_l}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)$$

to the Cauchy problem (1). Moreover, for all $s\ge 0$ and $l=1,\dots ,k$, the solution satisfies the following decay estimates:

$$\begin{array}{c}\hfill \parallel u_{\ell }{(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_{\ell }-{\mu }_{\ell }}{\parallel u_{0l}\parallel }_{L^{m_l}\cap L^{\infty }}forallq\in [m_l,\infty ].\end{array}$$

Theorem 2

(Loss of decay). Let us assume $0<{\alpha }_{\ell }<1$, ${\alpha }_{\ell }<{\mu }_{\ell }<1$, ${\sigma }_{\ell }\ge 1$, $m_l\ge 1$, and $M_{\ell }>0$ for all $\ell =1,\dots ,k$. Assume that for all $\delta >0$

$$\begin{array}{c}\hfill \max \left\{1,\frac{{\alpha }_1-{\mu }_1+1}{{\mu }_k-{\alpha }_k},\frac{m_k}{m_1}-\delta \right\}<p_1<\frac{1}{{\mu }_k-{\alpha }_k},\end{array}$$

$$\begin{array}{c}\hfill \max \left\{1,\frac{{\alpha }_2-{\mu }_2+1}{{\mu }_1-{\alpha }_1-{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)},\frac{m_1}{m_2}-\delta \right\}<p_2<\frac{1}{{\mu }_1-{\alpha }_1-{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}\end{array}$$

and for $l=3,\cdots ,k-1$

$$\begin{array}{cc}\hfill p_{\ell }& <\frac{1}{{\mu }_{\ell -1}-{\alpha }_{\ell -1}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})},\hfill \\ \hfill p_l>& \max \left\{1,\frac{{\alpha }_{\ell }-{\mu }_{\ell }+1}{{\mu }_{\ell -1}-{\alpha }_{\ell -1}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})},\frac{m_{l-1}}{m_l}-\delta \right\}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill p_k>\max \left\{\frac{m_k}{m_{k-1}}-\delta ,\frac{1}{{\mu }_{k-1}-{\alpha }_{k-1}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{k-2})}^{({\mu }_k,\dots ,{\mu }_{k-2})}(p_1,\dots ,p_{k-1})}\right\},\end{array}$$

where, for $l=3,\cdots ,k-1$

$$\begin{array}{c}\hfill \begin{array}{c}\left\{\begin{array}{c}{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)=1-p_1({\mu }_k-{\alpha }_k)\hfill \\ {\gamma }_{({\alpha }_k,{\alpha }_1)}^{({\mu }_k,{\mu }_1)}(p_1,p_2)=1-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)\hfill \\ {\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -1})}^{({\mu }_k,\dots ,{\mu }_{\ell -1})}(p_1,\dots ,p_{\ell })=1-p_{\ell }({\mu }_{\ell }-{\alpha }_{\ell })+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1}).\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(7)

Then, there exists a positive constant ε, such that for any data

$$(u_{01},\dots ,u_{0k})\in {\mathcal{A}}_k:=\prod _{\ell =1}^k\left(L^{m_l}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)with{\parallel (u_{01},\dots ,u_{0k})\parallel }_{{\mathcal{A}}_k}\le \epsilon ,$$

we have a uniquely determined global (in time) Sobolev solution

$$u\in \prod _{\ell =1}^k\mathcal{C}\left([0,\infty ),L^{m_l}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)$$

to the Cauchy problem (1). Moreover, for all $s\ge 0$ and $l=2,\cdots ,k-1$, the solution satisfies the decay estimate

$$\begin{array}{cc}\hfill & \parallel u_1{(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_1-{\mu }_1+{\gamma }_{{\alpha }_k}^{{\mu }_k}\left(p_1\right)}{\parallel u_{01}\parallel }_{L^{m_1}\cap L^{\infty }}forallq\in [m_1,\infty ],\hfill \\ & \parallel u_{\ell }{(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_{\ell }-{\mu }_{\ell }+{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -1})}^{({\mu }_k,\dots ,{\mu }_{\ell -1})}(p_1,\dots ,p_{\ell })}{\parallel u_{0l}\parallel }_{L^{m_l}\cap L^{\infty }}forallq\in [m_l,\infty ],\hfill \\ & \parallel u_k{(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_k-{\mu }_k}{\parallel u_{0k}\parallel }_{L^{m_k}\cap L^{\infty }}forallq\in [m_k,\infty ].\hfill \end{array}$$

We suppose $m_1=m_2=1$ in the following result.

Theorem 3

(Loss of decay). Let us assume $0<{\alpha }_{\ell }<1$, ${\alpha }_{\ell }<{\mu }_{\ell }<1$, ${\sigma }_{\ell }\ge 1$, $M_{\ell }>0$ for all $\ell =1,\dots ,k$. Assume that for all $\delta >0$

$$\begin{array}{c}\hfill p_1=\frac{1}{{\mu }_k-{\alpha }_k}.\end{array}$$

$$\begin{array}{c}\hfill p_{\ell }=\frac{1}{{\mu }_{\ell -1}-{\alpha }_{\ell -1}}.\ell =2,\dots ,k-1\end{array}$$

$$\begin{array}{c}\hfill p_k>\frac{1}{{\mu }_{k-1}-{\alpha }_{k-1}-\delta \gamma \left(p_{k-1}\right)},\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{c}\left\{\begin{array}{c}\gamma \left(p_1\right)=1\hfill \\ \gamma \left(p_l\right)=1+p_l\gamma \left(p_{l-1}\right),forl=2,\cdots ,k-1.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(8)

Then, there exists a positive constant ε, such that for any data

$$(u_{01},\dots ,u_{0k})\in {\mathcal{A}}_k=\prod _{\ell =1}^k\left(L^1\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)with{\parallel (u_{01},\dots ,u_{0k})\parallel }_{{\mathcal{A}}_k}\le \epsilon$$

we have a uniquely determined global (in time) Sobolev solution

$$u\in \prod _{\ell =1}^k\mathcal{C}\left([0,\infty ),L^1\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)$$

to the Cauchy problem (1). Moreover, for $l=1,\cdots ,k-1$ and for all $s\ge 0$, the solution satisfies the following decay estimate:

$$\begin{array}{cc}\hfill & \parallel u_{\ell }{(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_{\ell }-{\mu }_{\ell }}{\left(ln(2+s)\right)}^{\gamma \left(p_l\right)}{\parallel u_{0l}\parallel }_{L^1\cap L^{\infty }}forallq\in [1,\infty ],\hfill \\ & \parallel u_k{(s,·)\parallel }_{L^q}\lesssim {(1+s)}^{{\alpha }_k-{\mu }_k}{\parallel u_{0k}\parallel }_{L^1\cap L^{\infty }}forallq\in [1,\infty ].\hfill \end{array}$$

Remark 1.

The nonlinear term $F_{\mu ,p}(t,w)$ in (2) may be written as

$$F_{\mu ,p}(t,w)=\Gamma (1-\mu )I_t^{1-\mu }{\left(\right|w|}^p)$$

where Γ is the Euler Gamma function, and $I_t^{1-\mu }{\left(\right|w|}^p)$ is the fractional Riemann–Liouville integral of ${\left|w\right|}^p$ in $[0,t]$. Therefore, it is reasonable to expect that the relations with the power nonlinearities introduced in Proposition 1, Proposition 2, and Proposition 3 as ${\mu }_l$ tend to 1, for all $l=1,\cdots ,k$ and $k=2$.

