Finite and Infinte Time Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

Abstract

In this paper, we investigate the global behavior of the weak solutions to the initial boundary value problem for the nonlinear wave equation in a bounded domain. The nonlinearity includes a logarithmic term and several power-type terms with nonnegative variable coefficients. Two new necessary and sufficient conditions for blow up of the weak solutions are established. The first one addresses the blow up of the global weak solutions at infinity. The second necessary and sufficient condition is obtained in the case of strong superlinearity and concerns blow up of the weak solutions for a finite time. Additionally, we derive new sufficient conditions on the initial data that guarantee blow up for either finite or infinite time. A comparison with previous results is also given.

1. Introduction

We consider the following initial boundary value problem for the nonlinear wave equation:

$$u_{tt}-\Delta u=a_0{\left(x\right)uln\left|u\right|}^k+\sum _{i=1}^r{a}_i\left(x\right){\left|u\right|}^{p_i-1}u=f(x,u),t>0,x\in \Omega ,$$

(1)

$$\left\{\begin{array}{cc}u(0,x)=u_0\left(x\right),\hfill & u_t(0,x)=u_1\left(x\right),x\in \Omega ,\hfill \\ u(t,x)=0,\hfill & t\ge 0,x\in \partial \Omega ,\hfill \\ u_0\left(x\right)\in {\mathrm{H}}_0^1(\Omega ),\hfill & u_1\left(x\right)\in {\mathrm{L}}^2(\Omega ),\hfill \end{array}\right.$$

(2)

where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$$(n\ge 1)$ with smooth boundary $\partial \Omega$. We assume that

$$\begin{array}{cc}\hfill & k>0,a_i\left(x\right)\in \mathrm{C}\left(\overline{\Omega }\right),0<a\le a_0\left(x\right)\le A,0\le a_i\left(x\right)\le A_i,i=1,\dots ,r,\forall x\in \overline{\Omega }\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\displaystyle 1<p_1<\cdots <p_r,p_r<\infty \mathrm{if}n=1,2;p_r\le \frac{n}{n-2}\mathrm{if}n\ge 3.}\hfill \end{array}$$

(4)

At a later stage, we impose the following additional condition on the coefficients $a_i\left(x\right)$, $i=1,\cdots ,r$:

$$\sum _{i=1}^r{a}_i\left(x\right)\ge K>0\forall x\in \overline{\Omega }.$$

(5)

Assumption (5) guarantees the power-type superlinear growth of the nonlinearity $f(x,u)$. This assumption is satisfied, for instance, by a nonlinearity $f(x,u)$ involving a single power-type term with a strictly positive variable coefficient, i.e., $r=1$ and $a_1\left(x\right)>0$ for all $x\in \overline{\Omega }$.

Logarithmic nonlinearity has applications in many fields such as nuclear physics, inflation, cosmology, quantum optics, fluid dynamic, transport phenomena, and more; see e.g., refs. [1,2,3,4,5,6,7,8]. For this reason, partial differential equations with logarithmic nonlinearities have been the subject of intensive research in the last few decades.

After the pioneering papers of Gazenave [9] and Górka [10], the global behavior of the weak solution to the wave and Klein–Gordon equations with the logarithmic nonlinearity $f(x,u)={uln\left|u\right|}^k$, $k>0$ is considered in [11,12,13,14]. By means of the potential well method, the global existence and infinite time blow up of the weak solutions are proved in [12,13] for the wave equation and the Klein–Gordon equation, respectively. In the case of arbitrary positive initial energy, an infinite time blow up result is given in [12]. Moreover, a growth estimate of the infinite time blowup solution is obtained in [14] for any initial energy level.

Recently, a wave equation with combined logarithmic and power-type nonlinearity with variable coefficients was studied in our previous paper [15]. In [15], we established a necessary and sufficient condition for blow up at infinity of the global weak solutions to (1)–(4), applicable to arbitrary initial energy. Furthermore, we derive a growth estimate for the blowing up global solutions.

Let us also note that logarithmic nonlinearities have been studied in various types of equations, including the wave and the Klein–Gordon equations with damping terms [16,17,18,19,20], Boussinesq-type equations [21,22], plate equations [23,24], parabolic equations [25,26], wave equations with fractional boundary dissipation [27], etc.

The aim of this paper is twofold. First, we extend the research in [15] by deriving new necessary and sufficient conditions for infinite time blow up of the global weak solutions to (1)–(4); see Theorem 2. The newly proposed necessary and sufficient conditions require that the scalar product $(u\left(t\right),u_t\left(t\right))$ has a nonnegative sign at some time b.

The second goal of the paper is to investigate the impact of Assumption (5) on the behavior of the solutions. We demonstrate that Assumption (5) leads to the nonexistence of global solutions to problem (1)–(5). More precisely, we prove that the maximal existence time $T_m$ of the weak solutions is finite; see Theorems 3 and 4. However, when $T_m<\infty$ we do not know whether $u(t,x)$ blows up at $T_m$, i.e., ${\parallel u\left(t\right)\parallel }_{{\mathrm{L}}^2(\Omega )}\to \infty$ as $t\to T_m$ or ${\parallel \nabla u\left(t\right)\parallel }_{{\mathrm{L}}^2(\Omega )}\to \infty$ as $t\to T_m$. To exclude the second case, when ${\parallel u\left(t\right)\parallel }_{{\mathrm{L}}^2(\Omega )}$ is bounded in $[0,T_m)$ while ${\parallel \nabla u\left(t\right)\parallel }_{{\mathrm{L}}^p(\Omega )}$ blows up at $T_m$, we impose more restrictive conditions (41) on the power exponents in $f(x,u)$. In this manner, novel necessary and sufficient conditions for the finite time blow up of every local weak solution are established; see Theorems 5 and 6. This indicates that the combination of conditions (5) and (41) significantly alters the nature of the blow up type of the weak solutions: from infinite time blow up (under (3) and (4)) to finite time blow up (under (3)–(5) and (41)).

Indeed, for the pure logarithmic nonlinearity ${uln\left|u\right|}^k,k>0$, there is always an infinite time blow up of the weak solutions; see [12,13,14]. Conversely, for the nonlinearity with superlinear terms (such as ${a\left(x\right)\left|u\right|}^{p}ln{\left|u\right|}^k$ or ${a\left(x\right)\left|u\right|}^{p-1}u$, $k>0$, $p>1$, $a\left(x\right)>0$$\forall x\in \overline{\Omega }$), finite time blow up of the local weak solutions has been observed; see [28,29]. It is important to note that if Assumption (5) fails in some subdomains of $\Omega$, then the question whether problem (1)–(4) has a global weak solution that blows up at infinity or local weak solutions that blow up for a finite time, remains open.

The structure of the paper is as follows. In Section 2, some definitions and preliminary results are introduced. The main results are formulated and proved in Section 3 and Section 4. Section 3 addresses the infinite time blow up of the global weak solutions. Section 4 deals with the nonexistence of global weak solutions as well as the finite time blow up of local weak solutions. In Section 5, a comparison with previous results is provided.

2. Preliminary

Throughout the paper, we utilize the following short notation for the functions that are dependent on both t and x:

$$\begin{array}{c}\hfill {\parallel u\left(t\right)\parallel }_p={\parallel u(t,·)\parallel }_{{\mathrm{L}}^p(\Omega )},1<p\le \infty ,(u\left(t\right),v\left(t\right))={\int }_{\Omega }u(t,x)v(t,x)dx.\end{array}$$

For the sake of simplicity, we write $\parallel ·\parallel$ instead of ${\parallel ·\parallel }_2$.

