Destruction of solutions for class of wave $p(x)-$bi-Laplace equation with nonlinear dissipation

1.   Introduction and formulation of the problem

Let $\Omega$ is a bounded open set in $\mathbb{R}^{n}$ with a sufficiently smooth boundary $\partial \Omega$ and outward facing unit normal $n$, let $u(x, t) = u$. The purpose of this study is to obtain sufficient conditions to prove the global nonexistence result for initial boundary value problem of wave equation containing the $p(x)$-bi-Laplacian operator

where $\mu, b$ are positive constants, the spaces $\mathcal{V}(\Omega)$ and $L^{p(x)}(\Omega)$ are defined in Definition 1 and (2.1).

This problem is a mathematical model of wave processes in mathematical physics, taking into account dissipation and the relationship between the different parameters. Recently, new strongly nonlinear dissipative wave equations of the hyperbolic type have been intensively considered in mathematical physics. It should be mentioned that many authors have studied the question of existence, uniqueness, regularity and blow-up of weak solutions for parabolic and elliptic equations involving the $p(x)$-Laplacian view of its applications in the fields of nonlinear elasticity, fluid dynamics, elastic mechanics etc, see ^{[4,6,8,12,13,15,16,17,20,21]} and the references therein.

In ^[2], the author established the existence of weak solutions for $p(x, t)$-Laplacian equation with damping term

and proved the blow-up of weak solutions with negative initial energy, where $\alpha$ is a nonnegative constant and $a, b, p, \sigma$ are given functions. Such equations are usually referred as equations with nonstandard growth conditions. It is proved the blow-up result of weak solutions with negative initial energy as well as for certain solutions with positive initial energy to the following equation

In particular case $p(x) = 2$, the problem (1.1) is reduced to the Petrovsky type equation

It is studied where, the author established an existence result and proved that the solution continues to exist globally if $m \geq r$ and blows up in finite time if $m < r$ and the initial energy is negative. Motivated by the above work, we obtain the blow-up results of solution to problem (1.1) for nonpositive initial energy. In order to state our result, we use some ideas introduced in the work of ^[7,11,14].

2.   Main results

In this section, we recall some definitions and basic properties about the generalized Sobolev and Lebesgue spaces with variable exponents. The reader is referred to ^[3,5,9,10] for more detailes.

Denote

and

Then, the mesurable function

satisfies the log-Hölder continuity condition

For some $\lambda > 0$ the variable exponent Lebesgue space $L^{p(x)}(\Omega)$ is defined as the set of mesurable functions $u: \Omega \rightarrow \mathbb{R}$ such that $\mathcal{P}_{p(.)}(\lambda u) < \infty$ with respect to the Luxemburg norm

where

The space $(L^{p(x)}(\Omega), ||.||_{p(x)})$ is separable, uniformly convex, reflexive and its dual space is $L^{q(x)}(\Omega)$ where $\frac{1}{p(x)} + \frac{1}{q(x)} = 1, { for \ all\ } x \in \Omega$.

Morever if $u \in L^{p(x)}(\Omega)$ and $v \in L^{q(x)}(\Omega)$ then $u v \in L^{s(x)}(\Omega)$ and we have the generalised Hölder's type inequality

Lemma 1. If $p$ is a mesurable function on $\Omega$ then for any $u \in L^{p(x)}(\Omega)$ we have

For any nonnegative integer $k$ the variable exponent Sobolev space is defined

endowed with the norm

Then $W^{k, p(x)}(\Omega)$ is defined as the closure of $C^{\infty} _{0}(\Omega)$ with respect to the norm $\Vert u \Vert_{W^{k, p(x)}}$. In this way $L^{p(x)}(\Omega), W^{k, p(x)}(\Omega)$ and $W^{k, p(x)}_{0}(\Omega)$ are separable and reflexive Banach spaces.