3. Preliminaries

Let us consider the Cauchy problem

$$\begin{array}{c}{\partial }_t^{1+\alpha }v+{(-\Delta )}^{\sigma }v+m^{2}v=F(t,x)\hfill \\ v(x,0)=v_0\left(x\right),v_t(0,x)=0,\hfill \end{array}$$

(9)

With parameters $\alpha \in (0,1)$, $\sigma \ge 1$, and $m>0$, and under the data condition $v_t(0,x)=0$, the problem can be formally transformed into an integral equation. The solution of the problem is then given by:

$$\begin{array}{c}\hfill u(t,x)=G_{\sigma ,\alpha }^m(t,x)★v_0\left(x\right)+N_{\alpha ,\sigma }^m\left(v\right)(t,x)\end{array}$$

(10)

with

$$\begin{array}{cc}\hfill G_{\sigma ,\alpha }^m(t,x)& {\displaystyle ={\int }_{{\mathbb{R}}^d}{e}^{ix·\xi }{E}_{\alpha +1}\left(-t^{\alpha +1}{\langle \xi \rangle }_{m,\sigma }^2\right)d\xi ,}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill N_{\sigma ,\alpha }^m\left(v\right)(t,x)& ={\int }_0^t{G}_{\alpha ,\sigma }^M(t-s){★}_{\left(x\right)}{I}_s^{\alpha }\left(F\right)(t,s)ds,\hfill \end{array}$$

(12)

where the semigroup of operators ${G_{\sigma ,\alpha }^m(t,·)}_{t\ge 0}$ is defined through the Fourier transform as follows:

$$\widehat{\left(G_{\sigma ,\alpha }^m(t,·)f\right)}(t,\xi )=E_{\alpha +1}\left(-t^{\alpha +1}{\langle \xi \rangle }_{m,\sigma }^2\right)\widehat{f}\left(\xi \right)\mathrm{with}{\langle \xi \rangle }_{\sigma ,m}^2={\left|\xi \right|}^{2\sigma }+m^2.$$

Here, $E_{\beta }\left(z\right)={\displaystyle \sum _{k=0}^{\infty }}\frac{z^k}{\Gamma (\beta k+1)}$ denotes the Mittag-Leffler function (see [31]).

According to [14], a representation of solutions to the linear problem associated with Equation (9) (without the term $F(t,x)$) can be given as $v(t,x)=G_{\sigma ,\alpha }^m(t,x)★v_0(t,x)$. This representation involves convolving the initial data $v_0(t,x)$ with the semigroup of operators $G_{\sigma ,\alpha }^m(t,x)$.

In [14], the authors proved the following result.

Proposition 4

(see [14]). Let us assume that $\alpha \in (0,1)$, $r\ge 1$, $\sigma \ge 1$, and $v_0\in L^r\cap L^{\infty }$. Then, the solution of the linear Cauchy problem

$$\begin{array}{c}{\partial }_t^{1+\alpha }v+{(-\Delta )}^{\sigma }v+m^{2}v=0,\hfill \\ v(x,0)=v_0\left(x\right),v_t(0,x)=0,\hfill \end{array}$$

(13)

for all $t\ge 0$ and $1\le r\le q\le \infty$, satisfies the following $L^r-L^q$ estimates:

$${\parallel v(t,·)\parallel }_{L^q}\lesssim {(1+t)}^{-(1+\alpha )}{\parallel v_0\parallel }_{L^r\cap L^{\infty }}.$$

(14)

4. Analysis of Weakly Coupled Linear Systems

We will use the decay estimates for solutions to:

$$\left\{\begin{array}{c}{\partial }_t^{1+{\alpha }_1}{u}_1+{(-\Delta )}^{{\sigma }_1}{u}_{\ell }+M_1^2{u}_1=0,\hfill \\ {\partial }_t^{1+{\alpha }_2}{u}_2+{(-\Delta )}^{{\sigma }_2}{u}_2+M_2^2{u}_2=0,\hfill \\ .\hfill \\ .\hfill \\ {\partial }_t^{1+{\alpha }_k}{u}_k+{(-\Delta )}^{{\sigma }_k}{u}_k+M_k^2{u}_k=0,\hfill \\ u_{\ell }(0,x)=u_{0l}\left(x\right),{\partial }_t{u}_{\ell }(0,x)=0,\ell =1,2,\dots ,k.\hfill \end{array}\right.$$

(15)

In order to establish the global existence (over time) of Sobolev solutions with small initial data for the weakly coupled systems of semi-linear models (1), we express their solutions in the following form:

$$\begin{array}{c}{u}_l^{ln}(t,x):=G_{{\sigma }_l,{\alpha }_l}^{M_l}(t,x){\ast }_{\left(x\right)}{u}_{0l}\left(x\right),\mathrm{for}\mathrm{all}l=1,\cdots ,k.\hfill \end{array}$$

(16)

Proposition 5.

Let $u_{0l}\in L^{m_l}\cap L^{\infty }$ with $m_l\ge 1$ for all $l=1,\cdots ,k$. Then, the solution of the linear Cauchy problem (15) satisfies the following $L^{m_l}-L^q$ estimates:

$$\begin{array}{cc}\hfill & \parallel u_l^{ln}(t,·){\parallel }_{L^q}\lesssim {(1+t)}^{-(1+{\alpha }_l)}{\parallel u_{0l}\parallel }_{L^{m_l}\cap L^{\infty }}forallq\in [m_l,\infty ].\hfill \end{array}$$

By applying Duhamel’s principle and some fixed-point argument, we can derive the formal integral representation of solutions to (1) as follows:

$$\begin{array}{c}{u}_1(t,x):=u_1^{ln}(t,x)+{\int }_0^t{G}_{{\sigma }_1,{\alpha }_1}^{M_1}(t-\varrho ,·){\ast }_{\left(x\right)}{F}_{{\mu }_1,p_1}\left(u_k\right)d\varrho =(u_1^{ln}+u_1^{nl})(t,x),\hfill \\ u_l(t,x):=u_l^{ln}(t,x)+{\int }_0^t{G}_{{\sigma }_l,{\alpha }_l}^{M_l}(t-\varrho ,·){\ast }_{\left(x\right)}{F}_{{\mu }_l,p_l}\left(u_{l-1}\right)d\varrho =(u_l^{ln}+u_l^{nl})(t,x).\hfill \end{array}$$

(17)

for all $l=2,\cdots ,k$.