First, we recall the important functionals $J\left(w\right)$ and $I\left(w\right)$, defined for every $w\left(x\right)\in {\mathrm{H}}_0^1(\Omega )$:

$$\begin{array}{cc}\hfill J\left(w\right):=& {\displaystyle \frac{1}{2}}{\parallel \nabla w\parallel }^2-{\int }_{\Omega }{\int }_0^{w\left(x\right)}f(x,z)dzdx\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\parallel \nabla w\parallel }^2-{\displaystyle \frac{k}{2}}{\int }_{\Omega }{a}_0\left(x\right)w^{2}ln\left|w\right|dx\hfill \\ \hfill & +{\displaystyle \frac{k}{4}}{\int }_{\Omega }{a}_0\left(x\right)w^{2}dx-\sum _{i=1}^r{\displaystyle \frac{1}{p_i+1}}{\int }_{\Omega }{a}_i\left(x\right){\left|w\right|}^{p_i+1}dx,\hfill \end{array}$$

$$\begin{array}{cc}\hfill I\left(w\right):=& {\parallel \nabla w\parallel }^2-{\int }_{\Omega }wf(x,w)dx\hfill \\ \hfill =& {\parallel \nabla w\parallel }^2-k{\int }_{\Omega }{a}_0\left(x\right)w^{2}ln\left|w\right|dx-\sum _{i=1}^r{\int }_{\Omega }{a}_i\left(x\right){\left|w\right|}^{p_i+1}dx.\hfill \end{array}$$

For every $p\in [1,p_1]$, we have the following relation between the functionals $I\left(w\right)$ and $J\left(w\right)$:

$$\begin{array}{cc}\hfill J\left(w\right)=& {\displaystyle \frac{1}{p+1}}I\left(w\right)+{\displaystyle \frac{p-1}{2(p+1)}}{\parallel \nabla w\parallel }^2+{\displaystyle \frac{k}{4}}{\int }_{\Omega }{a}_0\left(x\right){\left|w\right|}^{2}dx\hfill \\ & -{\displaystyle \frac{k(p-1)}{2(p+1)}}{\int }_{\Omega }{a}_0\left(x\right)w^{2}ln\left|w\right|dx+\sum _{i=1}^r{\displaystyle \frac{p_i-p}{(p_i+1)(p+1)}}{\int }_{\Omega }{a}_i\left(x\right){\left|w\right|}^{p_i+1}dx.\hfill \end{array}$$

(6)

Formula (6) generalizes the relationship between the functionals $I\left(w\right)$ and $J\left(w\right)$ in the case of a combined logarithmic and power-type nonlinearity. We remark that expression (6) is applied with $p=1$ for a pure logarithmic nonlinearity; see, e.g., [12,15]. In contrast, for a pure power-type nonlinearity ($a_0\left(x\right)\equiv 0$), relation (6) is used with $p=p_1$; see, e.g., [29] and references therein.

In order to introduce the main results of the paper, let us recall some definitions.

Definition 1. 

The function $u(t,x)$ is a weak solution to problem (1)–(4) if

$$u(t,x)\in \mathrm{C}((0,T_m);{\mathrm{H}}_0^1(\Omega ))\cap {\mathrm{C}}^1((0,T_m);{\mathrm{L}}^2(\Omega ))\cap {\mathrm{C}}^2((0,T_m);{\mathrm{H}}^{-1}(\Omega ))$$

and the identity

$$\begin{array}{cc}{\int }_{\Omega }{u}_t(t,x)\mu \left(x\right)dx+{\int }_0^t{\int }_{\Omega }\nabla u(\tau ,x)\hfill & \nabla \mu \left(x\right)dxd\tau \hfill \\ & ={\int }_0^t{\int }_{\Omega }f(x,u(\tau ,x))\mu \left(x\right)dxd\tau +{\int }_{\Omega }{u}_1\left(x\right)\mu \left(x\right)dx\hfill \end{array}$$

holds for every $\mu \left(x\right)\in {\mathrm{H}}_0^1(\Omega )$ and every $t\in [0,T_m)$.

Note that $u(t,x)\in {\mathrm{L}}^p(\Omega )$ for every $t\in [0,T_m)$ according to the Sobolev inequality (13) and Assumption (4). Thus, the above definition is correct.

Definition 2. 

Suppose $u(t,x)$ is a weak solution to problem (1)–(4) in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$. Then, solution $u(t,x)$ blows up at $T_m$ if

$$\underset{t\to T_m,t<T_m}{lim\; sup}\parallel u\left(t\right)\parallel =\infty .$$

If $u(t,x)$ is a weak solution to problem (1)–(4) in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$, we define the energy functional as

$$\begin{array}{cc}\hfill E\left(u\right(t,·\left)\right):& ={\displaystyle \frac{1}{2}}{\parallel u_t\left(t\right)\parallel }^2+J\left(u(t,·)\right)=\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}\parallel u_t{\left(t\right)\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \nabla u\left(t\right)\parallel }^2-{\displaystyle \frac{k}{2}}{\int }_{\Omega }{a}_0\left(x\right)u^2(t,x)ln\left|u(t,x)\right|dx\hfill \\ & +{\displaystyle \frac{k}{4}}{\int }_{\Omega }{a}_0\left(x\right)u^2(t,x)dx-\sum _{i=1}^r{\int }_{\Omega }{\displaystyle \frac{a_i\left(x\right)}{p_i+1}}{\left|u(t,x)\right|}^{p_i+1}dx.\hfill \end{array}$$

(7)

Moreover, the following conservation law is valid:

$$E\left(0\right)=E\left(u(t,·)\right)\forall t\in [0,T_m).$$

(8)

For simplicity, we use the short notation $I\left(t\right)=I\left(u\right(t\left)\right)=I\left(u\right(t,·\left)\right)$, $J\left(t\right)=J\left(u\right(t\left)\right)=J\left(u\right(t,·\left)\right)$, $E\left(t\right)=E\left(u\right(t\left)\right)=E\left(u\right(t,·\left)\right)$.

Combining (6)–(8) for $p=1$, we obtain the following useful relation:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}I\left(t\right)& =E\left(0\right)-{\displaystyle \frac{ka}{4}}{\parallel u\left(t\right)\parallel }^2-{\displaystyle \frac{1}{2}}{\parallel u_t\left(t\right)\parallel }^2-B\left(t\right)\hfill \\ & =E\left(0\right)-{\displaystyle \frac{ka}{4}}{\parallel u\left(t\right)\parallel }^2-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u\left(t\right),u_t\left(t\right))}^2}{{\parallel u\left(t\right)\parallel }^2}}-{\displaystyle \frac{1}{2}}\left(\parallel u_t{\left(t\right)\parallel }^2-{\displaystyle \frac{{(u\left(t\right),u_t\left(t\right))}^2}{{\parallel u\left(t\right)\parallel }^2}}\right)-B\left(t\right),\hfill \end{array}$$

(9)

where

$$B\left(t\right)={\displaystyle \frac{k}{4}}{\int }_{\Omega }(a_0\left(x\right)-a){\left|u\right|}^{2}dx+\sum _{i=1}^r{\displaystyle \frac{p_i-1}{2(p_i+1)}}{\int }_{\Omega }{a}_i\left(x\right){\left|u\right|}^{p_i+1}dx\ge 0\forall t\in [0,T_m).$$

(10)

The nonnegativity of $B\left(t\right)$ is due to the conditions (3) and (4) on the functions $a_i\left(x\right)$ and the power exponents $p_i$ in $f(x,u)$.

The proofs of the main results are based on the behavior of the nonnegative smooth function $\psi \left(t\right):={\parallel u\left(t\right)\parallel }^2$. In the following, we recall some preliminary results for $\psi \left(t\right)$.

Definition 3. 

