On the Cauchy Problem for Pseudohyperbolic Equations with Lower Order Terms

Abstract

We consider a class of strictly pseudohyperbolic equations with lower order terms. The solvability of the Cauchy problem in Sobolev spaces with special weights is established. The uniqueness of the solution is proven and estimates are obtained.

1. Introduction

Pseudohyperbolic equations are equations that are unsolvable with respect to the highest derivative and have the form

$$L_0\left(D_x\right)D_t^{l}u+\sum _{k=0}^{l-1}{L}_{l-k}\left(D_x\right)D_t^{k}u=f(t,x),$$

(1)

where $L_0\left(D_x\right)$ is a quasielliptic operator. The class of pseudohyperbolic equations was introduced in [1]. Examples of such equations are equations arising in hydrodynamics (for example, the generalized Boussinesq equation [2,3,4,5,6,7]), in elasticity theory (for example, the Vlasov equation [8,9]), and in waveguide modeling (see, for example [10,11,12]).

The theory of partial differential equations of the form (1) began to develop after the publication of S. L. Sobolev’s works on the dynamics of a rotating fluid (see his works in [13]). These works were the first in-depth studies of differential equations unsolvable with respect to the highest derivative. Therefore, equations of the form (1) are often called Sobolev-type equations. Currently, there is a large number of publications devoted to the study of various problems for such equations. There are also more than dozen monographs on this theme (see, for example [1,14,15,16]). However, there are still few works on the theory of boundary value problems for pseudohyperbolic equations. The theory of the Cauchy problem is the most developed for such equations (see, for example [1,12,14,17,18,19]). Note that in these papers, the solvability of the Cauchy problem was studied in well-known Sobolev spaces $W_{2,\gamma }^r$ with the exponential weight $e^{-\gamma t}$. In the literature, such spaces are often used to prove the solvability of boundary value problems for parabolic and hyperbolic equations. However, when studying the solvability of the Cauchy problem for equations unsolvable with respect to the highest derivative in the spaces $W_{2,\gamma }^{l,r}$, essential differences from the classical equations may arise (see [1]). In particular, in [1,17,18,19], the unique solvability of the Cauchy problem for strictly pseudohyperbolic equations in $W_{2,\gamma }^{l,r}$ was proven under the condition that the right-hand sides of the equations have certain smoothness and are orthogonal to some monomials. In these papers, it was established that the number of the orthogonality conditions is finite and essentially depends on the lower order terms of the equations. Note that the requirements of the orthogonality of the right-hand sides to some monomials for the solvability of the Cauchy problem for equations of the form (1) differ significantly from the solvability conditions for the Cauchy problem for equations of the hyperbolic and parabolic type.

In this paper, we continue the study of the Cauchy problem for strictly pseudohyperbolic equations with constant coefficients and lower order terms. Here, we define a new class of weighted Sobolev spaces $W_{2,\gamma ,{\varkappa }_q,\sigma }^{l,r}$. This class contains the spaces $W_{2,\gamma }^{l,r}$, i.e., $W_{2,\gamma }^{l,r}\subset W_{2,\gamma ,{\varkappa }_q,\sigma }^{l,r}$. Functions from this class and their generalized derivatives belong to Lebesgue spaces with the exponential weight $e^{-\gamma t}$ and special power weights to $x\in {ℝ}^n$. We prove the existence and the uniqueness of the solution to the Cauchy problem in these spaces under minimal requirements on the right-hand sides. The obtained results strengthen well-known theorems from [1,17,18,19].

2. The Main Results

Recall the definition of pseudohyperbolic operators without lower order terms [1]:

$$L(D_t,D_x)=L_0\left(D_x\right)D_t^l+\sum _{k=0}^{l-1}{L}_{l-k}\left(D_x\right)D_t^k.$$

(2)

We assume that the operators satisfy the following conditions.

Condition 1. 

The symbol $L(i\eta ,i\xi )$ of the operator (2) is homogeneous with respect to some vector $({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n)$, ${\alpha }_0>0$, $1/{\alpha }_j\in \mathbb{N}$, $j=1,\dots ,n$, i.e.,

$$L(c^{{\alpha }_0}i\eta ,c^{{\alpha }_1}i{\xi }_1,\dots ,c^{{\alpha }_n}i{\xi }_n)=cL(i\eta ,i{\xi }_1,\dots ,i{\xi }_n),c>0.$$

Condition 2. 

The operator $L_0\left(D_x\right)$ is quasielliptic, i.e., $L_0\left(i\xi \right)=0$, $\xi \in {ℝ}^n$, if and only if $\xi =0$.

Condition 3. 

The equation

$${\left(i\eta \right)}^l+\sum _{k=0}^{l-1}\frac{L_{l-k}\left(i\xi \right)}{L_0\left(i\xi \right)}{\left(i\eta \right)}^k=0,\xi \in {ℝ}^n\setminus \left\{0\right\},$$

(3)

has only real roots ${\eta }_1\left(\xi \right),\dots ,{\eta }_l\left(\xi \right)$.

Definition 1. 

The differential operator $L(D_t,D_x)$ in (2) is called pseudohyperbolic if Conditions 1–3 hold. If the roots of (3) are distinct real numbers, then the operator $L(D_t,D_x)$ is called strictly pseudohyperbolic.

Consider strictly pseudohyperbolic operators with lower order terms of the form

$$\mathcal{L}(D_t,D_x)=(L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))D_t^l+\sum _{k=0}^{l-1}(L_{l-k}\left(D_x\right)+L_{l-k}^{\prime }\left(D_x\right))D_t^k,$$

(4)

i.e., $\mathcal{L}(D_t,D_x)$ is representable as

$$\mathcal{L}(D_t,D_x)=L(D_t,D_x)+L^{\prime }(D_t,D_x),$$

where the principal part $L(D_t,D_x)$ is a strictly pseudohyperbolic operator, and

$$L^{\prime }(D_t,D_x)=L_0^{\prime }\left(D_x\right)D_t^l+\sum _{k=0}^{l-1}{L}_{l-k}^{\prime }\left(D_x\right)D_t^k.$$

Here, the symbols $L_{l-k}^{\prime }\left(i\xi \right)$ of the operators $L_{l-k}^{\prime }\left(D_x\right)$, $k=0,\dots ,l$, satisfy the inequalities

$$c_1({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q})\le |L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)|\le c_2({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q}),$$

(5)

where

$${\langle \xi \rangle }^2=\sum _{i=1}^n{\xi }_i^{2/{\alpha }_i},\varkappa =1-l{\alpha }_0,{\varkappa }_q=q-l{\alpha }_0,0<{\varkappa }_q<\varkappa ,$$

$$\hspace{1em}\hspace{1em}|L_{l-k}^{\prime }\left(i\xi \right)|\le c_0{\langle \xi \rangle }^{{\epsilon }_k},{\varkappa }_q\le {\epsilon }_k<1-k{\alpha }_0,k=0,\dots ,l-1.$$

