Well defined extinction time of solutions for a class of weak-viscoelastic parabolic equation with positive initial energy

1.   Literature overview and new contributions

Let $1 < p < +\infty$ and $\Omega$ be an open bounded domain in $\mathbb{R}^n, n\geq 2$, with smooth boundary $\partial\Omega$. The $p$-Laplacian $\Delta_p$ is a nonlinear differential operator of order $2$ defined by

This differential operator intervenes in various domains among which we can mention: the modeling of the mechanical phenomena, the image processing and some physical problems. To model the movement of some non-Newtonian fluids, O. Ladyzhenskaya proposed in ^[17] the following system

Let $u(x, t)$ be the concentration of a particle component (or the density of heat) in $\Omega$. The equation

describes the evolution of the concentration during the propagation of particles in $\Omega$. Here $c(x, t, u)$ describes a source if it is positive or a bowl if it is negative, the diffusion coefficient $a(x, t, u, \nabla u)$ reflects the intrinsic ability of diffusion in particles in the medium. Needful to say, it has a numerous generalizations, we can also do with $p(x)$-Laplacian denoted by $\Delta_{p(x)}$, which has an exponent variable property.

The fact that $p(x)$-Laplacian is not homogeneous, makes non-linearities more complicated than the operator $p$-Laplacian. Studies of various mathematical systems with variable exponent growth conditions have received considerable attention in recent years, which is justified by their various physical applications. However, few papers have treated evolutionary equations of non-local $p(x)$-Laplacian type (Please see ^{[1,5,12,13,24,25]}). Viscoelastic materials demonstrate properties between those of elastic materials and viscous fluid. As a consequence of the widespread use of polymers and other modern materials which exhibit stress relaxation, the theory of visco-elasticity has provided important applications in materials science and engineering (Please see ^{[7,8,10,11,19,21,22]}).

The viscoelastic materials show a behavior which is something between that of elastic solids and Newtonian fluids. Indeed, the stresses in these media depend on the entire history of their deformation, not only on their current state of deformation or their current state of motion. This is the reason why they are called materials with memory. The viscoelastic equations with fading memory in a bounded space has been deeply studied by several authors, in view of its wide applicability.

The lack of stability of solutions of viscoelastic partial differential equations is a huge restriction for qualitative studies. In the present paper, we consider

for $x \in \Omega, \ 0 < t < \infty$ with initial and boundary conditions

where $q(\cdot)$ and $p(\cdot)$ are two continuous functions on $\overline{ \Omega}$ such that

with

The viscoelastic term is represented as $\int^{t}_{0}\mu \left(t-s\right)\Delta_x w \, ds$, it is called "weak-viscoelastic" when it comes with the time weighted function $\sigma(t)$, which is considered as a dissipative term and causes stability of systems. The nonlinear term $\vert w \vert ^{p(x)-2} w$ is known as the source of instability. The importance of our study lies in the study of the interaction between the exponents of source term and the Laplacian with the presence of weak-viscoelastic term. We take the exponents as a variable functions, with their difficulties in the mathematical point of view, to obtain a very large applications. These contributions extend the early results in literature.

We assume that $q(x)$ satisfies the Zhikov-Fan condition, i.e. for all $x, y \in \Omega$,

with $K > 0, \ 0 < \kappa < 1$ and

Since the relaxation function $\mu$ lies with the stability of solutions, we state assumptions on $\mu$ and $\sigma$ as: $\mu, \sigma \in C^1 (\mathbb{R}^{+}, \mathbb{R}^{+})$ satisfy

For positive constant $C$ depending only on $\Omega$ determined by Lemma 2.1, we set for some constant $\lambda > 0$ (will be specified later)

Problem (1.2)–(1.4) is related to the parabolic problem and when the exponents $q(x) = q, \ p(x) = p$, the existence/non-existence results have been extensively studied (please see ^{[2,3,4,14,15,23]}).

Extinction phenomenon for parabolic equation with nonlinearities in divergence form are investigated in ^[18], under nonlinear boundary flux in bounded star-shaped region. The authors assumed conditions on weight function to guarantee that the solution exists globally or blows up at finite time. Moreover, using the modified differential inequality, upper and lower bounds for the blow-up time of solutions were derived in higher dimensional spaces.

In the case where $q(\cdot)$ and $p(\cdot)$ are constants, the existence of local solutions of initial-boundary value problem

is proved by Akagi in ^[2] for initial data $u_{0} \in L^{r}(\Omega)$ and $2\leq q < p < +\infty$ and $\Omega$ is an open bounded domain in $\mathbb{R}^n$, under $r > n(p-q)/q$. For the case where $q(\cdot)$ and $p(\cdot)$ are two measurable functions, it is well known that some additional techniques must be used to study the existence/nonexistence of solutions for (1.2)–(1.4) and of course the classical methods may be failed unless some developments are made.