We say that a nonnegative function $\psi \left(t\right)\in {\mathrm{C}}^2\left([0,T_m)\right)$, $0<T_m\le \infty$ blows up at $T_m$ if

$$\underset{t\to T_m,t<T_m}{lim\; sup}\psi \left(t\right)=\infty .$$

Lemma 1 

(Lemma 2.2 in [30]). Suppose $\psi \left(t\right)\in C^2\left([0,T_m)\right)$, $0<T_m\le \infty$, is a nonnegative function and M is an arbitrary nonnegative constant. If $\psi \left(t\right)$ blows up at $T_m$, then there exists $t_0\in [0,T_m)$ such that $\psi \left(t_0\right)>M$ and ${\psi }^{\prime }\left(t_0\right)>0$.

The following lemma is a modified version of Theorem 3.2 in [31], formulated for a function $\psi \left(t\right)$ defined in $[t_0,T_m)$, $t_0\ge 0$ instead of $[0,T_m)$.

Lemma 2. 

Suppose $\psi \left(t\right)\in {\mathrm{C}}^2\left([t_0,T_m)\right)$, $t_0\ge 0$ is a nonnegative solution to the problem

$$\begin{array}{cc}& {\psi }^{″}\left(t\right)\psi \left(t\right)-\gamma {\psi }^{\prime 2}\left(t\right)=Q\left(t\right),t\in [t_0,T_m),t_0<T_m\le \infty ,\hfill \\ & \gamma >1,Q\left(t\right)\in \mathrm{C}\left([t_0,T_m)\right),Q\left(t\right)\ge 0fort\in [t_0,T_m).\hfill \end{array}$$

If $\psi \left(t\right)$ blows up at $T_m$, then $T_m<\infty$.

Let us recall the necessary and sufficient condition for blow up at infinity of the global solutions to problem (1)–(4), proved in [15].

Theorem 1 

(Theorem 1 in [15]). If $u(t,x)$ is a global weak solution to problem (1)–(4), then

(i) 

$u(t,x)$ blows up at infinity if and only if

$$\mathit{there}\mathit{exists}b\ge 0\mathit{such}\mathit{that}E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u\left(b\right)\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u\left(b\right),u_t\left(b\right));$$

(11)

(ii) 

if (11) fails, then the estimate

$${\parallel u\left(t\right)\parallel }^2\le \left(\parallel u_0{\parallel }^2-{\displaystyle \frac{4E\left(0\right)}{ka}}\right)e^{-\sqrt{ka}t}+{\displaystyle \frac{4E\left(0\right)}{ka}}$$

(12)

holds for every $t\ge 0$.

Below, we give some inequalities which are important for the proof of the main results. For $w\in {\mathrm{H}}_0^1(\Omega )$, we employ the Sobolev embedding theorem,

$${\parallel w\parallel }_q\le C_q\parallel \nabla w\parallel ,\mathrm{where}\left\{\begin{array}{ccc}& 1\le q\le \infty \hfill & \mathrm{if}n=1;\hfill \\ & 1\le q<\infty \hfill & \mathrm{if}n=2;\hfill \\ & 1\le q\le {\displaystyle \frac{2n}{n-2}}\hfill & \mathrm{if}n\ge 3\hfill \end{array}\right.$$

(13)

and the Gagliardo–Nirenberg inequality:

$$\parallel D^j{w\parallel }_p\le C\parallel D^m{w\parallel }_r^{\alpha }{\parallel w\parallel }_q^{1-\alpha }+C{\parallel w\parallel }_q.$$

(14)

Inequality (14) is valid for $1\le q,r\le \infty$, $m,j\in \mathbb{N},j<m$, and real numbers $\alpha$ and p which satisfy relations

$${\displaystyle \frac{1}{p}}={\displaystyle \frac{j}{n}}+\left({\displaystyle \frac{1}{r}}-{\displaystyle \frac{m}{n}}\right)\alpha +{\displaystyle \frac{1-\alpha }{q}},{\displaystyle \frac{j}{m}}\le \alpha \le 1.$$

The constant C in (14) depends on $\Omega$ and the parameters n, j, m, r, q, and $\alpha$ but not on w. For more details, we refer the reader to [32,33].

3. Infinite Time Blow Up

In this section, we formulate and prove a new necessary and sufficient condition for blow up at infinity of the global weak solutions to problem (1)–(4). In contrast to the necessary and sufficient condition for blow up at infinity in Theorem 1, this condition requires a nonnegative sign of the scalar product $(u\left(b\right),u_t\left(b\right))$ at some time b. As a consequence, new sufficient conditions for infinite time blow up are derived.

Theorem 2. 

Suppose (3), (4) hold and $u(t,x)$ is a global weak solution of problem (1), (2).

(i) 

If $E\left(0\right)>0$, then

$\left(i_1\right)$

$u(t,x)$ blows up at infinity if and only if

$$\begin{array}{cc}& \mathit{there}\mathit{exists}b\ge 0\mathit{such}\mathit{that}\hfill \\ & \parallel u\left(b\right)\parallel > 0,(u\left(b\right),u_t\left(b\right))\ge 0,E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u\left(b\right)\parallel }^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u\left(b\right),u_t\left(b\right))}^2}{{\parallel u\left(b\right)\parallel }^2}};\hfill \end{array}$$

(15)

$\left(i_2\right)$

if (15) fails, then $\parallel u\left(t\right)\parallel$ is globally bounded and the estimate

$${\parallel u\left(t\right)\parallel }^2\le max\left\{{\displaystyle \frac{4E\left(0\right)}{ka}},{\parallel u_0\parallel }^2\right\}$$

(16)

holds for $t\in [0,\infty )$.

(ii) 

If $E\left(0\right)=0$, then

$\left(i{i}_1\right)$

$u(t,x)$ blows up at infinity if and only if

$$\begin{array}{cc}& \mathit{there}\mathit{exists}b\ge 0\mathit{such}\mathit{that}(u\left(b\right),u_t\left(b\right))>0;\hfill \end{array}$$

(17)

$\left(i{i}_2\right)$

If (17) fails, then $\parallel u\left(t\right)\parallel$ is globally bounded and the estimate

$${\parallel u\left(t\right)\parallel }^2\le {\parallel u_0\parallel }^2$$

(18)

holds for $t\in [0,\infty )$.

(iii) 

If $E\left(0\right)<0$, then every global weak solution $u(t,x)$ blows up at infinity.

Proof. 

$\left(i\right)$ Let $E\left(0\right)>0$.

$\left(i_1\right)$ Sufficiency. The proof of sufficiency is similar to the proof of Theorem 6.2 in [29], but for the reader’s convenience we provide it below. Suppose that (15) holds. The proof of the infinite time blow up of $u(t,x)$ consists of three steps.

Step 1. Here, we prove that if $I\left(t\right)<0$ for $t\in [b,T)$, $b<T\le \infty$, and $(u\left(b\right),u_t\left(b\right))\ge 0$, then the functions

$$\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2\mathrm{and}\varphi \left(t\right)={\displaystyle \frac{{(u\left(t\right),u_t\left(t\right))}^2}{{\parallel u\left(t\right)\parallel }^2}}={\displaystyle \frac{1}{4}}{\displaystyle \frac{{\left({\psi }^{\prime }\left(t\right)\right)}^2}{\psi \left(t\right)}}$$

are strictly increasing for $t\in (b,T)$. Simple computations give us

$${\psi }^{\prime }\left(t\right)=2(u\left(t\right),u_t\left(t\right)),{\psi }^{″}\left(t\right)=2{\parallel u_t\left(t\right)\parallel }^2-2I\left(t\right)>0\mathrm{for}t\in [b,T).$$

(19)

