Existence, Decay, and Blow-up of Solutions for a Weighted -Biharmonic Equation with Nonlinear Damping and Source Terms

Abstract

In this paper, we consider the weighted -biharmonic equation with nonlinear damping and source terms. We proved the global existence of solutions. Later, the decay of the energy is established by using Nakao’s inequality. Finally, we proved the blow-up of solutions in finite time.

1. Introduction

In this work, we study the following weighted -biharmonic equation with initial-boundary value: where, is a domain with smooth boundary in . and the coefficient are assumed a strictly continuous and positive differentiable function in .

Freitas and Zuazua [1] considered the linear wave equation with indefinite damping of the form.

He proved the stability results.

In [2], Yu investigated the equation with constant coefficients.

He showed globality, boundedness, blow-up, convergence up to a subsequence towards the equilibria, and exponential stability. Gerbi and Said-Houari [3] proved the exponential decay of solutions (3) for .

Huang and Chen [4] considered the nonlinear Klein-Gordon equation with damping term.

Using potential well argument, they obtain global solutions and blow-up result in finite time.

Tahamtani [5] discussed with nonlinear hyperbolic equation with the Lewis function.

He considered a blow-up result.

Pişkin and Fidan [6] considered the variable coefficient wave equation.

They proved the blow-up of solutions.

Al-Gharabli and Al-Mahdi [7] investigated the following nonlinear plate equation.

They proved the local existence using the Faedo-Galerkin method.

Zheng et al. [8] considered the Petrovsky equation: in a bounded domain. They proved the blow-up of solutions.

Guo and Li [9] considered the Petrovsky equation with a strong damping term.

They utilized an energy estimation technique to derive the minimum possible blow-up time.

Wu [10] considered with variable coefficients and obtained the blow-up result with lower and upper boundedness.

Messaoudi [11] studied the following problem:

He studied the decay of solutions of the problem (11). Then, the problem (11) was studied by Wu and Xue [12] and Pişkin [13] under different conditions.

Boonaama et al. [14] have studied blow-up, decay, and existence of solutions of the following equation:

Later, the same authors [15] studied the following equation:

They established global existence.

Pişkin and Fidan [16] are concerned with the following problem:

They prove the blow-up of solutions for finite time with negative initial energy.

Then, Mokeddem [17] studied the global solutions and decay rate estimate for the energy of the following equation:

Motivated by the above-mentioned papers, in this paper, we investigate to prove the global existence, decay, and blow-up of solutions for problem (1), which was not previously studied, where we study weighted -biharmonic equation with nonlinear damping and source terms.

The rest of the work is as follows. In Section 2, we give some assumptions needed in this work. In Section 3, we prove the global existence theorem. In Section 4, we prove the decay of solutions by Nakao’s inequality. In Section 5, the blow-up result is proved for .

2. Preliminaries

In this part, we present certain lemmas and assumptions required for the formulation and proof of our results. Let and indicate the typical , , and norms (see [18, 19]).

To investigate eq. (1), we define the weighted Sobolev space as the closure of in the norm:

Lemma 1 (see [20]). Let be a nonincreasing and nonnegative function defined on , satisfying

is a nonnegative constant, and is a positive constant. Then, we get, for each , where and

Lemma 2 (see [21]). Let be a -function satisfying

If then for where is the smallest root of the equation

Lemma 3 (see [21]). If be a nonincreasing function on and supplies the differential inequality Here, and exist a finite time :

The upper limits for are estimated as given below: (i)If and (ii)If

Next, we prove the local existence theorem which may be proved by [22, 23].

Theorem 4 (local existence). We assume that , , and ; then, problem (1) is a unique local solution.

3. Global Existence

In this part, we show the global existence of the solution for problem (1). We define the following functionals:

The functional of problem (1) is as follows: and we denote the Nehari set

Lemma 5. Assume that is a solution to problem (1). Then, the energy of problem (1) defined by (30) satisfies and

Proof. Multiply eq. (1) by , integrate it over , and apply Green’s formula,

Lemma 6. Suppose that , , , and

Then, for each , .

Proof. Since and due to the continuity of , it follows that for some interval near . Let be the maximum time for which eq. (29) holds on the interval .
Thus, from (28) and (29), From , we get Then, using the definition and , we have Thanks to Lemma 6 and (37), we obtain We can conclude that based on reference (21), When by repeating the procedure, is extended to .

Lemma 7. If the conditions of Lemma 6 are satisfied, then there exists such that the

Proof. We get and when set , we may deduce that

Theorem 8. Assume that by Lemma 6, and , . Then, the solution for (1) is global.

Proof. We get by , then ; here, . From Theorem 4, we get the global existence result.

4. Decay

In this part, we show the decay of the solution for problem (1).

Theorem 9. Assume that . We assume that , , and . Thus, we have the following decay: where and and are positive constants.

Proof. Integrating over , , we obtain Therefore, by using and Hölder’s inequality, we get where .
Then, there exist and so that Multiply the first equation of (1) by , integrate it over , and apply Green’s formula; we get Now, we use the Cauchy-Schwarz inequality and Hölder inequality, and we get By using the Hölder inequality from the last term, we have Then, by (37), we have Next, we calculate the fourth term of the right-hand side of (48), and we obtain Later Then, We have where
Therefore, Moreover, we get Here,
Integrating over , we get Then, we get Next, integrating over , we obtain Since , we decide that Thus, As a result, Hence, Therefore, From that is a nonincreasing function and on , we have After that, Therefore, We obtain Case 1. When and and by Lemma 1, we obtain where
Case 2. When where , which complete the proof of Theorem 9.

5. Blow-up

In this part, we prove our main blow-up results to problem (1) for .

Definition 10. A solution to problem (1) is referred to as a blow-up if there exists a finite time so that

Then, we have

Lemma 11. We assume that , , , and . Then

Proof. By taking the first- and second-order derivative of (73), we get By (76) and (30), we have Here, , we have (74).

Lemma 12. Let and , then (i)If , then for , here (ii)When and . Then, for . We get

Proof. (i)When and for , thenand by integration over , we get Then, we get for , with (ii)When and . Then, for . We get ,

Theorem 13. Suppose that and , we obtain Case 1 and Case 2.

Case 1. If and the solution blows up in finite time in the sense of and Furthermore, if , we get where

Case 2. If and . The solution blows up in finite time in the sense of and With

Proof. Set where where is a strictly positive constant that will defined later. Then, by taking the first- and the second-order derivative of , we have where For simplicity of calculation, we define By (75) and Hölder’s inequality, we get By Case 1 and Lemma 12, we obtain Then, by (87), (90), and (93), we have By , we get and from , we get When is a nonnegative function, we obtain Therefore, from , we have We obtain Multiplying of (98) by , integrate it over , and we get with and defined.
Finally, utilizing Lemma 3, there exists a such that and is estimated based on the sign of . Thus, equation (72) is satisfied.

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.