(6)

We study the class of the operators (4) for which the equation

$${\left(i\eta \right)}^l+\sum _{k=0}^{l-1}\frac{L_{l-k}\left(i\xi \right)+L_{l-k}^{\prime }\left(i\xi \right)}{L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)}{\left(i\eta \right)}^k=0,\xi \in {ℝ}^n\setminus \left\{0\right\},$$

(7)

has distinct real roots ${\eta }_1\left(\xi \right),\dots ,{\eta }_l\left(\xi \right)$, and the function

$$m(i\eta +\gamma ,i\xi )=\sum _{k=1}^l\prod _{l\ne k}{|i\eta +\gamma -i{\eta }_j\left(\xi \right)|}^2$$

(8)

for all $(\eta ,\xi )\in {ℝ}^{n+1}$, $\gamma \ge 0$ satisfies the estimates

$$a_1{\left(\right|i\eta +\gamma |+\langle \xi \rangle }^{{\alpha }_0}{)}^{2(l-1)}\le m(i\eta +\gamma ,i\xi )\le a_2{\left(\right|i\eta +\gamma |+\langle \xi \rangle }^{{\alpha }_0}{)}^{2(l-1)},$$

(9)

where $a_1,a_2>0$ are some constants.

We consider the Cauchy problem for strictly pseudohyperbolic equations with lower order terms and zero initial conditions

$$\left\{\begin{array}{c}\mathcal{L}(D_t,D_x)u=f(t,x),t>0,x\in {ℝ}^n,\hfill \\ \\ D_t^{k}u{|}_{t=0}=0,k=0,\dots ,l-1.\hfill \end{array}\right.$$

(10)

To study the problem, we follow the scheme of [1,18].

Let $G\subseteq {ℝ}^{n+1}$. We use the symbol

$$W_{2,\gamma }^{l,r}\left(G\right),r=(1/{\alpha }_1,\dots ,1/{\alpha }_n),\gamma >0,$$

to denote the Sobolev space with the weight $e^{-\gamma t}$. The norm in $W_{2,\gamma }^{l,r}\left(G\right)$ is defined as follows:

$$\parallel u(t,x),W_{2,\gamma }^{l,r}\left(G\right)\parallel = \parallel e^{-\gamma t}u(t,x),W_2^{l,r}\left(G\right)\parallel .$$

The Cauchy problem (10) was studied in [18] in the case of

$$f(t,x)\in W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\cap L_{2,\gamma }({ℝ}_{+};L_1\left({ℝ}^n\right)),s=(1/{\alpha }_1,\dots ,1/{\alpha }_n),\gamma >0,$$

$\left|\alpha \right|/2>{\varkappa }_q$, $\left|\alpha \right|={\displaystyle \sum _{i=1}^n}{\alpha }_i$. The unique solvability of (10) in $W_{2,\gamma }^{l,r}\left({ℝ}_{+}^{n+1}\right)$ was proven under the assumption that $D_t^{k}u(t,x)\in W_{2,\gamma }^{0,(1-k{\alpha }_0)r}\left({ℝ}_{+}^{n+1}\right)$, $k=0,\dots ,l$.

In this article, we investigate the solvability of the Cauchy problem (10) in a wider scale of weighted Sobolev spaces

$$W_{2,\gamma ,{\varkappa }_q,\sigma }^{l,r}\left(G\right),G\subseteq {ℝ}^{n+1},r=(1/{\alpha }_1,\dots ,1/{\alpha }_n),\gamma >0,\sigma \ge 0.$$

A locally integrable function $u(t,x)$ belongs to the space $W_{2,\gamma ,{\varkappa }_q,\sigma }^{l,r}\left(G\right)$, if $u(t,x)$ has the generalized derivatives $D_t^{{\beta }_0}{D}_x^{\beta }u(t,x)$, ${\beta }_0/l+\beta \alpha \le 1$, in G; moreover,

$$D_t^{{\beta }_0}{D}_x^{\beta }u(t,x)\in L_{2,\gamma }\left(G\right),{\varkappa }_q\le \beta \alpha \le 1,$$

and

$${(1+\langle x\rangle )}^{-\sigma ({\varkappa }_q-\beta \alpha )}{D}_t^{{\beta }_0}{D}_x^{\beta }u(t,x)\in L_{2,\gamma }\left(G\right),{\langle x\rangle }^2=\sum _{i=1}^n{x}_i^{2/{\alpha }_i},0\le \beta \alpha <{\varkappa }_q.$$

The norm is defined as follows:

$$\parallel u(t,x),W_{2,\gamma ,{\varkappa }_q,\sigma }^{l,r}\left(G\right)\parallel =\sum _{{\beta }_0/l+\beta \alpha \le 1,{\varkappa }_q\le \beta \alpha \le 1}\parallel D_t^{{\beta }_0}{D}_x^{\beta }u(t,x),L_{2,\gamma }\left(G\right)\parallel$$

$$+\sum _{{\beta }_0/l+\beta \alpha \le 1,0\le \beta \alpha <{\varkappa }_q}\parallel {(1+\langle x\rangle )}^{-\sigma ({\varkappa }_q-\beta \alpha )}{D}_t^{{\beta }_0}{D}_x^{\beta }u(t,x),L_{2,\gamma }\left(G\right)\parallel .$$

Denote the Fourier transform of $u_{\gamma }(t,x)=e^{-\gamma t}u(t,x)\in L_2\left({ℝ}^{n+1}\right)$ by ${\tilde{u}}_{\gamma }(\eta ,\xi )$, its partial Fourier transform in x by ${\widehat{u}}_{\gamma }(t,\xi )$, and its partial Fourier transform in t by ${\widehat{u}}^{\gamma }(\eta ,x)$.

We seek a solution to (10) in $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ and assume that the following generalized derivatives exist in ${ℝ}_{+}^{n+1}$:

$$D_x^{\beta }{D}_t^{l}u(t,x)\in L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right),{\varkappa }_q\le \beta \alpha \le \varkappa .$$

We prove the following theorems.

Theorem 1. 

For every function $u(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}^{n+1}\right)$, such that

$$D_x^{\beta }{D}_t^{l}u(t,x)\in L_{2,\gamma }\left({ℝ}^{n+1}\right),{\varkappa }_q\le \beta \alpha \le \varkappa ,$$

we have the estimate

$$\gamma \parallel ({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q}){\left(\gamma +\left|\eta \right|+{\langle \xi \rangle }^{{\alpha }_0}\right)}^{l-1}{\tilde{u}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel$$

$$\hspace{1em} \le c\parallel \mathcal{L}(D_t,D_x)u(t,x),L_{2,\gamma }\left({ℝ}^{n+1}\right)\parallel ,$$

(11)

where the constant $c>0$ does not depend on $u(t,x)$.

Theorem 2. 