Here, $u_1^{nl}={\int }_0^t{G}_{{\sigma }_1,{\alpha }_1}^{M_1}(t-\varrho ,·){\ast }_{\left(x\right)}{F}_{{\mu }_1,p_1}\left(u_k\right)d\varrho$ is the solution to

$$\left\{\begin{array}{c}{\partial }_t^{1+{\alpha }_1}{u}_1+{(-\Delta )}^{{\sigma }_1}{u}_1+M_1^2{u}_l=F_{{\mu }_1,p_1}\left(u_k\right),\hfill \\ u_1(0,x)=0,{\partial }_t{u}_1(0,x)=0.\hfill \end{array}\right.$$

and $u_l^{nl}={\int }_0^t{G}_{{\sigma }_l,{\alpha }_l}^{M_l}(t-\varrho ,·){\ast }_{\left(x\right)}{F}_{{\mu }_l,p_l}\left(u_{l-1}\right)d\varrho$ is the solution to

$$\left\{\begin{array}{c}{\partial }_t^{1+{\alpha }_l}{u}_l+{(-\Delta )}^{{\sigma }_l}{u}_{\ell }+M_l^2{u}_l=F_{{\mu }_l,p_l}\left(u_{l-1}\right),\hfill \\ u_l(0,x)=0,{\partial }_t{u}_l(0,x)=0,\ell =2,\dots ,k.\hfill \end{array}\right..$$

5. Proof of Main Results

Before showing our results, we recall the following lemma from [32].

Lemma 1.

Let us consider $\theta \in [0,1)$, $a\ge 0$, and $b\ge 0$. There exists a constant $C=C(a,b,\theta )>0$, such that the following estimate holds for all $t>0$:

$$\begin{array}{c}\hfill \begin{array}{c}{\int }_0^t{(t-\varrho )}^{-\theta }{(1+t-\varrho )}^{-a}{(1+\varrho )}^{-b}d\varrho \hfill \\ \le \left\{\begin{array}{cc}C{(1+t)}^{-\min \{a+\theta ,b\}}\hfill & if\max \{a+\theta ,b\}>1,\hfill \\ C{(1+t)}^{-\min \{a+\theta ,b\}}\ln (2+t)\hfill & if\max \{a+\theta ,b\}=1,\hfill \\ C{(1+t)}^{1-a-\theta -b}\hfill & if\max \{a+\theta ,b\}<1.\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(18)

5.1. Proof of Theorem 1

Let $T>0$. We introduce the space $X^k\left(T\right)$ as follows:

$$\begin{array}{c}\hfill X^k\left(T\right):=\prod _{\ell =1}^{k}C\left([0,T],L^{m_l}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)\end{array}$$

with the norm

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{X^k\left(T\right)}:={\parallel (u_1,u_2,\dots ,u_k)\parallel }_{X^k\left(T\right)}:=& \underset{0⩽t⩽T}{sup}\left\{\sum _{\ell =1}^k{R}_{\ell }(t,u_l),\right\},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill R_{\ell }(t,u_l)={(1+t)}^{{\mu }_{\ell }-{\alpha }_{\ell }}(\parallel u_{\ell }(t,·){\parallel }_{L^{m_l}}+{\parallel u_{\ell }(t,·)\parallel }_{L^{\infty }}),\end{array}$$

and the operator P by

$$P:u=(u_1,u_2,\dots ,u_k)\in X^k\left(T\right)\to P\left(u\right)=P\left(u\right)(t,x):=u^{ln}(t,x)+u^{nl}(t,x).$$

In order to prove the global (in time) existence and uniqueness of Sobolev solutions in $X^k\left(T\right)$, we will demonstrate that the operator P satisfies the following two inequalities:

$$\begin{array}{cc}\hfill & {\parallel P\left(u\right)\parallel }_{X^k\left(T\right)}\lesssim \parallel (u_{01},u_{02},\dots u_{0k}){\parallel }_{{\mathcal{A}}_k}+\sum _{\ell =1}^{\ell =k}{\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }},\hfill \end{array}$$

(19)

$$\begin{array}{c}\hfill \parallel P\left(u\right)-P\left(\tilde{u}\right){\parallel }_{X^k\left(T\right)}\lesssim {\parallel u-\tilde{u}\parallel }_{X^k\left(T\right)}\sum _{\ell =1}^{\ell =k}\left({\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }-1}+{\parallel \tilde{u}\parallel }_{X^k\left(T\right)}^{p_l-1}\right).\end{array}$$

(20)

Using the definition of the norm in $X^k\left(T\right)$ and Proposition 5, we may conclude:

$$\parallel u^{ln}{\parallel }_{X^k\left(T\right)}\lesssim {\parallel (u_{01},u_{02},\dots ,u_{0k})\parallel }_{{\mathcal{A}}_k}.$$

Hence, in order to complete the proof of (19), it is reasonable to show the following inequality:

$$\parallel u^{nl}{\parallel }_{X^k\left(T\right)}\lesssim \sum _{\ell =1}^{\ell =k}{\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }}.$$

If $u:=(u_1,u_2,\cdots ,u_k)\in X^k\left(T\right)$, then by interpolation we derive for $l=1,\cdots ,k$

$$\begin{array}{c}\parallel u_{\ell }{(t,·)\parallel }_{L^q}\lesssim {(1+t)}^{({\alpha }_{\ell }-{\mu }_{\ell })}{\parallel u\parallel }_{X^k\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_l,\infty ].\hfill \end{array}$$

On the other hand, we also have

$$\begin{array}{cc}\hfill \parallel u_1^{nl}(t,·){\parallel }_{L^q}& \lesssim \parallel {\int }_0^t{G}_{{\sigma }_1,{\alpha }_1}^{M_1}(t-\varrho ,·){\ast }_{\left(x\right)}{I}_s^{{\alpha }_1}\left(F_{{\mu }_1,p_1}\left(u_k\right)\right){d\varrho \parallel }_{L^q}\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{\int }_0^s{(s-\eta )}^{-{\mu }_1}{\parallel u_k(\eta ,·)\parallel }_{L^{p_{1}q}}^{p_1}d\eta dsd\varrho }\hfill \\ & \lesssim {\parallel u\parallel }_{X^k\left(T\right)}^{p_1}{J}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{1}q\in [m_k,\infty ],\hfill \end{array}$$

where

$${\displaystyle J_1\left(t\right)={\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{\int }_0^s{(s-\eta )}^{-{\mu }_1}{(1+\eta )}^{-p_1({\mu }_k-{\alpha }_k)}d\eta dsd\varrho .}$$

(21)

We are interested in estimating the right-hand side of (21). For this we need Lemma 1. We put

$${\displaystyle \omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_1}{(1+\eta )}^{-p_1({\mu }_k-{\alpha }_k)}d\eta .}$$

By using Lemma 1, we obtain $\omega \left(s\right)\lesssim {(1+s)}^{-{\mu }_1}$, if we assume that $p_1>\frac{1}{{\mu }_k-{\alpha }_k}$. On the other hand, the conditions $q\in [m_1,\infty ]$ and $p_{1}q\in [m_k,\infty ]$ imply that $p_1\ge \frac{m_k}{m_1}$.