Therefore, ${\psi }^{\prime }\left(t\right)>{\psi }^{\prime }\left(b\right)\ge 0$ for $t\in (b,T)$, i.e., $\psi \left(t\right)$ is strictly increasing in $(b,T)$. The monotonicity of $\psi \left(t\right)$ guarantees that $\varphi \left(t\right)$ is also a strictly increasing function in $(b,T)$. Indeed, from the Cauchy–Schwarz inequality, the negative sign of $I\left(t\right)$, and the positive sign of ${\psi }^{\prime }\left(t\right)$, we obtain

$$\begin{array}{cc}\hfill {\varphi }^{\prime }\left(t\right)=& {\displaystyle \frac{1}{4}}{\displaystyle \frac{{\psi }^{\prime }\left(t\right)}{{\psi }^2\left(t\right)}}\left[2\psi \left(t\right){\psi }^{″}\left(t\right)-{\left({\psi }^{\prime }\left(t\right)\right)}^2\right]\hfill \\ & ={\displaystyle \frac{1}{4}}{\displaystyle \frac{{\psi }^{\prime }\left(t\right)}{{\psi }^2\left(t\right)}}\left\{{4\parallel u\left(t\right)\parallel }^2\left[\parallel u_t{\left(t\right)\parallel }^2-I\left(t\right)\right]-4{(u\left(t\right),u_t\left(t\right))}^2\right\}\hfill \\ & ={\displaystyle \frac{{\psi }^{\prime }\left(t\right)}{{\psi }^2\left(t\right)}}\left\{\left[{\parallel u\left(t\right)\parallel }^2{\parallel u_t\left(t\right)\parallel }^2-{(u\left(t\right),u_t\left(t\right))}^2\right]-{\parallel u\left(t\right)\parallel }^{2}I\left(t\right)\right\}>0\hfill \end{array}$$

for $t\in (b,T)$. Thus, Step 1 is proved.

Step 2. Now, we prove that if (15) holds, then $I\left(t\right)<0$ for every $t\in [b,\infty )$. First, from (9) and (10) and the Cauchy–Schwarz inequality we obtain that $I\left(b\right)<0$. Suppose by contradiction that there exists $T<\infty$ such that $I\left(t\right)<0$ for $t\in [b,T)$ and $I\left(T\right)=0$. Relations (9) and (10) and the monotonicity of $\psi \left(t\right)$ and $\varphi \left(t\right)$ in $(b,T)$ give us

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}I\left(t\right)\le & E\left(0\right)-{\displaystyle \frac{ka}{4}}{\parallel u\left(b\right)\parallel }^2-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u\left(b\right),u_t\left(b\right))}^2}{{\parallel u\left(b\right)\parallel }^2}}:=-{\displaystyle \frac{c_1}{2}}<0\hfill \end{array}$$

(20)

for $t\in [b,T)$. Finally, by the following impossible chain of inequalities

$$\begin{array}{cc}\hfill 0={\displaystyle \frac{1}{2}}I\left(T\right)& \le E\left(0\right)-{\displaystyle \frac{ka}{4}}{\parallel u\left(T\right)\parallel }^2-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u\left(T\right),u_t\left(T\right))}^2}{{\parallel u\left(T\right)\parallel }^2}}\hfill \\ & \le E\left(0\right)-{\displaystyle \frac{ka}{4}}{\parallel u\left(b\right)\parallel }^2-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u\left(b\right),u_t\left(b\right))}^2}{{\parallel u\left(b\right)\parallel }^2}}<0\hfill \end{array}$$

we arrive at a contradiction. Hence, $I\left(t\right)<0$ for every $t\in [b,\infty )$.

Step 3. In (20), from Step 2, we prove that there exists a constant $c_1$ such that

$$I\left(t\right)\le -c_1<0\mathrm{for}t\in [b,\infty ).$$

(21)

From (19) and (21), we have

$${\psi }^{″}\left(t\right)\ge -2I\left(t\right)\ge 2c_1>0\mathrm{for}t\in [b,\infty ).$$

(22)

Integrating (22) twice, it follows that

$$\psi \left(t\right)\ge c_1{(t-b)}^2+2(t-b)(u\left(b\right),u_t\left(b\right))+{\parallel u\left(b\right)\parallel }^2.$$

Hence, ${lim}_{t\to \infty }\psi \left(t\right)=\infty$, i.e., $u(t,x)$ blows up at infinity.

Necessity. Suppose that $u(t,x)$ blows up at infinity. Applying Lemma 1 for $T_m=\infty$ and $M=4E\left(0\right)/\left(ka\right)>0$ it follows that there exists $t_0\ge 0$ such that $\psi \left(t_0\right)={\parallel u\left(t_0\right)\parallel }^2\ge 4E\left(0\right)/\left(ka\right)>0$ and ${\psi }^{\prime }\left(t_0\right)=(u\left(t_0\right),u_t\left(t_0\right))>0$. Thus, condition (15) holds for $b=t_0$. Statement $\left(i_1\right)$ is proved.

$\left(i_2\right)$ If (15) fails, then we have that for every $t\ge 0$ at least one of the following three conditions is satisfied:

$$\parallel u\left(t\right)\parallel =0,$$

(23)

$$(u\left(t\right),u_t\left(t\right))<0$$

(24)

or

$$E\left(0\right)\ge {\displaystyle \frac{ka}{4}}{\parallel u\left(t\right)\parallel }^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u\left(t\right),u_t\left(t\right))}^2}{{\parallel u\left(t\right)\parallel }^2}}.$$

(25)

To prove estimate (16), we consider the following two cases:

$$\mathrm{Case}1:\parallel u_0{\parallel }^2\le {\displaystyle \frac{4E\left(0\right)}{ka}}$$

(26)

and

$$\mathrm{Case}2:\parallel u_0{\parallel }^2>{\displaystyle \frac{4E\left(0\right)}{ka}}.$$

(27)

Case 1. If $\psi \left(0\right)=\parallel u_0{\parallel }^2$ satisfies (26), then we will prove that

$$\psi \left(t\right)\le {\displaystyle \frac{4E\left(0\right)}{ka}}$$

(28)

for every $t\ge 0$. Suppose by contradiction that estimate (28) fails for some $t_1>0$, i.e.,

$$\psi \left(t_1\right)>{\displaystyle \frac{4E\left(0\right)}{ka}}.$$

From this assumption it follows that there exist $t_0$ and $t_{*}$ such that $0\le t_0<t_{*}<t_1$,

$$\psi \left(t_0\right)={\displaystyle \frac{4E\left(0\right)}{ka}},$$

$$(u\left(t_{*}\right),u_t\left(t_{*}\right))={\displaystyle \frac{1}{2}}{\psi }^{\prime }\left(t_{*}\right)>0,\mathrm{and}{\parallel u\left(t_{*}\right)\parallel }^2=\psi \left(t_{*}\right)>{\displaystyle \frac{4E\left(0\right)}{ka}}>0.$$

(29)

Inequalities (29) contradict our assumptions (23)–(25) at the time $t_{*}$. Therefore, in the hypothesis of Case 1, estimate (28) holds for every $t\ge 0$.

Case 2. In this case, we will show that

$${\parallel u\left(t\right)\parallel }^2=\psi \left(t\right)\le {\parallel u_0\parallel }^2$$

(30)

for every $t\ge 0$. Since (27) is fulfilled, then from (23)–(25) it follows that ${\psi }^{\prime }\left(t\right)=(u\left(t\right),u_t\left(t\right))<0$ as long as $\psi \left(t\right)>{\displaystyle \frac{4E\left(0\right)}{ka}}>0$, i.e., $\psi \left(t\right)$ is strictly decreasing.

If $\psi \left(t\right)>{\displaystyle \frac{4E\left(0\right)}{ka}}>0$ for every $t\ge 0$, then (30) is satisfied for every $t\ge 0$.

Suppose there exists a time $t_2>0$ for which $\psi \left(t_2\right)={\displaystyle \frac{4E\left(0\right)}{ka}}$. Then, applying Case 1 for $t_2$ instead of $t_0$ and $\psi \left(t_2\right)$ instead of $\parallel u_0{\parallel }^2$, we conclude that $\psi \left(t\right)\le {\displaystyle \frac{4E\left(0\right)}{ka}}<{\parallel u_0\parallel }^2$ for every $t\in [t_2,\infty )$. Combining estimates (28) from Case 1 and (30) from Case 2, we obtain (16). The proof of $\left(i_2\right)$ is completed.