We shall frequently use the generalized Poincaré's inequality in $W^{1, p(x)}_{0}(\Omega)$ given by

Definition 1. We define the function space of our problem and its norm as follows

with the norm

Lemma 2. [18,Theorem 4.4] Let $\Omega$ is a bounded domain with Lipschitz boundary. In the space $X = W^{2, p(x)}(\Omega) \cap W^{1, p(x)}_{0}(\Omega)$ the norm $\Vert. \Vert _{X}$ and $\Vert \Delta_x. \Vert_{L^{p(x)}(\Omega)}$ are equivalent norms.

Lemma 3. [1,Theorem 5.4] Let $\Omega$ be a domain in $\mathbb{R}^{n}$ that has the cone property then for $n > p$ and $p \leq q \leq \frac{np}{n-p}$ there exist the following imbeddings

Lemma 4. [19,Lemma 2.1] Assume that $L(t)$ is is a twice continuously differentiable function satisfying

where $C_{0}, \theta > 0$, $-1 < \beta \leq 0$, $\alpha > 0$ are constants. Then $L(t)$ blows up in finite time.

3.   Blow up result

Theorem 1. Let $u$ be an energy weak solution to problem (1.1). Suppose that

Assume further that

and

then the solution $u$ blows up on the finite interval $(0, t_{max})$.

Proof. Multiplying Eq (1.1) by $\partial_{t}u$, and integration by parts over $\Omega$, one has

So the corresponding energy of solution to (1.1) is defined by

In addition

Which gives in turn the following energy identity

We define the sets

and

So by applying Hölder and Young inequality we arrive at

Clearly since $2 \leq p^{-} \leq p(x) \leq p^{+} \leq \frac{2n}{n-2}$ then by exploiting lemma 3, we have

Because $\partial_{t}u$ is regular and by Young inequality we obtain

At this step we will assume that

where $k = \max(k_{0}, k_{1})$. We notice that estimate (3.5) will be important to prove the blow-up result. Therfore

Consequently by virtue of (3.4) we derive the following estimate for the energy functional

Suppose that $E(0) \leq 0$ then it follows from (3.6) that $E(t) \leq 0$. Define the auxiliary function $L(t)$ by the following formula

where $N > 0$ is to be specified later and $H(t)$ is given by

We differentiate (3.8) and use the Eq (3.4) to arrive at

Therfore

From (3.8) we see that $H$ is a nondecreasing function and

Differentiating (3.7) twice leads to

It's clear from (3.1) and (3.11) that

Now, by using Young's inequality we have

Again Young's inequality yields

where $\beta$ in an arbitrary nonnegative constant to be specified later. By combining (3.3) and (3.5) we get

Inserting (3.13) into (3.12) leads to

By virtue of (3.5) we have

We define the sets

and

So

We first note that

Therfore from (3.15) we have

Consequently

Combination of (3.15) and (3.2) leads to

Substituting (3.14), (3.17) and (3.19) into (3.18) we obtain

Now we pick $\beta$ so small that

Once $\beta$ is chosen we select $N$ large enough that

Therfore from (3.21) and (3.22) there exists a constant $\gamma$ such that (3.20) takes the form

Now we use Hölder inequality to estimate the term $\Vert u \Vert_{L^{r}(\Omega)}^{r}$ as follows

Hence

and from the definition of $L(t)$ in (3.7) we have

This gives

Combining (3.23) and (3.27) yields

We see that the requirements of theorem 1 are satisfied with

Therefore, $L$ blows up in finite time. This completes the proof.

4.   Conclusions

Let us pass to a survey of the results and methods of proving non-existence and blow-up theorems applicable to equations of hyperbolic type. Here it is necessary to clarify what is meant by the term "destruction of the solution". By this term, we understand the existence of a finite time moment at which the solution of the evolutionary problem leaves the smoothness class to which this solution belonged on the interval $(0, T_{max})$ (the smoothness class for which the local solvability theorem is formulated and proved). Looking ahead, we note that in all problems for nonlinear equations considered in the literature, the destruction of the solution is accompanied by the inversion of the norm of the latter to infinity (in the space where we are looking for a solution), however, such behavior of solution is not at all necessary in the concept of destruction.

Acknowledgments

The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.

Conflict of interest

The authors declare there is no conflict of interest.