Once more, we apply Lemma 1 to obtain

$$\begin{array}{cc}\hfill J_1\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{(1+s)}^{-{\mu }_1}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{(1+\varrho )}^{{\alpha }_1-{\mu }_1}d\varrho }\hfill \\ & \lesssim {(1+t)}^{{\alpha }_1-{\mu }_1}.\hfill \end{array}$$

For $l=2,\cdots ,k$ and $q\in [m_l,\infty ]$, we have

$$\begin{array}{cc}\hfill \parallel u_{\ell }^{nl}{(t,·)\parallel }_{L^q}& \lesssim {\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }}{J}_l\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{\ell }q\in [m_{l-1},\infty ],\hfill \end{array}$$

where

$${\displaystyle J_l\left(t\right)={\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}{\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{(1+\eta )}^{-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})}d\eta dsd\varrho .}$$

(22)

To estimate the right-hand side of (22), we require the use of Lemma 1. Let

$${\displaystyle \omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{(1+\eta )}^{-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})}d\eta .}$$

By using Lemma 1, we obtain $\omega \left(s\right)\lesssim {(1+s)}^{-{\mu }_{\ell }}$, if we assume that $p_{\ell }>\frac{1}{{\mu }_{\ell -1}-{\alpha }_{\ell -1}}$. On the other hand, the conditions $q\in [m_l,\infty ]$ and $p_{\ell }q\in [m_{l-1},\infty ]$ imply that $p_{\ell }\ge \frac{m_{l-1}}{m_l}$.

Once more, we apply Lemma 1 to obtain

$$\begin{array}{ccc}\hfill J_l\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}{(1+s)}^{-{\mu }_{\ell }}dsd\varrho }\hfill & \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{(1+s)}^{{\alpha }_{\ell }-{\mu }_{\ell }}d\varrho }\hfill \\ & \lesssim {(1+t)}^{{\alpha }_{\ell }-{\mu }_{\ell }}.\hfill \end{array}$$

In order to prove (24), let us consider two vector-functions u and $\tilde{u}$ belonging to $X^k\left(T\right)$. Then, we have

$$\begin{array}{cc}\hfill & P\left(u\right)-P\left(\tilde{u}\right)\hfill \\ & =({\int }_0^t{G}_{{\sigma }_1,{\alpha }_1}^{M_1}(t-s)★I_s^{{\alpha }_1}\left({\int }_0^s{(s-\eta )}^{-{\mu }_1}\left(|u_k{(\eta ,·)|}^{p_1}-{\left|{\tilde{u}}_k(\eta ,·)\right|}^{p_1}\right)d\eta \right)(t,s,x)ds,\cdots ,\hfill \\ & {\int }_0^t{G}_{{\sigma }_k,{\alpha }_k}^{M_k}(t-s)★I_s^{{\alpha }_k}\left({\int }_0^s{(s-\eta )}^{-{\mu }_k}\left(|u_{k-1}{(\eta ,·)|}^{p_k}-{|{\tilde{u}}_{k-1}(\eta ,·)|}^{p_k}\right)d\eta \right)(t,s,x)ds).\hfill \end{array}$$

We estimate, for $q\in [m_1,\infty ]$,

$$\begin{array}{cc}\hfill & \parallel {\int }_0^t{G}_{{\sigma }_1,{\alpha }_1}^{M_1}(t-s)★I_s^{{\alpha }_1}({\int }_0^s{(s-\eta )}^{-{\mu }_1}\left(|u_k{(\eta ,·)|}^{p_1}-{\left|{\tilde{u}}_k(\eta ,·)\right|}^{p_1}\right)d\eta )(t,s,·)ds{\parallel }_{L^q}\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{\int }_0^s{(s-\eta )}^{-{\mu }_1}\parallel |u_k{(\eta ,·)|}^{p_1}-{\left|{\tilde{u}}_k(\eta ,·)\right|}^{p_1}{\parallel }_{L^q}d\eta dsd\varrho .}\hfill \end{array}$$

Using Hölder’s inequality implies the inequality

$$\begin{array}{cc}\hfill \parallel |u_k{(s,·)|}^{p_1}& -|{\tilde{u}}_k{(s,·)|}^{p_1}{\parallel }_{L^q}\lesssim \parallel u_k(s,·)-{\tilde{u}}_k(s,·){\parallel }_{L^{q{p}_1}}\left(\parallel u_k{(s,·)\parallel }_{L^{q{p}_1}}^{p_1-1}+{\parallel {\tilde{u}}_k(s,·)\parallel }_{L^{q{p}_1}}^{p_1-1}\right).\hfill \end{array}$$

By using the definition of the norm of the solution space $X^k\left(T\right)$, for $p_1\ge \frac{m_k}{m_1}$ and $0⩽s⩽t$, we obtain the following estimates:

$$\begin{array}{ccc}& & \parallel u_k(s,·)-{\tilde{u}}_k(s,·){\parallel }_{L^{q{p}_1}}\lesssim {(1+s)}^{{\alpha }_k-{\mu }_k}{R}_k(s,u_k-{\tilde{u}}_k),\hfill \\ & & \parallel u_k{(s,·)\parallel }_{L^{q{p}_1}}^{p_1-1}\lesssim {(1+s)}^{(p_1-1)({\alpha }_k-{\mu }_k)}{\parallel u\parallel }_{X^k\left(T\right)}^{p_1-1},\hfill \\ & & \parallel {\tilde{u}}_k(s,·){\parallel }_{L^{q{p}_1}}^{p_1-1}\lesssim {(1+s)}^{(p_1-1)({\alpha }_k-{\mu }_k)}{\parallel u\parallel }_{X^k\left(T\right)}^{p_1-1}.\hfill \end{array}$$

Hence, we obtain

$$\begin{array}{cc}\hfill \parallel |u_k{(s,·)|}^{p_1}-{\left|{\tilde{u}}_k(s,·)\right|}^{p_1}{\parallel }_{L^q}& \lesssim {(1+s)}^{-p_1({\mu }_k-{\alpha }_k)}{R}_k(s,u_k-{\tilde{u}}_k)\left({\parallel u\parallel }_{X^k\left(T\right)}^{p_1-1}+{\parallel \tilde{u}\parallel }_{X^k\left(T\right)}^{p_1-1}\right)\hfill \\ & \lesssim {(1+s)}^{-p_1({\mu }_k-{\alpha }_k)}{\parallel u-\tilde{u}\parallel }_{X^k\left(T\right)}\left({\parallel u\parallel }_{X^k\left(T\right)}^{p_1-1}+{\parallel \tilde{u}\parallel }_{X^k\left(T\right)}^{p_1-1}\right).\hfill \end{array}$$

By the same argument, for $l=2,\dots ,k$ and $0⩽s⩽t$, we obtain the following estimate:

$$\begin{array}{cc}\hfill \parallel |u_{\ell }{(s,·)|}^{p_{\ell }}-{\left|{\tilde{u}}_{\ell }(s,·)\right|}^{p_{\ell }}{\parallel }_{L^q}& \lesssim {(1+s)}^{p_{\ell }({\alpha }_{\ell }-{\mu }_{\ell })}{R}_l(s,u_{\ell -1}-{\tilde{u}}_{\ell -1})\left({\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }-1}+{\parallel \tilde{u}\parallel }_{X^k\left(T\right)}^{p_{\ell }-1}\right)\hfill \\ & \lesssim {(1+s)}^{p_{\ell }({\alpha }_{\ell }-{\mu }_{\ell })}{\parallel u-\tilde{u}\parallel }_{X^k\left(T\right)}\left({\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }-1}+{\parallel \tilde{u}\parallel }_{X^k\left(T\right)}^{p_{\ell }-1}\right).\hfill \end{array}$$

So, for $p_1>\frac{1}{{\mu }_k-{\alpha }_k}$ and $p_l>\frac{1}{{\mu }_{l-1}-{\alpha }_{l-1}}$ for all $l=2,\cdots ,k$, we obtain the desired estimate (20).