$\left(ii\right)$ Let $E\left(0\right)=0$.

$\left(i{i}_1\right)$ Sufficiency. Since (17) holds at some time $t=b$, then $\parallel u\left(b\right)\parallel > 0$ and consequently (15) is satisfied. The rest of the proof is identical to the proof of sufficiency of statement $\left(i_1\right)$.

Necessity. As $u(t,x)$ blows up at infinity, then Lemma 1 for $T_m=\infty$ and $M=0$ gives us that there exists $t_0\ge 0$ such that $\psi \left(t_0\right)={\parallel u\left(t_0\right)\parallel }^2>0$ and ${\psi }^{\prime }\left(t_0\right)=(u\left(t_0\right),u_t\left(t_0\right))>0$. Thus, condition (17) holds for $b=t_0$ and $\left(i{i}_1\right)$ is proved.

$\left(i{i}_2\right)$ If (17) fails, then for every $t\ge 0$ we have

$$(u\left(t\right),u_t\left(t\right))\le 0.$$

Hence, the function $\psi \left(t\right)$ is decreasing for $t\in [0,\infty )$ and satisfies (18).

$\left(iii\right)$ Let $E\left(0\right)<0$.

The proof follows from Theorem 3 in [15] and we omit it. Thus, Theorem 2 is proved. □

Below, we give sufficient conditions on the initial data that guarantee infinite time blow up of the global solutions to problem (1)–(4).

Proposition 1 (Sufficient conditions).

Suppose $u(t,x)$ is a global weak solution to problem (1)–(4).

(i) 

If $E\left(0\right)>0$ and

$$\parallel u_0\parallel >0,(u_0,u_1)\ge 0,E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u_0\parallel }^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u_0,u_1)}^2}{\parallel u_0{\parallel }^2}},$$

(31)

then $u(t,x)$ blows up at infinity.

(ii) 

If $E\left(0\right)=0$ and $(u_0,u_1)\ge 0$, then every nontrivial solution $u(t,x)$ blows up at infinity.

Proof. 

$\left(i\right)$ The proof of this statement is a consequence of the sufficiency of Theorem 2$\left(i_1\right)$ with $b=0$.

$\left(ii\right)$ If $E\left(0\right)=0$ and $\parallel u_0\parallel > 0$, then from the sufficiency of Theorem 2$\left(i_1\right)$ with $b=0$, it follows that $u(t,x)$ blows up at infinity. If $\parallel u_0\parallel =0$, since $u(t,x)$ is a nontrivial solution, there exists $b>0$ such that $(u\left(b\right),u_t\left(b\right))>0$ and $u(t,x)$ blows up at infinity from the sufficiency of Theorem 2$\left(i{i}_1\right)$. □

The following result is important for the comparison of the new sufficient conditions with the previous ones; see Section 5.

Proposition 2. 

Under the conditions of Theorem 2 it follows that $I\left(t\right)<0$ for every $t\in [b,\infty )$.

The proof of this statement is given in Step 1 and Step 2 of the proof of sufficiency in Theorem 2$\left(i_1\right)$ and it is valid for arbitrary energy $E\left(0\right)$.

Remark 1. 

All results presented in Section 3 are valid for the case of pure logarithmic nonlinearity, i.e., when $a_i\left(x\right)\equiv 0$ for every $i=1,\cdots ,r$.

4. Nonexistence of Global Solutions

Our aim in this section is to study the nonexistence of global solutions to problem (1)–(4) provided the additional Assumption (5); see Theorems 3 and 4. Later on, we show that (5) together with (41) changes the blow up type of the solutions: from infinite time blow up (under (3) and (4)) to finite time blow up (under (3)–(5) and (41)); see Theorems 5 and 6.

Theorem 3. 

Suppose (3)–(5) hold and $u(t,x)$ is a weak solution of problem (1), (2) defined in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$.

(i) 

Let $E\left(0\right)>0$, then

$\left(i_1\right)$

If (15) is satisfied, then the maximal existence time of the solution $u(t,x)$ is finite, i.e., $T_m<\infty$.

$\left(i_2\right)$

If (15) fails, then $\parallel u\left(t\right)\parallel$ is bounded for $t\in [0,T_m)$ and estimate (16) holds for $t\in [0,T_m)$.

(ii) 

Let $E\left(0\right)=0$, then

$\left(i{i}_1\right)$

If (17) is satisfied, then the maximal existence time of $u(t,x)$ is finite, i.e., $T_m<\infty$.

$\left(i{i}_2\right)$

If (17) fails, then $\parallel u\left(t\right)\parallel$ is bounded for every $t\in [0,T_m)$ and the estimate (18) holds for $t\in [0,T_m)$.

(iii) 

If $E\left(0\right)<0$, then the maximal existence time of $u(t,x)$ is finite, i.e., $T_m<\infty$.

Proof. 

$\left(i\right)$ Let $E\left(0\right)>0$.

$\left(i_1\right)$ Assume by contradiction that $T_m=\infty$. From Theorem 2$\left(i_1\right)$ we obtain that $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$ blows up at infinity.

Now, we will prove that there exists $t_0\ge b$ such that $\psi \left(t\right)$ satisfies, for every $t\ge t_0$, the following inequality:

$${\psi }^{″}\left(t\right)\psi \left(t\right)-{\displaystyle \frac{(p+3)}{4}}{\psi }^{\prime 2}\left(t\right)\ge B_1\left(t\right),$$

(32)

where $p\in (1,p_1]$ and $B_1\left(t\right)$ is a suitable nonnegative function for $t_0\ge b$, which will be defined later on.

Indeed, from (6)–(8) and (19), we have for every $p\in (1,p_1]$ the identity

$$\begin{array}{cc}\hfill {\psi }^{″}\left(t\right)=& (p+3)\parallel u_t{\left(t\right)\parallel }^2-2(p+1)E\left(0\right)+{(p-1)\parallel \nabla u\left(t\right)\parallel }^2+{\displaystyle \frac{k(p+1)}{2}}{\int }_{\Omega }{a}_0\left(x\right){\left|u\right|}^{2}dx\hfill \\ & -k(p-1){\int }_{\Omega }{a}_0\left(x\right)u^{2}ln\left|u\right|dx+2\sum _{i=1}^r{\displaystyle \frac{p_i-p}{p_i+1}}{\int }_{\Omega }{a}_i\left(x\right){\left|u\right|}^{p_i+1}dx.\hfill \end{array}$$

(33)

Let us define the sets

$${\Omega }_1\left(t\right)=\{x\in \Omega :|u(t,x)|<1\},{\Omega }_2\left(t\right)=\{x\in \Omega :|u(t,x)|\ge 1\}.$$

Hence, $\Omega \left(t\right)={\Omega }_1\left(t\right)\cup {\Omega }_2\left(t\right)$.