For the case of nonlocal $p(x)$-Laplacian equations and in the absence of the memory term $(\mu\equiv0)$, problem (1.2)–(1.4) has been studied by Ôtani ^[20]. The author treated the question of existence and qualitative studies of solutions of (1.2)–(1.4) and showed that the difficulties come from the use of non-monotone perturbation theory. To complete these studies, the questions of blow-up of solutions for the same problem are discussed later by many authors.

In this paper we shall establish a blow-up result of solutions for problem (1.2)–(1.4) in the Lebesgue and Sobolev spaces with variable exponents and positive initial energy.

2.   Preliminary and well known results

We try to list here some useful mathematical tools.

First, let $p:\Omega \longrightarrow (1, \infty)$ be a measurable function. Denoting by

We define the $p(\cdot)$ modular of a measurable function $w:\Omega \longrightarrow \mathbb{R} \cup \{ \pm \infty \}$ as

where

The special Orlicz Musielak space $L^{p(\cdot)}(\Omega)$ is a Lebesgue space with variable-exponent and it consists of all the measurable function $w$ defined on $\Omega$ for which

Let

be the Luxembourg norm on this space (see ^[16]).

The Sobolev space $W^{1, q(\cdot)}\left(\Omega\right)$ consists of functions $w\in L^{q(\cdot) }(\Omega)$ whose distributional gradient $\nabla_x w$ exists and satisfies $|\nabla_x w|\in L^{q(\cdot)}(\Omega)$. This space is a Banach with respect to the norm

Lemma 2.1 (Corollaries $8$.$2$.$5$ and $8$.$3$.$2$ in ^[9]). $(1)$ If $(1.6)$ holds with $q(x)$, then

where $\Omega$ is a bounded domain and $C > 0$ is a constant. The norm of space $W^{1, q(\cdot)}_{0}(\Omega)$ is given by

$(2)$ If

is a measurable function and

with

then

with continuous and compact embedding and

where $C_{S} > 0$ is an embedding constant.

Proposition 2.2 (Section $1$ and Lemma $3$.$2$.$20$ in ^[9]). Let $1 < p_{-} \leq p_{+} < +\infty$. The spaces $L^{p(\cdot)}(\Omega)$ and $W^{1, p(\cdot)}(\Omega)$ are separable, uniformly convex, reflexive Banach spaces. The conjugate space of $L^{p(\cdot)}(\Omega)$ is $L^{p'(\cdot)}(\Omega)$, where

For $w\in L^{p(\cdot)}(\Omega)$ and $v \in L^{p'(\cdot)}(\Omega)$, we have

Lemma 2.3 (Lemma $3$.$2$.$4$ in ^[9]). If $p(.) \geq 1$ is a measurable function on $\Omega$ and $w\in L^{p(\cdot)}(\Omega)$, then $\left\Vert w \right\Vert_{p(\cdot)} \leq 1$ and $Q_{p(\cdot)}(w) \leq 1$ are equivalent. For $w \in L^{p(\cdot)}(\Omega)$, we have

$(1)$ $\left\Vert w \right\Vert_{p(\cdot)} \leq 1$ implies $Q_{p(\cdot)}(w) \leq \left\Vert w \right\Vert_{p(\cdot)}$.

$(2)$ $\left\Vert w \right\Vert_{p(\cdot)} > 1$ implies $Q_{p(\cdot)}(w) \geq \left\Vert w \right\Vert_{p(\cdot)}$.

Lemma 2.4 (Section $2$ in ^[6], Lemma $3$.$2$.$5$ in ^[9]). If $p(.) \in [1, \infty)$ is a measurable function on $\Omega$, then

for all $w\in L^{p(\cdot)}(\Omega)$.

Lemma 2.5 (Lemma $3$.$2$.$20$ in ^[9]). If $p_1(.)\geq p_2(.) \geq 1$ a.e. in $\Omega$, there is a continuous inclusion $L^{p_1(\cdot)}(\Omega)\subset L^{p_2(\cdot)}(\Omega)$ and for all $\, w\in L^{p_1(\cdot)}(\Omega)$,

where

The following notation will be used throughout this paper

for $v \in L^2(\Omega)$ and $t \geq 0$. We have the following technical Lemma.

Lemma 2.6. Let $\kappa \in {\mathbb N}$. For any $\Delta_{x}^p v \in C^{1}\left(0, T, H_{0}^{1}(\Omega) \right)$ with $p = 0, 1, \ldots, \kappa-1$, we have

Proof. Since

holds, we have

Consequently, we have

which implies

This completes the proof.