Remark 2.

All estimates (19) and (20) are uniform with respect to $T\in (0,\infty )$.

From (19), we can see that P maps $X^k\left(T\right)$ into itself for all T and for small data. By using standard contraction arguments, the estimates (19) and (20) lead to the existence of a unique solution to $u=P\left(u\right)$ and, consequently, to (1). This implies that the solution of (1) satisfies the desired decay estimate. Since all constants are independent of T, we can let T tend to ∞, which yields a global (in time) existence result for small data solutions to (1). This concludes the proof.

5.2. Proof of Theorem 2

Let $T>0$. We introduce the space $X^k\left(T\right)$ as follows:

$$\begin{array}{c}\hfill X^k\left(T\right):=\prod _{\ell =1}^{k}C\left([0,T],L^{m_l}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)\end{array}$$

with the norm

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{X^k\left(T\right)}& :=\underset{0⩽t⩽T}{sup}\left\{{(1+t)}^{-{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}{R}_1(t,u_1)+\sum _{l=2}^{k-1}{(1+t)}^{-{\gamma }_{({\alpha }_k,...,{\alpha }_{\ell -1})}^{({\mu }_k,...,{\mu }_{\ell -1})}(p_1,...,p_{\ell })}{R}_{\ell }(t,u_l)+M_k(t,u_k)\right\},\hfill \end{array}$$

where, for $l=1,\cdots ,k$,

$$\begin{array}{ccc}& R_{\ell }(t,u_l)={(1+t)}^{{\mu }_{\ell }-{\alpha }_{\ell }}\left(\parallel u_{\ell }{(t,·)\parallel }_{L^{m_l}}+{\parallel u_{\ell }(t,·)\parallel }_{L^{\infty }}\right),\mathrm{and}\hfill & \\ & \begin{array}{c}\left\{\begin{array}{c}{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)=1-p_1({\mu }_k-{\alpha }_k)\hfill \\ {\gamma }_{({\alpha }_k,{\alpha }_1)}^{({\mu }_k,{\mu }_1)}(p_1,p_2)=1-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)\hfill \\ {\gamma }_{({\alpha }_k,{\alpha }_1,{\alpha }_2)}^{({\mu }_k,{\mu }_1,{\mu }_2)}(p_1,p_2,p_3)=1-p_3({\mu }_2-{\alpha }_2)+p_3{\gamma }_{({\alpha }_k,{\alpha }_1)}^{({\mu }_k,{\mu }_1)}(p_1,p_2)\hfill \\ .\hfill \\ .\hfill \\ .\hfill \\ {\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -1})}^{({\mu }_k,\dots ,{\mu }_{\ell -1})}(p_1,\dots ,p_{\ell })=1-p_{\ell }({\mu }_{l-1}-{\alpha }_{l-1})+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1}),\hfill \end{array}\right.\hfill \end{array}\hfill \end{array}$$

for $l=3,\cdots ,k-1$. and the operator P by

$$P:u=(u_1,u_2,\dots ,u_k)\in X^k\left(T\right)\to P\left(u\right)=P\left(u\right)(t,x):=u^{ln}(t,x)+u^{nl}(t,x).$$

We will prove that, for $u=(u_1,u_2,\dots ,u_k);\tilde{u}=({\tilde{u}}_1,{\tilde{u}}_2,\dots ,{\tilde{u}}_k)$ in $X^k\left(T\right)$, the operator P satisfies the following two inequalities:

$$\begin{array}{cc}\hfill & {\parallel P\left(u\right)\parallel }_{X^k\left(T\right)}\lesssim \parallel (u_{01},u_{02},\dots u_{0k}){\parallel }_{{\mathcal{A}}_k}+\sum _{\ell =1}^{\ell =k}{\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }},\hfill \end{array}$$

(23)

$$\begin{array}{c}\hfill \parallel P\left(u\right)-P\left(\tilde{u}\right){\parallel }_{X^k\left(T\right)}\lesssim {\parallel u-\tilde{u}\parallel }_{X^k\left(T\right)}\sum _{\ell =1}^{\ell =k}\left({\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }-1}+{\parallel \tilde{u}\parallel }_{X^k\left(T\right)}^{p_l-1}\right).\end{array}$$

(24)

Using the definition of the norm in $X^k\left(T\right)$ and Proposition 5, we may conclude:

$$\parallel u^{ln}{\parallel }_{X^k\left(T\right)}\lesssim {\parallel (u_{01},u_{02},\dots ,u_{0k})\parallel }_{{\mathcal{A}}_k}.$$

Hence, in order to complete the proof of (19), it is reasonable to show the following inequality:

$$\parallel u^{nl}{\parallel }_{X^k\left(T\right)}\lesssim \sum _{\ell =1}^{\ell =k}{\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }}.$$

If $u:=(u_1,u_2,\cdots ,u_k)\in X^k\left(T\right)$, then, for $l=2,\cdots ,k-1$, by interpolation, we derive

$$\begin{array}{cc}\hfill & \parallel u_1{(t,·)\parallel }_{L^q}\lesssim {(1+t)}^{({\alpha }_1-{\mu }_1)+{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}{\parallel u\parallel }_{X^k\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_1,\infty ]\hfill \\ & \parallel u_{\ell }{(t,·)\parallel }_{L^q}\lesssim {(1+t)}^{({\alpha }_{\ell }-{\mu }_{\ell })+{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -1})}^{({\mu }_k,\dots ,{\mu }_{\ell -1})}(p_1,\dots ,p_{\ell })}{\parallel u\parallel }_{X^k\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_l,\infty ],\hfill \\ \hfill & \parallel u_k{(t,·)\parallel }_{L^q}\lesssim {(1+t)}^{{\alpha }_k-{\mu }_k}{\parallel u\parallel }_{X^k\left(T\right)}\mathrm{for}\mathrm{all}q\in [m_k,\infty ].\hfill \end{array}$$

On the other hand, for $q\in [m_1,\infty ]$, we have

$$\begin{array}{cc}\hfill & \parallel u_1^{nl}{(t,·)\parallel }_{L^q}\lesssim {\parallel u\parallel }_{X\left(T\right)}^{p_1}{J}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{1}q\in [m_k,\infty ],\hfill \end{array}$$

where

$${\displaystyle J_1\left(t\right)={\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{\int }_0^s{(s-\eta )}^{-{\mu }_1}{(1+\eta )}^{-p_1({\mu }_k-{\alpha }_k)}d\eta dsd\varrho .}$$

(25)

We are interested in estimating the right-hand side of (25). For this we need Lemma 1. We put

$${\displaystyle \omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_1}{(1+\eta )}^{-p_1({\mu }_k-{\alpha }_k)}d\eta .}$$

Thanks to Lemma 1, we obtain $\omega \left(s\right)\lesssim {(1+s)}^{1-{\mu }_1-p_1({\mu }_k-{\alpha }_k)}$, if we assume that $p_1<\frac{1}{{\mu }_k-{\alpha }_k}$.

