Blowing Up for the p -Laplacian Parabolic Equation with Logarithmic Nonlinearity

the original work cited. In this article, we are concerned with a problem for the p -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.


Introduction
In the current manuscript, we consider the following initial-boundary value problem for a nonlinear p-Laplacian equation: u t − div ∇u j j p−2 ∇u À Á + u j j p−2 u = u j j p−2 u ln u j j, x ∈ Ω, t > 0, where Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω and u 0 is the initial data p satisfying 2 < p<∞, ifn ≤ p, 2 < p < np n − p , if n > p: 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][2][3][4][5][6][7][8][9][10][11][12][13]).
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][17][18]).
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 p > 2: Also, in [14], the authors established the global boundedness and the blowup of the solution of the problem (5) for 1 < p < 2.
Motivated by the last recent mentioned works, here, we investigated problem (1) with the nonlinear diffusion Δ p = div ðj∇uj p−2 ∇uÞ and logarithmic nonlinearity juj p−2 u ln juj 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.

Preliminaries
As a starting point, we gave some essential definitions and lemmas.
for 1 < p < ∞, and we symbolize the positive constants by C and C i (i = 1, 2, ⋯).

Blowup
In this third section, we gave the proof of blowup of solution of our problem.

Theorem 3.
For any initial data u 0 ∈ H , the problem (1) has a unique weak solution: for some T > 0.
First, we introduce the energy functional in the following lemma.

Lemma 4.
Let uðtÞ be a solution of (1), then EðtÞ is nonincreasing; that is, Proof. Multiplying (1) by u t and integrating on Ω, we have Thus, ☐ To get to our goal of proving the main result, we define the functional Theorem 5. Assume that Eð0Þ < 0, then the solution of problem (1) blows up in finite time.

Data Availability
No data were used to support the study.

Conflicts of Interest
The author declares that he has no conflicts of interest.