For the last term on the right-hand side of (33) we obtain the estimate

$$\begin{array}{cc}\hfill 2\sum _{i=1}^r{\displaystyle \frac{p_i-p}{p_i+1}}{\int }_{\Omega }{a}_i\left(x\right){\left|u\right|}^{p_i+1}dx& \ge 2\sum _{i=1}^r{\displaystyle \frac{p_i-p}{p_i+1}}{\int }_{{\Omega }_2\left(t\right)}{a}_i\left(x\right){\left|u\right|}^{p_i+1}dx\hfill \\ & \ge 2\sum _{i=1}^r{\displaystyle \frac{p_i-p}{p_i+1}}{\int }_{{\Omega }_2\left(t\right)}{a}_i\left(x\right){\left|u\right|}^{p+1}dx\hfill \\ & \ge {\displaystyle \frac{2(p_1-p)}{p_r+1}}{\int }_{{\Omega }_2\left(t\right)}\left(\sum _{i=1}^r{a}_i\left(x\right)\right){\left|u\right|}^{p+1}dx\hfill \\ & \ge K_1{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{p+1}dx,\hfill \end{array}$$

(34)

where

$$K_1={\displaystyle \frac{2(p_1-p)}{p_r+1}}K>0.$$

To deal with the logarithmic term in (33) we use the trivial inequality

$$z^{2}ln\left|z\right|\le {\displaystyle \frac{1}{\gamma }}{\left|z\right|}^{2+\gamma }\mathrm{for}\left|z\right|\ge 1\mathrm{and}\mathrm{every}\gamma >0.$$

Since

$${\int }_{{\Omega }_1\left(t\right)}{a}_0\left(x\right)u^{2}ln\left|u\right|dx\le 0$$

(35)

and

$$0\le {\int }_{{\Omega }_2\left(t\right)}{a}_0\left(x\right)u^{2}ln\left|u\right|dx\le {\displaystyle \frac{A}{\gamma }}{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{2+\gamma }dx,$$

(36)

we have the estimate

$$\begin{array}{cc}\hfill -k(p-1)& {\int }_{\Omega }{a}_0\left(x\right)u^{2}ln\left|u\right|dx\hfill \\ & \ge -k(p-1){\int }_{{\Omega }_1\left(t\right)}{a}_0\left(x\right)u^{2}ln\left|u\right|dx-{\displaystyle \frac{k(p-1)A}{\gamma }}{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{2+\gamma }dx\hfill \\ & \ge -{\displaystyle \frac{k(p-1)A}{\gamma }}{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{2+\gamma }dx.\hfill \end{array}$$

(37)

If we choose

$$\gamma ={\displaystyle \frac{p-1}{2}},$$

then from (33), (34) and (37) it follows that

$$\begin{array}{cc}\hfill {\psi }^{″}\left(t\right)\ge & (p+3)\parallel u_t{\left(t\right)\parallel }^2-2(p+1)E\left(0\right)+{(p-1)\parallel \nabla u\left(t\right)\parallel }^2+{\displaystyle \frac{k(p+1)a}{2}}{\parallel u\left(t\right)\parallel }^2\hfill \\ & +K_1{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{p+1}dx-2kA{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{{\displaystyle \frac{p+3}{2}}}dx.\hfill \end{array}$$

(38)

Further, we represent ${\Omega }_2\left(t\right)={\Omega }_3\left(t\right)\cup {\Omega }_4\left(t\right)$, where

$${\Omega }_3\left(t\right)=\{x\in {\Omega }_2\left(t\right):\left|u(t,x)\right|<M\},{\Omega }_4\left(t\right)=\{x\in {\Omega }_2\left(t\right):\left|u(t,x)\right|\ge M\}$$

and

$$M={\left({\displaystyle \frac{2kA}{K_1}}\right)}^{{\displaystyle \frac{2}{p-1}}}.$$

Straightforward computations give us

$$\begin{array}{cc}& K_1{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{p+1}dx-2kA{\int }_{{\Omega }_2\left(t\right)}{\left|u\right|}^{{\displaystyle \frac{p+3}{2}}}dx\hfill \\ & \ge K_1{\int }_{{\Omega }_4\left(t\right)}{\left|u\right|}^{p+1}dx-2kA{\int }_{{\Omega }_3\left(t\right)}{\left|u\right|}^{{\displaystyle \frac{p+3}{2}}}dx-2kA{\int }_{{\Omega }_4\left(t\right)}{\left|u\right|}^{{\displaystyle \frac{p+3}{2}}}dx\hfill \\ & \ge {\int }_{{\Omega }_4\left(t\right)}{\left|u\right|}^{{\displaystyle \frac{p+3}{2}}}(K_1{\left|u\right|}^{{\displaystyle \frac{p-1}{2}}}-2kA)dx-2kA{M}^{{\displaystyle \frac{p+3}{2}}}|\Omega |\ge -2kA{M}^{{\displaystyle \frac{p+3}{2}}}|\Omega |.\hfill \end{array}$$

(39)

Finally, combining (38) and (39), we conclude that $\psi \left(t\right)$ satisfies (32) with

$$\begin{array}{cc}\hfill B_1\left(t\right):=& {(p-1)\parallel \nabla u\left(t\right)\parallel }^2{\parallel u\left(t\right)\parallel }^2+{\displaystyle \frac{1}{2}}k(p+1)a{\parallel u\left(t\right)\parallel }^4\hfill \\ & +(p+3)\left(\parallel u_t{\left(t\right)\parallel }^2{\parallel u\left(t\right)\parallel }^2-{(u\left(t\right),u_t\left(t\right))}^2\right)\hfill \\ & -{2(p+1)E\left(0\right)\parallel u\left(t\right)\parallel }^2-2kA{M}^{{\displaystyle \frac{p+3}{2}}}{|\Omega |\parallel u\left(t\right)\parallel }^2.\hfill \end{array}$$

Since $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$ blows up at infinity, there exists a sufficiently large time $t_0$ such that $B_1\left(t\right)\ge 0$ for every $t\ge t_0$. Then, according to Lemma 2, we obtain that $T_m<\infty$, which contradicts our assumption $T_m=\infty$. Thus, the solution $u(t,x)$ exists for a finite time and $\left(i_1\right)$ is proved.

$\left(i_2\right)$ The proof coincides with the proof of Theorem 2$\left(i_2\right)$ and we omit it.

$\left(ii\right)$ Let $E\left(0\right)=0$.

$\left(i{i}_1\right)$ Suppose that $T_m=\infty$. Then, from Theorem 2$\left(i{i}_1\right)$ it follows that $u(t,x)$ blows up at infinity. The rest of the proof is similar to the proof of Theorem 3$\left(i_1\right)$.

$\left(i{i}_2\right)$ As in the proof of Theorem 2$\left(i{i}_2\right)$, we obtain that function $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$ is a decreasing function for $t\in [0,T_m)$. Hence, the estimate (18) holds for $t\in [0,T_m)$.

$\left(iii\right)$ Let $E\left(0\right)<0$. If we suppose by contradiction that $T_m=\infty$, then it follows from Theorem 2$\left(iii\right)$ that $u(t,x)$ blows up at infinity. The rest of the proof is similar to the proof of Theorem 3$\left(i_1\right)$. □

Theorem 4. 

Suppose (3)–(5) hold and $u(t,x)$ is a weak solution of problem (1), (2) defined in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$.

(i) 

If (11) is satisfied, i.e.,

$$\mathit{there}\mathit{exists}b\ge 0\mathit{such}\mathit{that}E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u\left(b\right)\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u\left(b\right),u_t\left(b\right)),$$

then the maximal existence time is finite, i.e., $T_m<\infty$.

(ii) 

If (11) fails, then $\parallel u\left(t\right)\parallel$ is bounded for $t\in [0,T_m)$ and estimate (12) holds for $t\in [0,T_m)$, i.e.,

$${\parallel u\left(t\right)\parallel }^2\le \left(\parallel u_0{\parallel }^2-{\displaystyle \frac{4E\left(0\right)}{ka}}\right)e^{-\sqrt{ka}t}+{\displaystyle \frac{4E\left(0\right)}{ka}}.$$

holds for every $t\in [0,T_m)$.

Proof. 

$\left(i\right)$ Assume by contradiction that $T_m=\infty$. Then, according to Theorem 1$\left(i\right)$, we obtain that $\psi \left(t\right)={\parallel u\left(t\right)\parallel }^2$ blows up at infinity. The rest of the proof is identical to the proof of Theorem 3$\left(i_1\right)$.

