Extinction Phenomenon and Decay Estimate for a Quasilinear Parabolic Equation with a Nonlinear Source

Dengming Liu,Luo Yang,Kamyar Hosseini

doi:10.1155/2021/5569043

Dengming Liu, Luo Yang + Show 1 more

Open Access

https://doi.org/10.1155/2021/5569043

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Abstract

By energy estimate approach and the method of upper and lower solutions, we give the conditions on the occurrence of the extinction and nonextinction behaviors of the solutions for a quasilinear parabolic equation with nonlinear source. Moreover, the decay estimates of the solutions are studied.

Highlights

The main goal of this article is to investigate the extinction behavior and decay estimate of the following parabolic initial boundary value problem 8 >>>>< ut = div Àuαj∇ujm−1∇uÁ + ð λup uqdx, Ω>>>>: uðx, uðx, tÞ 0Þ = =0, u0ðxÞ, ðx, tÞ ∈ Ω × ð0,+∞Þ, ðx, tÞ ∈ ∂Ω × ð0,+∞Þ, x ∈ Ω : ð1Þ
Uðx, tÞ is the density of the fluid, uαj∇ujm−1∇u denotes the momentum velocity, and λupÐ Ωuqdx stands for the nonlinear nonlocal source
We mainly focus on the extinction phenomenon and the decay estimates of the solution to a quasilinear parabolic equation with a coupled nonlinear source

Summary

Introduction

Ω ⊂ RN , N ≥ m + 1, is an open bounded domain with smooth boundary ∂Ω, m, p, q, and λ that are positive parameters, 0 < m + α < 1, and um0 +α/m ∈ L∞ðΩÞ ∩ W10,m+1ðΩÞ is a nonzero nonnegative function. Problem (1) can be used to describe the nonstationary flows in a porous medium of fluids with a power dependence of the tangential stress on the velocity of displacement under polytropic conditions In this physical context, uðx, tÞ is the density of the fluid, uαj∇ujm−1∇u denotes the momentum velocity, and λupÐ Ωuqdx stands for the nonlinear nonlocal source. The nonnegative weak solution of problem (1) vanishes in finite time provided that the nonnegative initial datum u0ðxÞ is sufficiently small. For any nonnegative initial datum u0ðxÞ, problem (1) admits at least one nonextinction weak solution. (1) The nonnegative weak solution of problem (1) vanishes in finite time provided that λ is sufficiently small. (2) Problem (1) admits at least one non-extinction weak solution for any nonnegative initial datum u0ðxÞ provided that λ is sufficiently large

Proofs of the Main Results

A: ð28Þ

Conclusion