3.   Main results

Here, we present without proof, the first known result concerning local existence (in time) for problem (1.2)–(1.4). (see ^[22])

Theorem 3.1. Assume that $(1.5)$, $(1.6)$ and $(1.7)$ hold. Then problem $(1.2)–(1.4)$ has a unique local solution $w$ satisfying

for $T_{0} > 0$ depending on $\Vert w_0 \Vert_{1, q(\cdot)}$.

Now, to prove the blow up result, we should define the energy functional $E\left(t\right)$, associated with our problem by

Lemma 3.2. Let $w$ be a solution of $(1.2)–(1.4)$ with $(1.5)–(1.7)$. Then the energy functional satisfies

Proof. Multiplying (1.2) by $\partial_t w$, integrating by parts over $\Omega$, using (1.7) and Lemma 2.6, we get the desired result.

By using conditions (1.6), (1.8) and thanks to

there exists a constant

by (1.5) and (1.7), we have

such that

Let $f$ be a function defined by

Then $f$ is increasing in $(0, \alpha)$, decreasing for $\psi > \alpha$, $f\left(\psi\right) \longrightarrow -\infty$ as $\psi \longrightarrow +\infty$

and

for $\alpha = \left(\frac{q_{-}}{Cp_{+}} \right)^{\frac{q_{-}}{p_{+}-q_{-}}}.$

Lemma 3.3. Let $w$ be strong solution of (1.2)–(1.4) with (1.5)–(1.7) and initial condition satisfying

Then there exists a constant $\beta > \alpha$ such that

and

for all $t \in [0, T_{0})$.

Proof. We define the sets

and

By (3.2) and the Sobolev embedding, we get

Since $E(0) < E_1$, there exists $\beta > \alpha$ such that $f(\beta) = E(0)$, by using (3.7) we have $f\Big(\int_{\Omega} \left\vert\nabla_x w(x, 0) \right\vert^{p(x)}\, dx\Big)\leq E(0)$, which implies that $\int_{\Omega} \left\vert\nabla_x w(x, 0) \right\vert^{p(x)}\, dx\geq \beta$.

Now, to establish (3.5), we suppose by contradiction that

for some $t_0\geq 0$ and by continuity of

we have to choose $t_0$ such that

Again the use of (3.7) leads to

which is not possible, since $E (t) \leq E (0), \forall t \in[0, T)$. Then (3.5) is proved.

We use (3.2) and (3.7) to prove (3.6)

this imply

Therefore the desired results are proved.

Combining with the estimates obtained in the above Lemmas, we state the main result concerned with finite time blow-up.

Theorem 3.4. Assume that $(1.5)–(1.7)$ hold. Given $w_{0} \in W^{1, q(\cdot)}_{0}\left(\Omega\right)$ satisfying $(3.4)$, any solution of $(1.2)$ with $(1.3)$ and $(1.4)$ blows up in finite time $t^{*} < \infty$.

Proof. Set $H(t)\leq E_1-E(t)$. We define

By differentiating $L$, we get

Using Cauchy Schwarz's inequality and Lemma 2.6 to obtain

for some positive constant $C_{0} > 0$ (to be determined later). Then, we have

By using Young's inequality, (1.8) and Lemma 2.6 for some constant $c > 0$, we obtain

We then substitute for $\sigma(t) \left(\mu \circ \nabla_x w \right) (t)$ from (3.2), hence (3.8) becomes

where $0\leq H(t)\leq E_1-E(t)$. By using (1.8) and (3.6), the estimate (3.9) takes the form

This implies that, we can choose $c > C_0+\lambda$ to get

where $C_0 = 1-\frac{2}{p_-}$. Then

Since

Then

This implies that

and

Then

and

By addition, it leads to

where $C_3 = \min \{C^{2/p_-}_2, C^{2/p_+}_2\}$. Or

Using (3.10) and since $0 < L(0) \leq L(T)$, we have for any $t > 0$ either

Or

which gives, in turn

Or

Therefore $\partial_t L \left(t \right) \geq \gamma$, where $\gamma = \min\Big\{\Big(\frac{C_3}{2}\Big)^{p_-/2}, \Big(\frac{C_3}{2} \Big)^{p_+/2}\Big\}$, since $\frac{1}{p_+}-\frac{1}{p_-}\leq 0$ and (1.6), (3.11) yields

where

A direct integration of (3.16) over $\left[0, t\right]$ yields

which implies that

along with $1-\frac{q_{-}}{2} < 0$. Finally, we have

Acknowledgments

The author would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.

Conflict of interest

The author agree with the contents of the manuscript, and there is no conflict of interest among the author.