Once more, we apply Lemma 1 to obtain

$$\begin{array}{cc}\hfill J_1\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{(1+s)}^{1-{\mu }_1-p_1({\mu }_k-{\alpha }_k)}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{(1+\varrho )}^{1+{\alpha }_1-{\mu }_1-p_1({\mu }_k-{\alpha }_k)}d\varrho }\hfill \\ & \lesssim {(1+t)}^{1+{\alpha }_1-{\mu }_1-p_1({\mu }_k-{\alpha }_k)}\hfill \\ & \lesssim {(1+t)}^{{\alpha }_1-{\mu }_1+{\gamma }_{{\alpha }_k}^{{\mu }_k}\left(p_1\right)}.\hfill \end{array}$$

On the other hand, the conditions $q\in [m_1,\infty ]$ and $p_{1}q\in [m_k,\infty ]$ imply that $p_1\ge \frac{m_k}{m_1}$.

For $l=2$ and $q\in [m_2,\infty ]$, we have

$$\begin{array}{cc}\hfill \parallel u_2^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_2)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_2-1}\parallel |u_1(s,·){{|}^{p_2}\parallel }_{L^q}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_2)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_2-1}{\int }_0^s{(s-\eta )}^{-{\mu }_2}{\parallel u_1(\eta ,·)\parallel }_{L^{p_{2}q}}^{p_2}d\eta dsd\varrho }\hfill \\ & \lesssim {\parallel u\parallel }_{X\left(T\right)}^{p_2}{J}_2\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{2}q\in [m_1,\infty ],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill J_2\left(t\right)& {\displaystyle ={\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_2)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_2-1}{\int }_0^s{(s-\eta )}^{-{\mu }_2}{(1+\eta )}^{-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}d\eta dsd\varrho .}\hfill \end{array}$$

(26)

We are interested in estimating the right-hand side of (26). For this, we need Lemma 1. We put

$$\omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_2}{(1+\eta )}^{-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}d\eta .$$

Thanks to Lemma 1, we obtain

$$\omega \left(s\right)\lesssim {(1+s)}^{1-{\mu }_2-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)},$$

if we assume that $p_2<\frac{1}{{\mu }_1-{\alpha }_1-{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}$ and ${\mu }_1-{\alpha }_1-{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)>0$.

The condition ${\mu }_1-{\alpha }_1-{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)>0$ is equivalent to $p_1>\frac{1+{\alpha }_1-{\mu }_1}{{\mu }_k-{\alpha }_k}$.

Once more, we apply Lemma 1 to obtain

$$\begin{array}{cc}\hfill J_2\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_2)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_2-1}{(1+s)}^{1-{\mu }_2-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_2)}{(1+\varrho )}^{1+{\alpha }_2-{\mu }_2-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}d\varrho }\hfill \\ & \lesssim {(1+t)}^{1+{\alpha }_2-{\mu }_2-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}\hfill \\ & \lesssim {(1+t)}^{1+{\alpha }_2-{\mu }_2-p_2({\mu }_1-{\alpha }_1)+p_2{\gamma }_{\left({\alpha }_k\right)}^{\left({\mu }_k\right)}\left(p_1\right)}\hfill \\ & \lesssim {(1+t)}^{{\alpha }_2-{\mu }_2+{\gamma }_{({\alpha }_k,{\alpha }_1)}^{({\mu }_k,{\mu }_1)}(p_1,p_2)}.\hfill \end{array}$$

On the other hand, the conditions $q\in [m_2,\infty ]$ and $p_{2}q\in [m_1,\infty ]$ imply that $p_2\ge \frac{m_1}{m_2}$.

For $l=3,\cdots ,k-1$ and $q\in [m_l,\infty ]$, we have

$$\begin{array}{cc}\hfill \parallel u_{\ell }^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}\parallel |u_{\ell -1}(s,·){{|}^{p_{\ell }}\parallel }_{L^q}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}{\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{\parallel u_{\ell -1}(\eta ,·)\parallel }_{L^{p_{\ell }q}}^{p_{\ell }}d\eta dsd\varrho }\hfill \\ & \lesssim {\parallel u\parallel }_{X\left(T\right)}^{p_{\ell }}{J}_l\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{\ell }q\in [m_{l-1},\infty ],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\displaystyle J_l\left(t\right)={\int }_0^t}& {\displaystyle {(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}}\hfill \\ & \times {\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{(1+\eta )}^{-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})}d\eta dsd\varrho .\hfill \end{array}$$

(27)

On the other hand, we are interested in estimating the right-hand side of (27). For this, we need Lemma 1. We put

$${\displaystyle \omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{(1+\eta )}^{-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})}d\eta .}$$

Thanks to Lemma 1, we obtain

$$\omega \left(s\right)\lesssim {(1+s)}^{1-{\mu }_{\ell }-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})},$$

if we assume that

$$p_{\ell }<\frac{1}{{\mu }_{\ell -1}-{\alpha }_{\ell -1}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})}$$

and

$${\mu }_{\ell -1}-{\alpha }_{\ell -1}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})>0.$$

The last condition is equivalent to

$$p_{l-1}>\frac{1+{\alpha }_{l-1}-{\mu }_{l-1}}{{\mu }_{\ell -2}-{\alpha }_{\ell -2}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -3})}^{({\mu }_k,\dots ,{\mu }_{\ell -3})}(p_1,\dots ,p_{\ell -2})}$$

On the other hand, the conditions $q\in [m_l,\infty ]$ and $p_{\ell }q\in [m_{l-1},\infty ]$ imply that $p_{\ell }\ge \frac{m_l-1}{m_l}$.

Once more, we apply Lemma 1 to obtain

$$\begin{array}{ccc}\hfill J_l\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}{(1+s)}^{1-{\mu }_{\ell }-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})}dsd\varrho }\hfill & \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{(1+s)}^{{\alpha }_{\ell }-{\mu }_{\ell }+1-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})}d\varrho }\hfill \\ & \lesssim {(1+t)}^{{\alpha }_{\ell }-{\mu }_{\ell }+1-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})+p_{\ell }{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -2})}^{({\mu }_k,\dots ,{\mu }_{\ell -2})}(p_1,\dots ,p_{\ell -1})}\hfill \\ & \lesssim {(1+t)}^{{\alpha }_{\ell }-{\mu }_{\ell }+{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{\ell -1})}^{({\mu }_k,\dots ,{\mu }_{\ell -1})}(p_1,\dots ,p_{\ell })}.\hfill \end{array}$$

Finally, for $q\in [m_k,\infty ]$, we have

$$\begin{array}{cc}\hfill \parallel u_k^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}\parallel |u_{k-1}(s,·){{|}^{p_k}\parallel }_{L^q}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}{\int }_0^s{(s-\eta )}^{-{\mu }_k}{\parallel u_{k-1}(\eta ,·)\parallel }_{L^{p_{k}q}}^{p_k}d\eta dsd\varrho }\hfill \\ & \lesssim {\parallel u\parallel }_{X\left(T\right)}^{p_k}{J}_k\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{k}q\in [m_{k-1},\infty ],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\displaystyle J_k\left(t\right)={\int }_0^t}& {\displaystyle {(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}}\hfill \\ & \times {\int }_0^s{(s-\eta )}^{-{\mu }_k}{(1+\eta )}^{-p_k\left({\mu }_{k-1}-{\alpha }_{k-1}-{\gamma }_{({\alpha }_1,\dots ,{\alpha }_{k-2})}^{({\mu }_1,\dots ,{\mu }_{k-2})}(p_1,\dots ,p_{k-1})\right)}d\eta dsd\varrho .\hfill \end{array}$$

