Well-Posedness and Blow-Up of Solutions for a Variable Exponent Nonlinear Petrovsky Equation

Abstract

In this article, we investigate a nonlinear Petrovsky equation with variable exponent and damping terms. First, we establish the local existence using the Faedo–Galerkin approximation method under the conditions of positive initial energy and appropriate constraints on the variable exponents and . Finally, we prove a finite-time blow-up result for negative initial energy.

1. Introduction

In this work, we investigate the following initial-boundary value problem:where () is a bounded domain with smooth boundary :and are initial conditions. and are given measurable functions on satisfying the following equations:andwhereand the log-Hölder continuity condition for :(i)This kind of Equation (1) without variable exponent has its origin in the canonical model introduced by Petrovsky [1, 2]. Petrovsky [1, 2] type equation originated from the study of plate and beams, and it can also be used in many branches of science, such as ocean acoustics, geophysics, optics, and acoustics [3].(ii)The problems with variable exponents arise in many branches in science such as the image processing, filtration processes in porous media, flow of electrorheological fluids, and nonlinear viscoelasticity [4–6].

In the study of Ouaoua and Boughamsa [7], they looked into the following equation:They showed the local existence and also proved that the local solution is global. Antontsev et al. [8] studied the following a nonlinear Petrovsky equation:Under suitable assumptions on the variable exponents and initial data, they obtain local weak solutions and established a blow-up result. Tebba et al. [9] discussed a new class of nonlinear wave equation:Under appropriate assumptions on the variable exponents, they demonstrated the existence of a unique weak solution using the Faedo–Galerkin method. They also proved the finite time blow-up of solutions.

Moreover, numerous researchers have studied the mathematical behavior of equations using the Faedo–Galerkin and the perturbed energy method [10–14].

In this work, we are concerned the existence and blow-up of the problem (1). The obtained existence and blow-up results improve and generalize many results in the literature.

This work is composed of three sections in addition to the introduction. Part 2 presents preliminary information regarding variable exponents Lebesgue and Sobolev spaces. Additionally, we outline significant lemmas and assumptions. Part 3 focuses on proving the local existence of solutions. In Part 4, we establish the blow-up of solutions with a positive initial energy.

2. Preliminaries

Throughout this work, we present some important facts about Lebesgue and Sobolev spaces with variable exponents (see [5, 15]).

Let be a measurable function, where is a domain of We define the variable exponent Lebesgue space by the following equation:where Equipped with the following Luxembourg-type norm:The space is a Banach space.

The variable-exponent Sobolev space is defined as follows:This is a Banach space with respect to the norm

Furthermore, we set to be the closure of in the space . Let us note that the space has a differenet definition in the case of variable exponents.

However, under the log-Hölder continuity condition, both definitions are equivalent [5]. The space , dual of , is defined in the same way as the classical Sobolev spaces, where .

Lemma 1 (Diening et al. [5]). If: then we have: for any 

Lemma 2 (Diening et al. [5]). Let  be measurable functions defined on  such that: If  and , then , with: 

Lemma 3 (Diening et al. [5]). If  is a measurable function on  satisfying (6), then the embedding  is continuous and compact. Then, the embedding  is continuous and compact.

As per Lemma 3, there exists a positive constant denoted as that fulfills the following condition:

Lemma 4 (Komornik [16]). Let  be a nonincreasing function and assume that there are two constants  and  in the following equation: Then, we have the following equation: 

To articulate and demonstrate our outcome, we define the subsequent functionals:

Lemma 5. Let  be a solution of problem (1). Then, the energy functional satisfies the following equation: and 

Proof. Multiplying the first equation in Equation (1) by and integrating over yields the following equation:then:Integrating Equation (26) over , we obtain the following equation:

Lemma 6. Under the assumptions of Theorem 5 and  hold: and where with , and are the bests embedding constants of  and , respectively, then , for all .

Proof. Due to continuity, there exists , such that:Now, for all , we have the following equation:Using Equation (31), we obtain the following equation:By Lemma 5, we get the following equation:Moreover, according to Lemma 1, we obtain the following equation:By the embedding of and , we obtain the following equation:Then, we have the following equation:Since then, we obtain the following equation:This implies that:By repeating the aforementioned process, we can extend to .

3. Local Existence

This section is dedicated to establishing the local existence of problem (1). We will employ the Faedo–Galerkin method approximation.