$\left(ii\right)$ If (11) fails, then for $t\in [0,T_m)$ we have

$$\begin{array}{c}\hfill E\left(0\right)\ge {\displaystyle \frac{ka}{4}}{\parallel u\left(t\right)\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u\left(t\right),u_t\left(t\right))={\displaystyle \frac{ka}{4}}\psi \left(t\right)+{\displaystyle \frac{\sqrt{ka}}{4}}{\psi }^{\prime }\left(t\right).\end{array}$$

(40)

Integrating (40), we obtain (12). The proof of Theorem 4 is completed. □

In Theorems 3$\left(i_1\right)$ and 4$\left(i\right)$, we prove that the maximal existence time of the weak solution to problem (1)–(5) is finite ($T_m<\infty$). However, we do not know whether $\parallel u\left(t\right)\parallel \to \infty$ as $t\to T_m$ or $\parallel \nabla u\left(t\right)\parallel \to \infty$ as $t\to T_m$. To exclude the case when $\parallel u\left(t\right)\parallel$ is bounded in $[0,T_m)$ but $\parallel \nabla u\left(t\right)\parallel$ blows up at $T_m$, we impose the following additional restrictions on the power exponents $p_i$, $i=1,..,r$:

$$\begin{array}{cccc}& p_i<min\left\{{\displaystyle \frac{n+4}{n}},{\displaystyle \frac{n}{n-2}}\right\}={\displaystyle \frac{n+4}{n}},\hfill & \hfill & \mathrm{for}n=1,2,3,4;\hfill \\ & p_i\le min\left\{{\displaystyle \frac{n+4}{n}},{\displaystyle \frac{n}{n-2}}\right\}={\displaystyle \frac{n}{n-2}},\hfill & \hfill & \mathrm{for}n\ge 5.\hfill \end{array}$$

(41)

In the following two theorems, we formulate and prove our main assertions, namely, that under the additional restriction (41) the local weak solutions to (1)–(5) blow up at the maximal existence time $T_m<\infty$, i.e., $lim\; sup\parallel u\left(t\right)\parallel \to \infty$ as $t\to T_m$. Moreover, conditions (11), (15), and (17) become necessary and sufficient ones for finite time blow up of the solutions.

Theorem 5. 

Suppose (3)–(5) and (41) hold. Let $u(t,x)$ be a local weak solution of problem (1), (2), defined in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$.

(i) 

If $E\left(0\right)>0$, then $u(t,x)$ blows up at finite time $T_m$ if and only if assumption (15) is satisfied.

(ii) 

If $E\left(0\right)=0$, then $u(t,x)$ blows up at finite time $T_m$ if and only if assumption (17) is satisfied.

(iii) 

If $E\left(0\right)<0$, then every local weak solution $u(t,x)$ blows up at finite time $T_m$.

Proof. 

$\left(i\right)$ Let $E\left(0\right)>0$.

Sufficiency. Suppose (15) holds. Then, from Theorem 3$\left(i_1\right)$ it follows that $T_m<\infty$. We assume by contradiction that $u(t,x)$ does not blow up at $T_m$, i.e., for every $t\in [0,T_m)$ and some constant $C_1$ we have

$$\underset{t\to T_m,t<T_m}{lim\; sup}\parallel u\left(t\right)\parallel <C_1<\infty .$$

(42)

We will show that if (42) holds, then $\parallel \nabla u\left(t\right)\parallel$ and $\parallel u_t\left(t\right)\parallel$ are bounded in the maximal existence time interval $[0,T_m)$.

Now, the expression (7) and the conservation law (8) give us

$$\begin{array}{cc}\hfill {\parallel \nabla u\left(t\right)\parallel }^2\le & 2E\left(0\right)+k{\int }_{\Omega }{a}_0\left(x\right)u^2(t,x)ln\left|u(t,x)\right|dx+2\sum _{i=1}^r{\int }_{\Omega }{\displaystyle \frac{a_i\left(x\right)}{p_i+1}}{\left|u(t,x)\right|}^{p_i+1}dx.\hfill \end{array}$$

(43)

We estimate the logarithmic term on the right-hand side of (43) by means of the technique proposed in the proof of Theorem 3(i).

Using (35) and (36) with $\gamma =p_r-1$, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{a}_0\left(x\right)u^{2}ln\left|u(t,x)\right|dx& \le {\int }_{{\Omega }_1\left(t\right)}{a}_0\left(x\right)u^{2}ln\left|u(t,x)\right|dx+{\int }_{{\Omega }_2\left(t\right)}{a}_0\left(x\right)u^{2}ln\left|u(t,x)\right|dx\hfill \\ & \le {\displaystyle \frac{A}{\gamma }}{\int }_{{\Omega }_2\left(t\right)}{\left|u(t,x)\right|}^{2+\gamma }dx\le {\displaystyle \frac{A}{\gamma }}{\parallel u\left(t\right)\parallel }_{2+\gamma }^{2+\gamma }\hfill \\ & ={\displaystyle \frac{A}{p_r-1}}{\parallel u\left(t\right)\parallel }_{p_r+1}^{p_r+1}.\hfill \end{array}$$

(44)

Then, from (43) and (44), we obtain the estimate

$$\begin{array}{c}\hfill {\parallel \nabla u\left(t\right)\parallel }^2\le 2E\left(0\right)+{\displaystyle \frac{kA}{p_r-1}}{\parallel u\left(t\right)\parallel }_{p_r+1}^{p_r+1}+2\sum _{i=1}^r{\displaystyle \frac{A_i}{p_i+1}}{\parallel u\left(t\right)\parallel }_{p_i+1}^{p_i+1}.\end{array}$$

(45)

For every $p_i$, $i=1,\cdots ,r$ satisfying (41), we apply the Gagliardo–Nirenberg inequality (14) with $j=0$, $m=1$, $r=q=2$, and $\alpha =(p_i-1)n/\left(2(p_i+1)\right)$. Thus, using (14) and (42), we have the estimate

$${\parallel u\left(t\right)\parallel }_{p_i+1}\le C_{n,p_i}{\parallel \nabla u\left(t\right)\parallel }^{{\displaystyle \frac{n(p_i-1)}{2(p_i+1)}}}{\parallel u\left(t\right)\parallel }^{1-{\displaystyle \frac{n(p_i-1)}{2(p_i+1)}}}\le C_{n,p_i}{C}_1^{1-{\displaystyle \frac{n(p_i-1)}{2(p_i+1)}}}{\parallel \nabla u\left(t\right)\parallel }^{{\displaystyle \frac{n(p_i-1)}{2(p_i+1)}}},$$

(46)

where the constant $C_{n,p_i}$ depends only on n and $p_i$.

Applying (46) and the Young’s inequality for

$$p_i^{*}={\displaystyle \frac{4}{n(p_i-1)}},q_i^{*}={\displaystyle \frac{4}{4-n(p_i-1)}}$$

and every $ϵ>0$, we obtain

$${\parallel u\left(t\right)\parallel }_{p_i+1}^{p_i+1}\le ϵ{\parallel \nabla u\left(t\right)\parallel }^2+{\displaystyle \frac{4+n-n{p}_i}{4}}{\left(ϵp_i^{*}\right)}^{-{\displaystyle \frac{q_i^{*}}{p_i^{*}}}}{\left(C_{n,p_i}{C}_1^{1-{\displaystyle \frac{n(p_i-1)}{2(p_i+1)}}}\right)}^{(p_i+1)q_i^{*}}.$$

(47)