Let $\left|\alpha \right|/2>{\varkappa }_q>0$ and

$$f(t,x)\in W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right),s=({\alpha }_0/{\alpha }_1,\dots ,{\alpha }_0/{\alpha }_n),\gamma >0.$$

Then, the Cauchy problem (10) is uniquely solvable in $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ and the solution satisfies the estimate

$$\parallel u(t,x),W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)\parallel +\sum _{{\varkappa }_q\le \beta \alpha \le \varkappa }\parallel D_x^{\beta }{D}_t^{l}u(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$\hspace{1em}\hspace{1em} \le c\parallel f(t,x),W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\parallel ,$$

(12)

where the constant $c>0$ does not depend on $f(t,x)$.

Remark 1. 

The estimate (11) is the core in the proof of the uniqueness of the solution to the Cauchy problem under consideration in the spaces $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$. An estimate of such type is called the energy inequality for strictly hyperbolic operators [20,21].

Remark 2. 

Note that in the case of ${\varkappa }_q=0$, the theorem of the unique solvability of (10) was proven in [18].

Example 1. 

We consider the Cauchy problem for the pseudohyperbolic equation

$$\begin{array}{c}({\alpha }_{1}I-{\alpha }_2\mathsf{\Delta }+{\mathsf{\Delta }}^2)D_t^{2}u-{\mathsf{\Delta }}^{3}u=f(t,x),\hfill \\ \\ {u|}_{t=0}=0,D_t{u|}_{t=0}=0,\hfill \end{array}$$

(13)

where ${\alpha }_1\ge 0$, ${\alpha }_2\ge 0$. It follows from [1,17] that the Cauchy problem (13) with ${\alpha }_1={\alpha }_2=0$ is uniquely solvable in $W_{2,\gamma }^{2,\mathbf{6}}\left({ℝ}_{+}^{n+1}\right)$, $n\ge 9$, for every

$$f(t,x)\in W_{2,\gamma }^{0,\mathbf{1}}\left({ℝ}_{+}^{n+1}\right)\cap L_{2,\gamma }({ℝ}_{+};L_1\left({ℝ}^n\right)),\gamma >0.$$

It follows from [18] that the Cauchy problem (13) with ${\alpha }_1>0$, ${\alpha }_2\ge 0$ is uniquely solvable in $W_{2,\gamma }^{2,\mathbf{6}}\left({ℝ}_{+}^{n+1}\right)$ for arbitrary n and under less restrictions on the right-hand side $f(t,x)$:

$$f(t,x)\in W_{2,\gamma }^{0,\mathbf{1}}\left({ℝ}_{+}^{n+1}\right),\gamma >0.$$

However, in the case of ${\alpha }_1=0$, ${\alpha }_2>0$, there exists a unique solution to the Cauchy problem (13) in $W_{2,\gamma }^{2,\mathbf{6}}\left({ℝ}_{+}^{n+1}\right)$, $n\ge 5$, for every

$$f(t,x)\in W_{2,\gamma }^{0,\mathbf{1}}\left({ℝ}_{+}^{n+1}\right)\cap L_{2,\gamma }({ℝ}_{+};L_1\left({ℝ}^n\right)),\gamma >0.$$

Theorem 2 gives us a result on the unique solvability of the Cauchy problem (13) with ${\alpha }_1=0$, ${\alpha }_2>0$ in $W_{2,\gamma ,{\varkappa }_q,1}^{2,\mathbf{6}}\left({ℝ}_{+}^{n+1}\right)$, $\gamma >0$, ${\varkappa }_q=1/3$, $n\ge 5$, under less restrictions on the right-hand side $f(t,x)$. Namely, for solvability, it suffices to require that

$$f(t,x)\in W_{2,\gamma }^{0,\mathbf{1}}\left({ℝ}_{+}^{n+1}\right),\gamma >0.$$

3. Uniqueness of the Solution to the Cauchy Problem

As is known, the uniqueness of the solution to the Cauchy problem for hyperbolic equations follows from energy estimates [20,21]. Using an analog of such energy estimates, the uniqueness of the solution to the Cauchy problem for strictly pseudohyperbolic equations in $W_{2,\gamma }^{l,r}\left({ℝ}_{+}^{n+1}\right)$, $\gamma >0$, was proven in [1,17,18]. Let us show that one can prove the uniqueness of the solution to the Cauchy problem in $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$, $\left|\alpha \right|/2>{\varkappa }_q$, in a similar way.

Taking into account (5), (8), (9), the following lemma was proven in [18].

Lemma 1. 

There is a constant $c>0$ such that

$$c\gamma ({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q})\left(\right|\eta |+\gamma +{\langle \xi \rangle }^{{\alpha }_0}{)}^{l-1}\le \left|\mathcal{L}(i\eta +\gamma ,i\xi )\right|$$

(14)

for all $\gamma \ge 0$ and $(\eta ,\xi )\in {ℝ}^{n+1}$.

Using Lemma 1, we prove Theorem 1. Due to Parseval’s equality, we have

$$\parallel \mathcal{L}(D_t,D_x)u(t,x),L_{2,\gamma }\left({ℝ}^{n+1}\right)\parallel = \parallel \mathcal{L}(D_t+\gamma ,D_x)u_{\gamma }(t,x),L_2\left({ℝ}^{n+1}\right)\parallel$$

$$= \parallel (\mathcal{L}(i\eta +\gamma ,i\xi ){\tilde{u}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel .$$

Hence, (11) directly follows from (14). Theorem 1 is proven.

Note that, using Theorem 1, we can establish the uniqueness of the solution to the Cauchy problem (10) in $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$, $\gamma >0$, for $\left|\alpha \right|/2>{\varkappa }_q$. Indeed, if $u(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ is a solution to the Cauchy problem with $f(t,x)=0$, then, extending it by zero on $t<0$, we obtain the function $\overline{u}(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}^{n+1}\right)$ satisfying the conditions of Theorem 1. Therefore, (11) yields

$$\parallel ({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q}){\left(\left|\eta \right|+\gamma +{\langle \xi \rangle }^{{\alpha }_0}\right)}^{l-1}{\tilde{\overline{u}}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel =0.$$

Then,

$$\parallel ({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q}){\tilde{\overline{u}}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel =0.$$

Taking into account (5), by Parseval’s equality, we obtain

$$(L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))u_{\gamma }=0,t\in {ℝ}_{+},x\in {ℝ}^n.$$

(15)

By Condition 1, the symbol $L_0\left(i\xi \right)$ is homogeneous with respect to the vector ${\alpha }^{\prime }=\alpha /\varkappa$. Since

$$c_1{\langle \xi \rangle }^{\varkappa }\le {\langle \xi \rangle }^{\prime }\le c_2{\langle \xi \rangle }^{\varkappa },{\langle \xi \rangle }^{\prime }=\sum _{i=1}^n{\xi }^{2/{\alpha }_i^{\prime }},$$

with constants $c_j>0$ independent of $\xi \in {ℝ}^n$, then, from (5), we have

$$\begin{array}{c}{c}_1({\langle \xi \rangle }^{\prime }+{\langle \xi \rangle }^{\prime q^{\prime }})\le |L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)|\le c_2({\langle \xi \rangle }^{\prime }+{\langle \xi \rangle }^{\prime q^{\prime }}),\\ \\ q^{\prime }={\varkappa }_q/\varkappa ,0<q^{\prime }<1.\end{array}$$