(28)

We are interested in estimating the right-hand side of (28). For this we need Lemma 1. We put

$${\displaystyle \omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_k}{(1+\eta )}^{-p_k\left({\mu }_{k-1}-{\alpha }_{k-1}-{\gamma }_{({\alpha }_1,\dots ,{\alpha }_{k-1})}^{({\mu }_1,\dots ,{\mu }_{k-2})}(p_1,\dots ,p_{k-1})\right)}d\eta .}$$

Thanks to Lemma 1 we obtain $\omega \left(s\right)\lesssim {(1+s)}^{-{\mu }_k}$, if we assume that

$$p_k>\frac{1}{{\mu }_{k-1}-{\alpha }_{k-1}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{k-2})}^{({\mu }_k,\dots ,{\mu }_{k-2})}(p_1,\dots ,p_{k-1})}$$

and

$${\mu }_{k-1}-{\alpha }_{k-1}-{\gamma }_{({\alpha }_k,\dots ,{\alpha }_{k-2})}^{({\mu }_k,\dots ,{\mu }_{k-2})}(p_1,\dots ,p_{k-1})>0$$

which equivalent to

$$p_{k-1}>\frac{{\alpha }_{k-1}-{\mu }_{k-1}+1}{{\mu }_{k-2}-{\alpha }_{k-2}-{\gamma }_{{\alpha }_k,\dots ,{\alpha }_{k-3}}^{({\mu }_k,\dots ,{\mu }_{k-3})}(p_1,\dots ,p_{k-2})}.$$

Once more, we apply Lemma 1 to obtain

$$\begin{array}{cc}\hfill J_k\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}{(1+s)}^{-{\mu }_k}d\eta dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{(1+\varrho )}^{{\alpha }_k-{\mu }_k}d\varrho }\hfill \\ & \lesssim {(1+t)}^{{\alpha }_k-{\mu }_k}.\hfill \end{array}$$

The proof of (24) is similar to the proof of (20) of Theorem 1. This completes the proof.

5.3. Proof of Theorem 3

Let $T>0$. We introduce the space $X^k\left(T\right)$ as follows:

$$\begin{array}{c}\hfill X^k\left(T\right):=\prod _{\ell =1}^{k}C\left([0,T],L^{m_l}\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)\right)\end{array}$$

with the norm

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{X^k\left(T\right)}& :=\underset{0⩽t⩽T}{\sup }\left\{\sum _{l=1}^{k-1}{(1+t)}^{-\gamma \left(p_l\right)}{R}_l(t,u_l)+R_k(t,u_k)\right\},\hfill \end{array}$$

where, for $l=1,\cdots ,k$,

$$\begin{array}{ccc}& R_{\ell }(t,u_l)={(1+t)}^{{\mu }_{\ell }-{\alpha }_{\ell }}\left(\parallel u_{\ell }{(t,·)\parallel }_{L^{m_l}}+{\parallel u_{\ell }(t,·)\parallel }_{L^{\infty }}\right),\hfill & \\ & \begin{array}{c}\left\{\begin{array}{c}\gamma \left(p_1\right)=1\hfill \\ \gamma \left(p_l\right)=1+p_l\gamma \left(p_{\ell -1}\right),\mathrm{for}l=2,\cdots ,k-1.\hfill \end{array}\right.\hfill \end{array}\hfill \end{array}$$

The operator P is defined by

$$P:u=(u_1,u_2,\dots ,u_k)\in X^k\left(T\right)\to P\left(u\right)=P\left(u\right)(t,x):=u^{ln}(t,x)+u^{nl}(t,x).$$

We will prove that, for $u=(u_1,u_2,\dots ,u_k);\tilde{u}=({\tilde{u}}_1,{\tilde{u}}_2,\dots ,{\tilde{u}}_k)$ in $X^k\left(T\right)$, the operator P satisfies the following two inequalities:

$$\begin{array}{cc}\hfill & {\parallel P\left(u\right)\parallel }_{X^k\left(T\right)}\lesssim \parallel (u_{01},u_{02},\dots u_{0k}){\parallel }_{{\mathcal{A}}_k}+\sum _{\ell =1}^{\ell =k}{\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }},\hfill \end{array}$$

(29)

$$\begin{array}{c}\hfill \parallel P\left(u\right)-P\left(\tilde{u}\right){\parallel }_{X^k\left(T\right)}\lesssim {\parallel u-\tilde{u}\parallel }_{X^k\left(T\right)}\sum _{\ell =1}^{\ell =k}\left({\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }-1}+{\parallel \tilde{u}\parallel }_{X^k\left(T\right)}^{p_l-1}\right).\end{array}$$

(30)

Using the definition of the norm in $X^k\left(T\right)$ and Proposition 5, we may conclude:

$$\parallel u^{ln}{\parallel }_{X^k\left(T\right)}\lesssim {\parallel (u_{01},u_{02},\dots ,u_{0k})\parallel }_{{\mathcal{A}}_k}.$$

Hence, in order to complete the proof of (29), it is reasonable to show the following inequality:

$$\parallel u^{nl}{\parallel }_{X^k\left(T\right)}\lesssim \sum _{\ell =1}^{\ell =k}{\parallel u\parallel }_{X^k\left(T\right)}^{p_{\ell }}.$$

If $u:=(u_1,u_2,\cdots ,u_k)\in X^k\left(T\right)$, then by interpolation, we derive, for $l=1,\cdots ,k-1$,

$$\begin{array}{cc}\hfill & \parallel u_{\ell }{(t,·)\parallel }_{L^q}\lesssim {(1+t)}^{({\alpha }_{\ell }-{\mu }_{\ell })+\gamma \left(p_{\ell }\right)}{\parallel u\parallel }_{X^k\left(T\right)}\mathrm{for}\mathrm{all}q\in [1,\infty ],\hfill \\ \hfill & \parallel u_k{(t,·)\parallel }_{L^q}\lesssim {(1+t)}^{{\alpha }_k-{\mu }_k}{\parallel u\parallel }_{X^k\left(T\right)}\mathrm{for}\mathrm{all}q\in [1,\infty ].\hfill \end{array}$$

On the other hand, for $q\in [1,\infty ]$, we have

$$\begin{array}{cc}\hfill & \parallel u_1^{nl}{(t,·)\parallel }_{L^q}\lesssim {\parallel u\parallel }_{X\left(T\right)}^{p_1}{J}_1\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T],\hfill \end{array}$$

where

$${\displaystyle J_1\left(t\right)={\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{\int }_0^s{(s-\eta )}^{-{\mu }_1}{(1+\eta )}^{-p_1({\mu }_k-{\alpha }_k)}d\eta dsd\varrho .}$$

(31)

We are interested in estimating the right-hand side of (31). For this, we need Lemma 1. We put

$${\displaystyle \omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_1}{(1+\eta )}^{-p_1({\mu }_k-{\alpha }_k)}d\eta .}$$

Thanks to Lemma 1, we obtain $\omega \left(s\right)\lesssim {(1+s)}^{-{\mu }_1}ln(1+s)$, if we assume that $p_1=\frac{1}{{\mu }_k-{\alpha }_k}$.