Theorem 7. Suppose that  and satisfies Equation (6). Then, for any  problem (1) has a unique weak local solution: 

Proof. Let be a basis of that forms a complete orthonormal system in . Denote as the subspace generated by the first vectors from the basis . Due to normalization, we have . For a given integer , we consider the approximated solution:where is the solutions to the following Cauchy problem:It is worth noting that the systems (42)–(44) can be solved using Picard’s iteration method for ordinary differential equations. As a result, a solution exists within the interval for some , and we can extend this solution to the whole interval for any given by utilizing the a priori estimates provided below.
The first estimate: Multiplying Equation (42) by and summing over from to :Then, we obtain the following equation:By integrating Equation (45) over the interval we derive the estimate the following equation:Then, from Equation (38), the inequality (47) becomes:From Equation (48), we conclude that:Since is uniformly bounded in then is bounded in hence, up to a subsequence, weakly in . As in Messaoudi et al.’s [17] study, we have to show that
Furthermore, from Lemma 3 and Equation (49), we obtain the following equation:From Equations (49) and (50), we deduce the existence of a subsequence of (still denoted by the same symbol) and a function such that:By the Aubin–Lions compactness Lemma [18], we conclude from Equation (51) that:which implies:It follow from Equations (51) and (53) that:The second estimate: Now, we would like to get more estimates. In doing so, differentiating Equation (42) with respect to , we get the following equation:Next, multiplying Equation (55) by and summing over from to , we get the following equation:We have the following equation from Hölder’s inequality:We have, then then the following equation:since, The inequality (57), becomes the equation as follows:We have the following equation from Young’s inequality and Poincáre’s inequality:Substituting Equation (60) into Equation (56) and integrating over for all , we obtain the following equation:It follows from Equation (44) and the fact that:where is a positive constant independent of .
By multiplying both sides of Equation (42) by , summing over from to and setting , we obtain the following equation:Utilizing Young’s inequality along with Equations (43) and (44), we have:where is a positive constant independent of .
By Equations (62) and (64), Equation (61) becomes:We deduce from Equation (65) and Gronwall’s lemma that:for all , where is a positive constant independent of .
We can infer from Equation (66) that:Similarly, we have the following equation:Setting up and passing to the limit in Equation (42), we obtain the following equation:Given that is a basis of , we can deduce that satisfies Equation (1). From Equation (51), Equation (68) and Lemma 3.1.7 in Zheng’s [19] study with and , respectively, we infer that:We get from Equations (43), (44), and (70) that , .
Consequently, the proof of existence is now concluded.
Uniqueness of the solution: Now it remains to prove uniqueness. Let and be two solutions in the class described in the statement of this theorem, and .
Then, satisfies the following equation:andMultiplying Equation (71) by , then integrating with respect to , we get the following equation:By using the inequality:for all and a.e. .
This implies:By repeating the estimate as in Messaoudi’s [20] study, we arrive the following equation:Then:Gronwall’s inequality yields the following equation:Thus, . The shows the uniqueness.

4. Blow-Up

In this section, we examine the blow-up of the solution to problem (1). To begin, we introduce the following [20].

Lemma 8. If  is a measurable function and holds. Then, we have the following inequalities:

Lemma 9. Suppose the conditions of Lemma 8 hold and let  be the solution of Equation (1). Then:

 (i)

 (ii)

 (iii)

 (iv)

for any and Where and a positive constant and

Then, functions will be defined later. Now, we state and prove our blow-up result.

Theorem 10. Under the conditions of Lemma 9. Also, let initial energy satisfy  and the exponents  and  satisfy the following equation: Then, the solution of Equation (1) blows up in finite time , in the following sense: here , and  will given later in Equations (91) and (94), respectively.

Proof. When we multiply both sides by and integrate over the domain , the result is as follows:whereBy setting , we establish that Referring to Equation (88), it follows that We then define the following equation:for small that will be selected later, andBy deriving Equation (91) and applying Equation (1), we acquire the following equation:We subsequently utilize Young’s inequality for all, for all :to estimate the last term in Equation (94) as follows:which yields, by substitution in Equation (94):Therefore, by taking so that for large to be specified later, and substituting in Equation (97), we arrive at the following equation:Adding and subtracting from the right-hand side of Equation (98), we obtain the following equation:By exploiting Equation (99) and the inequality Lemma 9, we obtain the following equation:hence, Equation (100) yields the following equation:We then use Lemma 8 and Equation (92), for and , to deduce from Equation (101):where By noting that:and writing yields the following equation:whereAt this point, we choose large enough so that the coefficients of , and in Equation (104) are strictly positive; hence, we get the following equation:where is the minimum of these coefficients. Once is fixed (hence ), we pick small enough so that andTherefore, Equation (106) takes the following form:Consequently, we have the following equation:Next, we would like to show the following equation:where Γ is a positive constant depending on and (the constant of Equation (81)). Once Equation (110) is established, we obtain in a standard way the finite time blow-up of , hence of (see Batle et al. [21] for instance).
To prove Equation (110), we first estimate the following equation:which implies:Again Young’s inequality gives the following equation:for We take to get by Equation (92). Therefore, Equation (113) becomes:where . By using Equation (83), we obtain for all :Finally, by noting the following equation:and combining it with Equations (108) and (115), the inequality (110) is established. A simple integration of Equation (110) over , then yields the following equation:Therefore, Equation (117) shows that blows up in finite time:where Γ and are positive constant with and is given by Equation (91). This completes the proof.

Data Availability

There are no underlying data supporting the results of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.