(16)

By the conditions of Theorem 2, $\left|\alpha \right|/2>{\varkappa }_q$, then $|{\alpha }^{\prime }|/2>q^{\prime }$. Consequently, by Theorem 2 from [22], the quasielliptic equation with lower order terms $(L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))v=0$, $x\in {ℝ}^n$, has only the zero solution from the Sobolev space with the power weight

$$W_{2,q^{\prime },{\sigma }^{\prime }}^{r^{\prime }}\left({ℝ}^n\right),r^{\prime }=(1/{\alpha }_1^{\prime },\dots ,1/{\alpha }_n^{\prime }),{\alpha }_j^{\prime }={\alpha }_j/\varkappa ,q^{\prime }={\varkappa }_q/\varkappa ,{\sigma }^{\prime }=1,$$

$|{\alpha }^{\prime }|/2>q^{\prime }$, the norm in which is defined as follows

$$\parallel v\left(x\right),W_{2,q^{\prime },{\sigma }^{\prime }}^{r^{\prime }}\left({ℝ}^n\right)\parallel =\sum _{0\le \beta {\alpha }^{\prime }\le q^{\prime }}\parallel {(1+{\langle x\rangle }^{\prime })}^{-{\sigma }^{\prime }(q^{\prime }-\beta {\alpha }^{\prime })}{D}_x^{\beta }v\left(x\right),L_2\left({ℝ}^n\right)\parallel$$

$$+\sum _{q^{\prime }<\beta {\alpha }^{\prime }\le 1}\parallel D_x^{\beta }v\left(x\right),L_2\left({ℝ}^n\right)\parallel ,$$

where

$${\langle x\rangle }^{\prime }={\left(\sum _{i=1}^n{x}_i^{2/{\alpha }_i^{\prime }}\right)}^{1/2}.$$

Since

$$c_1{\langle x\rangle }^{\varkappa }\le {\langle x\rangle }^{\prime }\le c_2{\langle x\rangle }^{\varkappa }$$

(17)

with constants $c_j>0$ independent of $x\in {ℝ}^n$, then the belonging of the solution to the Cauchy problem (10) to $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$, $r=(1/{\alpha }_1,\dots ,1/{\alpha }_n)$, is equivalent to its belonging to $W_{2,\gamma ,q^{\prime },1}^{l,r^{\prime }}\left({ℝ}_{+}^{n+1}\right)$, $r^{\prime }=\varkappa r$, $q^{\prime }={\varkappa }_q/\varkappa$. Hence, $u_{\gamma }(t,x)\in W_{2,q^{\prime },1}^{r^{\prime }}\left({ℝ}^n\right)$ for $t>0$ and (15) has only the solution $u_{\gamma }(t,x)=0$, $t>0$, $x\in {ℝ}^n$. Thus, if the solution $u(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ to (10) exists, then it is uniquely determined.

In the next three sections, we prove that, under the conditions of Theorem 2, the Cauchy problem has a solution in $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$.

4. Construction of Approximate Solutions to the Cauchy Problem

In this section, following [1,18], we give formulas for approximate solutions to the Cauchy problem (10). Let

$$f(t,x)\in W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right),s=({\alpha }_0/{\alpha }_1,\dots ,{\alpha }_0/{\alpha }_n),\gamma >0.$$

We consider the Cauchy problem for an ordinary differential equation with a real parameter $\xi$, which is obtained by formally applying the Fourier transform in x to the problem (10)

$$\begin{array}{c}\mathcal{L}(D_t,i\xi )v=\widehat{f}(t,\xi ),t>0,\hfill \\ \\ D_t^k{v|}_{t=0}=0.\hfill \end{array}$$

(18)

Since ${\varkappa }_q>0$, the coefficient $L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)$ at the highest derivative has a singularity at $\xi =0$. We study the problem (18) for $\xi \in {ℝ}^n\setminus \left\{0\right\}$.

The solution to this problem can be represented as

$$v(t,\xi )=\underset{0}{\overset{t}{\int }}J(t-\tau ,\xi )\widehat{f}(\tau ,\xi )d\tau ,$$

(19)

where

$$J(t,\xi )=\frac{1}{2\pi }\underset{\Gamma \left(\xi \right)}{\int }\frac{e^{it\lambda }}{\mathcal{L}(i\lambda ,i\xi )}d\lambda ,$$

(20)

$\Gamma \left(\xi \right)$ is a contour in the complex plane surrounding all the roots of Equation (7).

Note that the integral $J(t,\xi )$ is a solution to the following Cauchy problem:

$$\begin{array}{c}\mathcal{L}(D_t,i\xi )J=0,t>0,\hfill \\ \\ D_t^k{J|}_{t=0}=0,k=0,\dots ,l-2,\hfill \\ \\ D_t^{l-1}{J|}_{t=0}={(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right))}^{-1}.\hfill \end{array}$$

(21)

Since the roots ${\eta }_k\left(\xi \right)$ of (7) are real and distinct, then the following lemmas hold (see [1,18]).

Lemma 2. 

The representation holds

$$J(t,\xi )=i^{1-l}{(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right))}^{-1}\sum _{k=1}^l{a}_k\left(\xi \right)e^{it{\eta }_k\left(\xi \right)},\xi \in {ℝ}^n\setminus \left\{0\right\},$$

where $a_1\left(\xi \right)=1$ for $l=1$,

$$a_k\left(\xi \right)=\prod _{j\ne k}{({\eta }_k\left(\xi \right)-{\eta }_j\left(\xi \right))}^{-1},k=1,\dots ,l,forl>1.$$

Proof. 

The proof of the lemma follows directly from (20) since

$$J(t,\xi )=\frac{1}{2\pi }\underset{\Gamma \left(\xi \right)}{\int }\frac{e^{it\lambda }}{(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)){\displaystyle \prod _{j=1}^l}i(\lambda -{\eta }_j\left(\xi \right))}d\lambda .$$

Taking into account that the roots ${\eta }_1\left(\xi \right),\dots ,{\eta }_l\left(\xi \right)$ are different and using the residue theorem, we obtain the required representation. □

Lemma 3. 

The estimate holds

$$\left|J(t,\xi )\right|\le |L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right){|}^{-1}\frac{t^{l-1}}{(l-1)!},t\ge 0,\xi \in {ℝ}^n\setminus \left\{0\right\}.$$

Proof. 