Once more, we apply Lemma 1 to obtain

$$\begin{array}{cc}\hfill J_1\left(t\right)& {\displaystyle \lesssim \ln (2+t){\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_1-1}{(1+s)}^{-{\mu }_1}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_1)}{(1+\varrho )}^{{\alpha }_1-{\mu }_1}d\varrho }\hfill \\ & \lesssim {(1+t)}^{{\alpha }_1-{\mu }_1}\ln (2+t)\hfill \\ & \lesssim {(1+t)}^{{\alpha }_1-{\mu }_1}{\left(\ln (2+t)\right)}^{\gamma \left(p_1\right)}.\hfill \end{array}$$

For $l=2,\cdots ,k-1$ and $q\in [1,\infty ]$, we have

$$\begin{array}{cc}\hfill \parallel u_{\ell }^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}\parallel |u_{\ell -1}(s,·){{|}^{p_{\ell }}\parallel }_{L^q}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}{\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{\parallel u_{\ell -1}(\eta ,·)\parallel }_{L^{p_{\ell }q}}^{p_{\ell }}d\eta dsd\varrho }\hfill \\ & \lesssim {\parallel u\parallel }_{X\left(T\right)}^{p_{\ell }}{J}_l\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\mathrm{and}{p}_{\ell }q\in [m_{l-1},\infty ],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\displaystyle J_l\left(t\right)={\int }_0^t}& {\displaystyle {(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}}\hfill \\ \hfill & \times {\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{(1+\eta )}^{-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})}{\left(\ln (1+\eta )\right)}^{p_{\ell }\gamma \left(p_{\ell -1}\right)}d\eta dsd\varrho .\hfill \end{array}$$

We remark that

$$\begin{array}{cc}\hfill J_l\left(t\right)\lesssim {\left(\ln (2+t)\right)}^{p_{\ell }\gamma \left(p_{\ell -1}\right)}& {\displaystyle {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}}\hfill \\ & \times {\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{(1+\eta )}^{-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})}d\eta dsd\varrho .\hfill \end{array}$$

(32)

On the other hand, we are interested in estimating the right-hand side of (32). For this, we need Lemma 1. We put

$${\displaystyle \omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_{\ell }}{(1+\eta )}^{-p_{\ell }({\mu }_{\ell -1}-{\alpha }_{\ell -1})}d\eta .}$$

Thanks to Lemma 1, we obtain

$$\omega \left(s\right)\lesssim {(1+s)}^{-{\mu }_{\ell }},$$

if we assume that

$$p_{\ell }=\frac{1}{{\mu }_{\ell -1}-{\alpha }_{\ell -1}}.$$

Once more, we apply Lemma 1 to obtain

$$\begin{array}{ccc}\hfill J_l\left(t\right)& {\displaystyle \lesssim {\left(\ln (2+t)\right)}^{p_{\ell }\gamma \left(p_{\ell -1}\right)}{\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_{\ell }-1}{(1+s)}^{-{\mu }_{\ell }}dsd\varrho }\hfill & \\ & {\displaystyle \lesssim {\left(\ln (2+t)\right)}^{p_{\ell }\gamma \left(p_{\ell -1}\right)}{\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_{\ell })}{(1+\varrho )}^{{\alpha }_{\ell }-{\mu }_{\ell }}d\varrho }\hfill \\ & \lesssim {(1+t)}^{{\alpha }_{\ell }-{\mu }_{\ell }}{\left(\ln (2+t)\right)}^{p_{\ell }\gamma \left(p_{\ell -1}\right)}\hfill \end{array}$$

Finally, for $q\in [1,\infty ]$, we have

$$\begin{array}{cc}\hfill \parallel u_k^{nl}{(t,·)\parallel }_{L^q}& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}\parallel |u_{k-1}(s,·){{|}^{p_k}\parallel }_{L^q}dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}{\int }_0^s{(s-\eta )}^{-{\mu }_k}{\parallel u_{k-1}(\eta ,·)\parallel }_{L^{p_{k}q}}^{p_k}d\eta dsd\varrho }\hfill \\ & \lesssim {\parallel u\parallel }_{X\left(T\right)}^{p_k}{J}_k\left(t\right)\mathrm{for}\mathrm{all}t\in [0,T]\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\displaystyle J_k\left(t\right)={\int }_0^t}& {\displaystyle {(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}}\hfill \\ & \times {\int }_0^s{(s-\eta )}^{-{\mu }_k}{(1+\eta )}^{-p_k({\mu }_{k-1}-{\alpha }_{k-1})}{\left(\ln (1+\eta )\right)}^{p_k\gamma \left(p_{k-1}\right)}d\eta dsd\varrho .\hfill \end{array}$$

(33)

We are interested in estimating the right-hand side of (33). Let $\delta >0$ be small enough and we use the fact that $ln(2+t)\lesssim {(1+t)}^{\delta }$ to obtain

$$\begin{array}{cc}\hfill {\displaystyle J_k\left(t\right)\lesssim {\int }_0^t}& {\displaystyle {(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}}\hfill \\ & \times {\int }_0^s{(s-\eta )}^{-{\mu }_k}{(1+\eta )}^{-p_k({\mu }_{k-1}-{\alpha }_{k-1}-\delta \gamma \left(p_{k-1}\right))}d\eta dsd\varrho .\hfill \end{array}$$

(34)

For this, we need Lemma 1. We put

$$\omega \left(s\right)={\int }_0^s{(s-\eta )}^{-{\mu }_k}{(1+\eta )}^{-p_k({\mu }_{k-1}-{\alpha }_{k-1}-\delta \gamma \left(p_{k-1}\right))}d\eta .$$

Thanks to Lemma 1, we obtain $\omega \left(s\right)\lesssim {(1+s)}^{-{\mu }_k}$, if we assume that

$$p_k>\frac{1}{{\mu }_{k-1}-{\alpha }_{k-1}-\delta \gamma \left(p_{k-1}\right)}.$$

Once more, we apply Lemma 1 to obtain

$$\begin{array}{cc}\hfill J_k\left(t\right)& {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{\int }_0^{\varrho }{(\varrho -s)}^{{\alpha }_k-1}{(1+s)}^{-{\mu }_k}d\eta dsd\varrho }\hfill \\ & {\displaystyle \lesssim {\int }_0^t{(1+t-\varrho )}^{-(1+{\alpha }_k)}{(1+\varrho )}^{{\alpha }_k-{\mu }_k}d\varrho }\hfill \\ & \lesssim {(1+t)}^{{\alpha }_k-{\mu }_k}.\hfill \end{array}$$

The proof of (30) is similar to the proof of (20) of Theorem 1. This completes the proof.

6. Conclusions

In the present paper, we proved the global (in time) existence of small data Sobolev solutions to the weakly coupled system of k semi-linear fractional $\sigma$-evolution equations with mass and different memory terms. We studied the relationship between the regularity assumptions for the data, the memory terms, and the admissible range of exponents $(p_1,p_2,\dots ,p_k)$ in Equation (1). In a forthcoming paper, we will study the blow-up of solutions to (1).