If we choose

$$ϵ={\left(4\sum _{i=1}^r{\displaystyle \frac{A_i}{p_i+1}}+{\displaystyle \frac{2kA}{p_r-1}}\right)}^{-1},$$

we have from (45) and (47) the estimate

$$\begin{array}{c}\hfill {\parallel \nabla u\left(t\right)\parallel }^2\le 2E\left(0\right)+{\displaystyle \frac{1}{2}}{\parallel \nabla u\left(t\right)\parallel }^2+C_2,\end{array}$$

where the constant $C_2$ is given by the expression

$$\begin{array}{cc}\hfill C_2=& 2\sum _{i=1}^r{\displaystyle \frac{A_i}{p_i+1}}{\displaystyle \frac{4+n-n{p}_i}{4}}{\left(ϵp_i^{*}\right)}^{-{\displaystyle \frac{q_i^{*}}{p_i^{*}}}}{\left(C_{n,p_i}{C}_1^{1-{\displaystyle \frac{n(p_i-1)}{2(p_i+1)}}}\right)}^{(p_i+1)q_i^{*}}\hfill \\ \hfill & +{\displaystyle \frac{kA}{p_r-1}}{\displaystyle \frac{4+n-n{p}_r}{4}}{\left(ϵp_r^{*}\right)}^{-{\displaystyle \frac{q_r^{*}}{p_r^{*}}}}{\left(C_{n,p_r}{C}_1^{1-{\displaystyle \frac{n(p_r-1)}{2(p_r+1)}}}\right)}^{(p_r+1)q_r^{*}}<\infty .\hfill \end{array}$$

Hence,

$$\underset{t\to T_m,t<T_m}{lim\; sup}\parallel \nabla u\left(t\right)\parallel \le 4E\left(0\right)+2C_2<\infty .$$

(48)

Now, we will prove that $\parallel u_t\left(t\right)\parallel$ is bounded for every $t\in [0,T_m)$. Since (4) holds, it follows from the embedding theorem (13) and (48) that

$$\underset{t\to T_m,t<T_m}{lim\; sup}{\parallel u\left(t\right)\parallel }_{p_i+1}<\infty \mathrm{for}i=1,\cdots r.$$

(49)

Then, using (7), the conservation law (8), the estimates (44), and (49), we obtain

$$\underset{t\to T_m,t<T_m}{lim\; sup}\parallel u_t\left(t\right)\parallel <\infty .$$

From the Continuation Principle, based on the local existence result, see [9,29], it follows that problem (1), (2) has a local weak solution in the interval $[0,T_m+\delta )$, $\delta >0$. This contradicts that $[0,T_m)$ is the maximal existence time interval. Hence, $u(t,x)$ blows up for at $T_m<\infty$.

Necessity. The proof is identical to the proof of Necessity in Theorem 2$\left(i_1\right)$.

(ii) Let $E\left(0\right)=0$.

Sufficiency. If (17) holds, then Theorem 3$\left(i{i}_1\right)$ gives us that $T_m<\infty$. Analogously to the proof of the sufficiency of Theorem 5$\left(i\right)$, we conclude that $u(t,x)$ blows up at $T_m<\infty$.

The proof of Necessity is identical to the proof of necessity in Theorem 2$\left(i{i}_1\right)$.

(iii) Let $E\left(0\right)<0$. From Theorem 3$\left(iii\right)$ it follows that $T_m<\infty$. The rest of the proof follows from the sufficiency of Theorem 5$\left(i\right)$. Theorem 5 is proved. □

Theorem 6. 

Suppose (3)–(5) and (41) hold. Then, every local weak solution $u(t,x)$ of problem (1), (2), defined in the maximal existence time interval $[0,T_m)$, blows up at finite time $T_m$ if and only if (11) is satisfied.

The proof follows the same ideas as the proof of Theorem 5. The only difference is that the conclusion $T_m<\infty$ is made by means of Theorem 4(i) instead of Theorem 3$\left(i_1\right)$.

In the following proposition, we summarize our sufficient conditions for finite time blow up, obtained from Theorems 5 and 6.

Proposition 3 (Sufficient conditions).

Suppose (3)–(5) and (41) hold. If $u(t,x)$ is a local weak solution of problem (1), (2), defined in the maximal existence time interval $[0,T_m)$, $0<T_m\le \infty$, then $u(t,x)$ blows up at finite time $T_m$ if at least one of the following assumptions (50)–(52) or (53) holds:

$$\parallel u_0\parallel >0,(u_0,u_1)\ge 0,0<E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u_0\parallel }^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{(u_0,u_1)}^2}{\parallel u_0{\parallel }^2}},$$

(50)

$$E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u_0\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u_0,u_1),$$

(51)

$$E\left(0\right)=0,(u_0,u_1)\ge 0$$

(52)

or

$$E\left(0\right)<0.$$

(53)

5. Comparison and Discussion

Firstly, we comment on the results for blowing up at infinity of the global weak solutions to problem (1)–(4). We compare the result of Theorem 1, which is proved in [15], with the result of Theorem 2. Since the conditions of Theorem 1 and Theorem 2 are necessary and sufficient conditions for infinite time blow up, both results are equivalent. The only difference is the time $b\ge 0$, where these conditions hold. If the conditions in Theorem 1 are satisfied at some time $b_1\ge 0$, then there exists time $b_2\ge 0$ for which the conditions in Theorem 2 hold and vice versa.

As for the sufficient conditions given at $t=0$, they are different. We compare the sufficient condition for arbitrary positive energy from Proposition 1 in [15], namely,

$$E\left(0\right)<{\displaystyle \frac{ka}{4}}{\parallel u_0\parallel }^2+{\displaystyle \frac{\sqrt{ka}}{2}}(u_0,u_1),$$

(54)

with the sufficient condition (31) from Proposition 1. If

$$(u_0,u_1)>\sqrt{ka}{\parallel u_0\parallel }^2,$$

then condition (31) is better then (54). If the opposite inequality is satisfied, i.e., $(u_0,u_1)<\sqrt{ka}{\parallel u_0\parallel }^2,$ then condition (54) is more general than (31). Finally, both conditions coincide when $(u_0,u_1)=\sqrt{ka}{\parallel u_0\parallel }^2.$

In order to compare our results with the results in [14], we consider problem (1), (2) for the following special case of the nonlinearity

$$f(x,u)={uln\left|u\right|}^k,k>1.$$

(55)

Now, we show that for arbitrary positive energy, $E\left(0\right)>0$, sufficient condition (31) in Proposition 1 is more general than condition (ii)(c) of Theorem 3.1 in [14], namely,

$$E\left(0\right)>0,I\left(u_0\right)<0,{\parallel u_0\parallel }^2>{\displaystyle \frac{4}{k}}E\left(0\right)\mathrm{and}(u_0,u_1)>0.$$

(56)

Indeed, if initial data $u_0$, $u_1$ satisfy (56), then they necessarily satisfy condition (31). Note that the assumption $I\left(u_0\right)<0$ in (56) is superfluous. In fact, if $\parallel u_0{\parallel }^2>{\displaystyle \frac{4}{k}}E\left(0\right)$, we conclude from (9) that $I\left(u_0\right)<0$. Furthermore, the inequality $I\left(u_0\right)<0$ results from the even more general assumptions (31); see Proposition 2.

Finally, let us mention that the problem (1), (2) with nonlinearity (55) is studied for different intervals of k. For example, the case $k>1$ is considered in [12,14], the case $k\in (0,1)$ is investigated in [10,19], while in [11] some additional restrictions on k are imposed. In contrast to the papers cited above, our results are valid for any $k>0$.

6. Conclusions

• In this paper, we study the wave equation with combined logarithmic and power-type nonlinearities with nonnegative variable coefficients.
• New necessary and sufficient conditions for infinite time blow up of the global weak solutions to problem (1)–(4) are proved for every initial energy level; see Theorem 2.
• By introducing the additional assumptions (5) and (41), we establish novel necessary and sufficient conditions for finite time blow up of the local weak solutions to problem (1)–(5), (41); see Theorems 5 and 6. This demonstrates that the combination of conditions (5) and (41) significantly alters the blow up type of the weak solutions: from infinite time blow up (under (3) and (4)) to finite time blow up (under (3)–(5) and (41)).
• For arbitrary initial energy, sufficient conditions for infinite and finite time blow up are derived; see Propositions 1 and 3.
• All the results presented in the paper can be applied to other hyperbolic and parabolic equations with nonlinearity $f(x,u)$ considered in (1).