Let ${\eta }_1\left(\xi \right),\dots ,{\eta }_l\left(\xi \right)$ be the roots of (7) for $\xi \in {ℝ}^n\setminus \left\{0\right\}$. We introduce the functions

$${\psi }_1(t,\xi )=e^{i{\eta }_1\left(\xi \right)t},{\psi }_j(t,\xi )=\underset{0}{\overset{t}{\int }}{e}^{i{\eta }_j\left(\xi \right)(t-s)}{\psi }_{j-1}(s,\xi )ds,j=2,\dots ,l.$$

(22)

It is easy to verify that ${\psi }_l(t,\xi )$ is a solution to the Cauchy problem

$$\begin{array}{cc}\mathcal{L}(D_t,i\xi )\psi =0,\hfill & t>0,\hfill \\ \\ D_t^k{\psi |}_{t=0}=0,\hfill & k=0,\dots ,l-2,\hfill \\ \\ D_t^{l-1}{\psi |}_{t=0}=1.\hfill \end{array}$$

Comparing it with (21), due to the uniqueness of the solution, we obtain the identity

$$J(t,\xi )\equiv {(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right))}^{-1}{\psi }_l(t,\xi ).$$

Since

$$\left|J(t,\xi )\right|\le |{(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right))}^{-1}\left|\right|{\psi }_l(t,\xi )|,$$

(23)

to prove the lemma, it suffices to estimate $|{\psi }_l(t,\xi )|$ for $t\ge 0$.

Taking into account the recurrence relations (22) and the realness of the roots ${\eta }_k\left(\xi \right)$, we have

$$|{\psi }_1(t,\xi )|\equiv 1,|{\psi }_2(t,\xi )|\le \underset{0}{\overset{t}{\int }}\left|{\psi }_1(s,\xi )\right|ds=t,$$

$$|{\psi }_3(t,\xi )|\le \underset{0}{\overset{t}{\int }}|{\psi }_2(s,\xi )|ds=\frac{t^2}{2},\dots ,\left|{\psi }_l(t,\xi )\right|\le \frac{t^{l-1}}{(l-1)!}.$$

Consequently, the required inequality follows from (23). □

Lemma 4. 

The identities are valid

$$\underset{0}{\overset{\infty }{\int }}{e}^{-(i\eta +\gamma )t}{D}_t^{k}J(t,\xi )dt\equiv {(i\eta +\gamma )}^k{\left(\mathcal{L}(i\eta +\gamma ,i\xi )\right)}^{-1},k=0,\dots ,l-1,$$

(24)

$$\underset{0}{\overset{\infty }{\int }}{e}^{-(i\eta +\gamma )t}{D}_t^{l}J(t,\xi )dt\equiv {(i\eta +\gamma )}^l{\left(\mathcal{L}(i\eta +\gamma ,i\xi )\right)}^{-1}-{(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right))}^{-1}$$

(25)

for $\gamma >0$, $\xi \in {ℝ}^n\setminus \left\{0\right\}$.

Proof. 

The proof of the lemma follows from (20), (21).

At first, construct a solution to the Cauchy problem (10). Applying the inverse Fourier operator in $\xi$ to $v(t,\xi )$ in (19), we can obtain a formal solution to the problem (10). However, Lemma 2 implies that the function defined by (20) increases unboundedly as $\left|\xi \right|\to 0$ and $v(t,\xi )$ can have a nonintegrable singularity at $\xi =0$. Hence, to obtain a formula for a solution to (10), it is necessary to apply some regularization of the inverse Fourier operator. To this end, we consider the sequence of the functions $\left\{v^m(t,\xi )\right\}$, where

$$v^m(t,\xi )={\chi }^m\left(\xi \right)v(t,\xi ),$$

$${\chi }^m\left(\xi \right)=\left\{\begin{array}{cc}1,\hfill & \langle \xi \rangle >1/m,\hfill \\ \\ 0,\hfill & \langle \xi \rangle \le 1/m.\hfill \end{array}\right.$$

By (19) and (20), $v^m(t,\xi )$ have no singularities at $\xi =0$. Since $f(t,x)\in W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)$, then the inverse Fourier transform operator $F^{-1}$ in $\xi$ is applicable to $v^m(t,\xi )$ and we can define the sequence of the functions $\left\{u^m(t,x)\right\}$, where

$$u^m(t,x)=F^{-1}\left[v^m\right](t,x).$$

Further, we show that the functions $u^m(t,x)$ for $m\gg 1$ can be considered as approximate solutions to the problem (10). □

5. Estimates of Approximate Solutions to the Cauchy Problem

In this section, we estimate $\left\{u^m(t,x)\right\}$ in the norm of $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ and prove that this sequence is fundamental.

Lemma 5. 

Let $s=({\alpha }_0/{\alpha }_1,\dots ,{\alpha }_0/{\alpha }_n)$. Then,

$$\sum _{{\varkappa }_q\le \beta \alpha \le 1}\parallel D_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le c\parallel f(t,x),W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\parallel$$

with a constant $c>0$ independent of m and $f(t,x)$; moreover, for every $k\ge 1$, we have

$$\sum _{{\varkappa }_q\le \beta \alpha \le 1}\parallel D_x^{\beta }{u}^{m+k}(t,x)-D_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0$$

as $m\to \infty$.

Proof. 

By Parseval’s equality, we have

$$\parallel D_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel = \parallel {\left(i\xi \right)}^{\beta }{\chi }^m\left(\xi \right)v(t,\xi ),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$${= \parallel \left(i\xi \right)}^{\beta }{\chi }^m\left(\xi \right)\underset{0}{\overset{t}{\int }}J(t-\tau ,\xi )\widehat{f}(\tau ,\xi )d\tau ,L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel .$$

We extend the function $\widehat{f}(\tau ,\xi )$ by zero for $t<0$, keeping the notation. Then, taking into account (24), we obtain

$$\parallel D_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel = \parallel {\left(i\xi \right)}^{\beta }{\chi }^m\left(\xi \right){\left(\mathcal{L}(i\eta +\gamma ,i\xi )\right)}^{-1}{\tilde{f}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel .$$

By (14), from the equality, it follows that

$$\parallel D_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le \frac{1}{c\gamma }\parallel \frac{\left(\right|\eta |+\gamma +{\langle \xi \rangle }^{{\alpha }_0}{)}^{1-l}}{({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q})}{\left(i\xi \right)}^{\beta }{\chi }^m\left(\xi \right){\tilde{f}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel .$$

Since ${\varkappa }_q\le \beta \alpha \le 1$, we obtain the required estimate. Convergence is proven in the same way.

The lemma is proven. □

Lemma 6. 

