Abstract
In this article, we are concerned with a problem for the -Laplacian parabolic equation with logarithmic nonlinearity; the blow-up result of the solution is proven. This work is completed Boulaaras’ work in Math. Methods Appl. Sci., (2020), where the author did not study the blowup of the solution.
1. Introduction
In the current manuscript, we consider the following initial-boundary value problem for a nonlinear -Laplacian equation: where is a bounded domain with smooth boundary and is the initial data satisfying
The terminology of nonlinear polynomials is among the work that researchers have focused on recently. For example, it is found in edge detection and optical elasticity, materials science, engineering, physics, and photonics. In addition, many works and problems in applied sciences have been designed and proposed by means of partial differential equations, including the modeling of some dynamic systems in physics and engineering ([1–13]).
The same is said for the evolutionary partial differential equations associated with -Laplacian (see [8, 14, 15]).
We also note that logarithmic nonlinearity has been concerned by many scientists and researchers, and it has introduced many issues, including the wave equation (see [3, 16–18]).
And for more information on some of the other works to which this term was introduced, we refer the reader to [13, 14, 16–24].
Later on, in [25], the authors by the multiplier method gave the energy decay of the solution of the following problem:
In addition, the authors in [14] proved the decay rate of solutions (exponential and polynomial) by using the inequality of Nakao for the seminar problem (3).
On the other hand, for the Laplacian parabolic equation with the logarithmic source term in [21], Chen et al. studied the following problem:
Then, in [23], the authors proved the global existence, the decay, and the blowup of the solutions of the problem: where
Also, in [14], the authors established the global boundedness and the blowup of the solution of the problem (5) for .
Motivated by the last recent mentioned works, here, we investigated problem (1) with the nonlinear diffusion and logarithmic nonlinearity which extends problem in [14]. Our goal is to blow up solutions for problem (1) in order to put some preliminaries. More precisely, we give the blow-up result.
2. Preliminaries
As a starting point, we gave some essential definitions and lemmas. for , and we symbolize the positive constants by and ().
Lemma 1 [7] (logarithmic Sobolev inequality). Let be all function . Then, for , , where
Remark 2. Let , and by defining for , we can write
3. Blowup
In this third section, we gave the proof of blowup of solution of our problem.
Theorem 3. For any initial data , the problem (1) has a unique weak solution: for some .
First, we introduce the energy functional in the following lemma.
Lemma 4. Let be a solution of (1), then is nonincreasing; that is, satisfies
Proof. Multiplying (1) by and integrating on , we have Thus, ☐
To get to our goal of proving the main result, we define the functional
Theorem 5. Assume that , then the solution of problem (1) blows up in finite time.
Proof. From (12), we have
Hence,
We set
where and
Multiplying (1) by and the derivative of (18) gives
Adding and subtracting into (20) (), we obtain
Applying the logarithmic Sobolev inequality gives
Setting and taking give
since
Consequently, for some , inequality (25) gives
Next, by (18), we have
Therefore,
where ,
Hence,
According to (25) and (30), we get
where , depending only on and .
Finally, by integrating (31), we obtain
Hence, blows up in time:
As a result, the proof is completed. ☐
Data Availability
No data were used to support the study.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
The researcher would like to thank the Deanship of Scientific Research, Qassim University, for funding the publication of this project.