Let $s=({\alpha }_0/{\alpha }_1,\dots ,{\alpha }_0/{\alpha }_n)$. Then,

$$\sum _{{\varkappa }_q\le \beta \alpha \le 1,{\beta }_0/l+\beta \alpha \le 1}\parallel D_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le c\parallel f(t,x),W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\parallel$$

with a constant $c>0$ independent of m and $f(t,x)$; moreover, for every $k\ge 1$, we have

$$\sum _{{\varkappa }_q\le \beta \alpha \le 1,{\beta }_0/l+\beta \alpha \le 1}\parallel D_t^{{\beta }_0}{D}_x^{\beta }{u}^{m+k}(t,x)-D_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0$$

as $m\to \infty$.

Proof. 

We assume that $f(t,x)$ is extended by zero on $t<0$.

Using Parseval’s equality and taking into account the properties of the function $J(t,\xi )$ from (20), and also that ${\beta }_0<l$, we have

$$\parallel D_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel = \parallel {\left(i\xi \right)}^{\beta }{\chi }^m\left(\xi \right)D_t^{{\beta }_0}v(t,\xi ),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$${= \parallel \left(i\xi \right)}^{\beta }{\chi }^m\left(\xi \right)\underset{0}{\overset{t}{\int }}{D}_t^{{\beta }_0}J(t-\tau ,\xi )\widehat{f}(\tau ,\xi )d\tau ,L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel .$$

Using the formula of the Fourier transform of convolution and Lemma 4, we obtain

$$\parallel D_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel = \parallel {\left(i\xi \right)}^{\beta }{\chi }^m\left(\xi \right){(i\eta +\gamma )}^{{\beta }_0}{\left(\mathcal{L}(i\eta +\gamma ,i\xi )\right)}^{-1}$$

$$\times {\tilde{f}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel .$$

By Lemma 1, the inequality holds

$$\begin{array}{c}{\displaystyle \parallel D_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le \frac{c_1}{\gamma }\parallel \frac{\left(\right|\eta |+\gamma +{\langle \xi \rangle }^{{\alpha }_0}{)}^{1-l}}{({\langle \xi \rangle }^{\varkappa }+{\langle \xi \rangle }^{{\varkappa }_q})}}\hfill \\ \\ \times {\left(i\xi \right)}^{\beta }{(i\eta +\gamma )}^{{\beta }_0}{\tilde{f}}_{\gamma }(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel ,\hfill \end{array}$$

(26)

where ${\varkappa }_q\le \beta \alpha \le 1,{\beta }_0/l+\beta \alpha \le 1$. Taking into account Parseval’s equality, we have

$$\parallel D_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le \frac{c_2}{{\gamma }^{l-1-{\beta }_0}}\parallel f(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

for $\beta \alpha ={\varkappa }_q$ and

$$\parallel D_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le \frac{c_3}{\gamma }\parallel f(t,x),W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\parallel$$

for $\beta \alpha =1$. By (26), we obtain similar estimates for ${\varkappa }_q<\beta \alpha <1$.

Convergence can be proven in the same way.

The lemma is proven. □

Lemma 7. 

Let $\left|\alpha \right|/2>{\varkappa }_q>0$. Then,

$$\sum _{\beta \alpha <{\varkappa }_q}\parallel {(1+\langle x\rangle )}^{-({\varkappa }_q-\beta \alpha )}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$\le \frac{c}{{\gamma }^{l-1}}\parallel f(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

(27)

with a constant $c>0$ independent of m and $f(t,x)$; moreover, for every $k\ge 1$, we have

$$\sum _{\beta \alpha <{\varkappa }_q}\parallel {(1+\langle x\rangle )}^{-({\varkappa }_q-\beta \alpha )}(D_x^{\beta }{u}^{m+k}(t,x)-D_x^{\beta }{u}^m(t,x)),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0$$

(28)

as $m\to \infty$.

Proof. 

Note that by Condition 1, the quasielliptic operator $L_0\left(D_x\right)$ is homogeneous with respect to the vector ${\alpha }^{\prime }=({\alpha }_1/\varkappa ,\dots ,{\alpha }_n/\varkappa )$, and the symbol of the differential operator $L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right)$ satisfies the inequality (16). By the lemma condition, $\left|\alpha \right|/2>{\varkappa }_q$, i.e., $|{\alpha }^{\prime }|/2>q^{\prime }$. It follows from [22] that the quasielliptic operator

$$((L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right)):W_{2,q^{\prime },1}^{r^{\prime }}\left({ℝ}^n\right)\to L_2\left({ℝ}^n\right),r^{\prime }=\left(\varkappa /{\alpha }_1,\dots ,\varkappa /{\alpha }_n\right),$$

establishes an isomorphism.

Therefore, for every $w\left(x\right)\in W_{2,q^{\prime },1}^{r^{\prime }}\left({ℝ}^n\right)$, the estimate holds

$$\parallel w\left(x\right),W_{2,q^{\prime },1}^{r^{\prime }}\left({ℝ}^n\right)\parallel \le c\parallel (L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))w\left(x\right),L_2\left({ℝ}^n\right)\parallel$$

with a constant $c>0$ independent of $w\left(x\right)$. Taking into account the definition of the norm of the weighted Sobolev space, we have

$$\sum _{0\le \beta {\alpha }^{\prime }<q^{\prime }}\parallel {(1+{\langle x\rangle }^{\prime })}^{-(q^{\prime }-\beta {\alpha }^{\prime })}{D}_x^{\beta }w\left(x\right),L_2\left({ℝ}^n\right)\parallel$$

$$\le c\parallel (L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))w\left(x\right),L_2\left({ℝ}^n\right)\parallel .$$

By (17), since $q^{\prime }={\varkappa }_q/\varkappa$, ${\alpha }^{\prime }=\alpha /\varkappa$, we obtain

$$\sum _{0\le \beta \alpha <{\varkappa }_q}\parallel {(1+\langle x\rangle )}^{-({\varkappa }_q-\beta \alpha )}{D}_x^{\beta }w\left(x\right),L_2\left({ℝ}^n\right)\parallel$$

$$\le c_1\parallel (L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))w\left(x\right),L_2\left({ℝ}^n\right)\parallel .$$

(29)

We take

$$w\left(x\right)=u^m(t,x)$$

(30)

for arbitrary fixed $t>0$. Substituting $w\left(x\right)$ defined by (30) to (29), we have

$$\sum _{0\le \beta \alpha <{\varkappa }_q}\parallel {(1+\langle x\rangle )}^{-({\varkappa }_q-\beta \alpha )}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$\le c_1\parallel (L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))u^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$\le c_2\sum _{{\varkappa }_q\le \nu \alpha \le \varkappa }\parallel D_x^{\nu }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel .$$

This implies the required inequality (27). The convergence (28) can be proven in a similar way.

The lemma is proven. □

Lemma 8. 

Let $\left|\alpha \right|/2>{\varkappa }_q>0$. Then,

$$\sum _{0\le \beta \alpha <{\varkappa }_q,{\beta }_0/l+\beta \alpha \le 1}\parallel {(1+\langle x\rangle )}^{-({\varkappa }_q-\beta \alpha )}{D}_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$\le \frac{c}{{\gamma }^{l-1}}\parallel f(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

with a constant $c>0$ independent of m and $f(t,x)$; moreover, for every $k\ge 1$, we have

$$\sum _{0\le \beta \alpha <{\varkappa }_q,{\beta }_0/l+\beta \alpha \le 1}\parallel {(1+\langle x\rangle )}^{-({\varkappa }_q-\beta \alpha )}(D_t^{{\beta }_0}{D}_x^{\beta }{u}^{m+k}(t,x)$$

$$-D_t^{{\beta }_0}{D}_x^{\beta }{u}^m(t,x)),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0$$

as $m\to \infty$.

The proof of the lemma is carried out according to the scheme of the proof of the previous lemma.

Lemma 9. 

Let $s=({\alpha }_0/{\alpha }_1,\dots ,{\alpha }_0/{\alpha }_n)$. Then,

$$\sum _{{\varkappa }_q\le \beta \alpha \le \varkappa }\parallel D_t^l{D}_x^{\beta }{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le c\left(\gamma \right)\parallel f(t,x),W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\parallel$$

(31)

with a constant $c\left(\gamma \right)>0$ independent of m and $f(t,x)$; moreover, for every $k\ge 1$, we have

$$\sum _{{\varkappa }_q\le \beta \alpha \le \varkappa }\parallel (D_t^l{D}_x^{\beta }{u}^{m+k}(t,x)-D_t^l{D}_x^{\beta }{u}^m(t,x)),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0$$

(32)

as $m\to \infty$.

Proof. 

For the function $v(t,\xi )$ defined in (19), by (21), the following equality is valid:

$$D_t^{l}v(t,\xi )=\frac{\widehat{f}(t,\xi )}{L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)}+\underset{0}{\overset{t}{\int }}{D}_t^{l}J(t-\tau ,\xi )\widehat{f}(\tau ,\xi )d\tau .$$

Using Parseval’s equality and the Heaviside function $\theta \left(t\right)$, we conclude that

$$\parallel D_x^{\beta }{D}_t^l{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel = \parallel {\chi }^m\left(\xi \right){\left(i\xi \right)}^{\beta }{D}_t^{l}v(t,\xi ),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$= \parallel {\chi }^m\left(\xi \right){\left(i\xi \right)}^{\beta }\frac{\widehat{f_{\gamma }}(t,\xi )\theta \left(t\right)}{L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)}$$

$$+{\chi }^m\left(\xi \right){\left(i\xi \right)}^{\beta }\underset{-\infty }{\overset{\infty }{\int }}{e}^{-\gamma (t-\tau )}{D}_t^{l}J(t-\tau ,\xi )\theta (t-\tau )\widehat{f_{\gamma }}(\tau ,\xi )\theta \left(\tau \right)d\tau ,L_2\left({ℝ}^{n+1}\right)\parallel .$$

By the property of the Fourier transform of convolution, we have

$$\parallel D_x^{\beta }{D}_t^l{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel = \parallel {\chi }^m\left(\xi \right){\left(i\xi \right)}^{\beta }\frac{\widetilde{f_{\gamma }}(\eta ,\xi )}{L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right)}$$

$$+{\chi }^m\left(\xi \right){\left(i\xi \right)}^{\beta }\left(\underset{0}{\overset{\infty }{\int }}{e}^{-(i\eta +\gamma )t}{D}_t^{l}J(t,\xi )dt\right)\widetilde{f_{\gamma }}(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel .$$

Using the representation (25), we obtain

$$\parallel D_x^{\beta }{D}_t^l{u}^m(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$= \parallel {\chi }^m\left(\xi \right){\left(i\xi \right)}^{\beta }{(i\eta +\gamma )}^l{\left(\mathcal{L}(i\eta +\gamma ,i\xi )\right)}^{-1}\widetilde{f_{\gamma }}(\eta ,\xi ),L_2\left({ℝ}^{n+1}\right)\parallel .$$

The identity

$${(i\eta +\gamma )}^l{\left(\mathcal{L}(i\eta +\gamma ,i\xi )\right)}^{-1}\equiv {(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right))}^{-1}-\sum _{k=0}^{l-1}\frac{(L_{l-k}\left(i\xi \right)+L_{l-k}^{\prime }\left(i\xi \right)){(i\eta +\gamma )}^k}{(L_0\left(i\xi \right)+L_0^{\prime }\left(i\xi \right))\mathcal{L}(i\eta +\gamma ,i\xi )},$$

estimates (5), (6), and Lemma 1 ensure the inequality (31).

Similar arguments imply (32).

The lemma is proven. □

Lemma 10. 

For every function

$$u(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}^{n+1}\right)$$

such that

$$D_x^{\beta }{D}_t^{l}u(t,x)\in L_{2,\gamma }\left({ℝ}^{n+1}\right),\beta \alpha =\varkappa or\beta \alpha ={\varkappa }_q,$$

we have

$$D_x^{\beta }{D}_t^{k}u(t,x)\in L_{2,\gamma }\left({ℝ}^{n+1}\right),{\varkappa }_q\le \beta \alpha \le 1-k{\alpha }_0,k=1,\dots ,l-1.$$

Proof. 

Since $u(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}^{n+1}\right)$, then

$$\langle \xi \rangle {\tilde{u}}_{\gamma }(\tau ,\xi )\in L_2\left({ℝ}^{n+1}\right),$$

(33)

$${\langle \xi \rangle }^{{\varkappa }_q}{\tilde{u}}_{\gamma }(\tau ,\xi )\in L_2\left({ℝ}^{n+1}\right).$$

(34)

By the condition of the lemma, we also have

$${\left|\tau \right|}^l{\langle \xi \rangle }^{\varkappa }{\tilde{u}}_{\gamma }(\tau ,\xi )\in L_2\left({ℝ}^{n+1}\right),$$

(35)

$${\left|\tau \right|}^l{\langle \xi \rangle }^{{\varkappa }_q}{\tilde{u}}_{\gamma }(\tau ,\xi )\in L_2\left({ℝ}^{n+1}\right).$$

(36)

Using Young’s inequality,

$$uv\le \frac{u^p}{p}+\frac{v^{p^{\prime }}}{p^{\prime }},u,v>0,1/p+1/p^{\prime }=1,$$

(37)

when $p=l/k$, $p^{\prime }=l/(l-k)$, we have

$${\left|\tau \right|}^k{\langle \xi \rangle }^{1-k{\alpha }_0}={\langle \xi \rangle }^{\varkappa }\left[{\left|\tau \right|}^k{\langle \xi \rangle }^{(l-k){\alpha }_0}\right]\le {\langle \xi \rangle }^{\varkappa }\left[\frac{k}{l}{\left|\tau \right|}^l+\frac{l-k}{l}{\langle \xi \rangle }^{l{\alpha }_0}\right]$$

(38)

and

$${\left|\tau \right|}^k{\langle \xi \rangle }^{{\varkappa }_q}\le {\langle \xi \rangle }^{{\varkappa }_q}\left[\frac{k}{l}{\left|\tau \right|}^l+\frac{l-k}{l}\right].$$

(39)

Taking into account (33), (35), and the notation $\varkappa =1-l{\alpha }_0$, from (38), it follows that

$$D_x^{\beta }{D}_t^{k}u(t,x)\in L_{2,\gamma }\left({ℝ}^{n+1}\right),\beta \alpha =1-k{\alpha }_0,k=1,\dots ,l-1.$$

Using (34), (36), from (39), we have

$$D_x^{\beta }{D}_t^{k}u(t,x)\in L_{2,\gamma }\left({ℝ}^{n+1}\right),\beta \alpha ={\varkappa }_q,k=1,\dots ,l-1.$$

Let us show that

$$D_x^{\beta }{D}_t^{k}u(t,x)\in L_{2,\gamma }\left({ℝ}^{n+1}\right),\beta \alpha ={\epsilon }_k,{\varkappa }_q<{\epsilon }_k<1-k{\alpha }_0,k=1,\dots ,l-1.$$

Rewriting ${\epsilon }_k$ in the form

$${\epsilon }_k=(1-\lambda ){\varkappa }_q+\lambda (1-k{\alpha }_0),0<\lambda <1,$$

and using Young’s inequality (37) with $p=1/(1-\lambda )$, $p^{\prime }=1/\lambda$, we have

$${\left|\tau \right|}^k{\langle \xi \rangle }^{{\epsilon }_k}={\left|\tau \right|}^k{\langle \xi \rangle }^{(1-\lambda ){\varkappa }_q}{\langle \xi \rangle }^{\lambda (1-k{\alpha }_0)}\le {\left|\tau \right|}^k\left[(1-\lambda ){\langle \xi \rangle }^{{\varkappa }_q}+\lambda {\langle \xi \rangle }^{(1-k{\alpha }_0)}\right].$$

(40)

Taking into account (33)–(36), from (38)–(40), we obtain the required result. □

Lemma 11. 

For every function

$$u(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$$

such that $D_t^k{u|}_{t=0}=0$, $k=0,\dots ,l-1$,

$$D_x^{\beta }{D}_t^{l}u(t,x)\in L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right),\beta \alpha =\varkappa or\beta \alpha ={\varkappa }_q,$$

we have

$$D_x^{\beta }{D}_t^{k}u(t,x)\in L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right),{\varkappa }_q\le \beta \alpha \le 1-k{\alpha }_0,k=1,\dots ,l-1.$$

The proof follows directly from Lemma 10.

In the next section, relying on Lemmas 5–11, we prove that the sequence $\left\{u^m(t,x)\right\}$ is convergent in $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$, and the limit function $u(t,x)$ is a solution to the Cauchy problem (10) and satisfies (12).

6. Solvability of the Cauchy Problem

As noted above, the uniqueness of the solution to the Cauchy problem (10) in the space $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ follows from the energy inequality when $\left|\alpha \right|/2>{\varkappa }_q$. Let us show the existence of a solution under the conditions specified in Theorem 2.

The proof of the solvability of the Cauchy problem in $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ is carried out in accordance with the scheme described in [1,17]. Note that, in contrast to [1,17], we study the solvability of the problem in wider weighted spaces and for equations containing lower order terms. Therefore, we use stronger estimates for approximate solutions established in the previous sections.

From Lemmas 5–8, it follows that the sequence of the functions $\left\{u^m(t,x)\right\}$ is fundamental in the weighted Sobolev space $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ and

$$\parallel u^m(t,x),W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)\parallel \le c\left(\gamma \right)\parallel f(t,x),W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\parallel ,$$

where $c\left(\gamma \right)>0$ is a constant independent of m and $f(t,x)$. Since the space $W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ is complete, there exists a limit function $u(t,x)\in W_{2,\gamma ,{\varkappa }_q,1}^{l,r}\left({ℝ}_{+}^{n+1}\right)$ such that

$$D_t^{k}u{|}_{t=0}=0,k=0,\dots ,l-1,$$

and a similar estimate holds.

By Lemma 9 and the properties of generalized differentiation, the generalized derivatives

$$D_x^{\beta }{D}_t^{l}u(t,x),\beta \alpha =\varkappa or\beta \alpha ={\varkappa }_q,$$

exist; moreover,

$$\parallel D_x^{\beta }{D}_t^l{u}^m(t,x)-D_x^{\beta }{D}_t^{l}u(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0,m\to \infty ,$$

$$\parallel D_x^{\beta }{D}_t^{l}u,L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \le c\left(\gamma \right)\parallel f(t,x),W_{2,\gamma }^{0,s}\left({ℝ}_{+}^{n+1}\right)\parallel .$$

According to Lemma 11, if $u(t,x)$ is such that

$$(L_0\left(D_x\right)+L_0^{\prime }\left(D_x\right))D_t^{l}u(t,x)\in L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right),$$

then we have

$$(L_{l-k}\left(D_x\right)+L_{l-k}^{\prime }\left(D_x\right))D_t^{k}u(t,x)\in L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)$$

for $k=1,\dots ,l-1$. Consequently, taking into account the construction of the functions $u^m(t,x)$, we obtain

$$\parallel \mathcal{L}(D_t,D_x)u^m(t,x)-f(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$= \parallel {\chi }^m\left(\xi \right)\mathcal{L}(D_t,i\xi )v(t,\xi )-\widehat{f}(t,\xi ),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel$$

$$= \parallel ({\chi }^m\left(\xi \right)-1)\widehat{f}(t,\xi ),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0,m\to \infty ,$$

and

$$\parallel \mathcal{L}(D_t,D_x)u^m(t,x)-\mathcal{L}(D_t,D_x)u(t,x),L_{2,\gamma }\left({ℝ}_{+}^{n+1}\right)\parallel \to 0,m\to \infty .$$

Hence, $u(t,x)$ is the solution to the Cauchy problem (10) and (12) holds.

Theorem 2 is proven.

7. Conclusions

The Cauchy problem was studied for a class of strictly pseudohyperbolic equations with lower order terms. A new class of weighted Sobolev spaces $W_{2,\gamma ,{\varkappa }_q,\sigma }^{l,r}$ was introduced. By definition, functions from this class and their generalized derivatives belong to Lebesgue spaces with the exponential weight $e^{-\gamma t}$ and special power weights in $x\in {ℝ}^n$. In these spaces, we established new results on the unique solvability of the considered Cauchy problem under minimal conditions on the right-hand sides of the equations. Theorem 2 strengthens the well-known results of the works [1,17,18,19] in which theorems on the solvability of the Cauchy problem were established in known Sobolev spaces with an exponential weight in t. Theorem 2 can be considered as an analog of theorems on the solvability of the Cauchy problem for strictly hyperbolic equations.