Blow-Up Results for a Class of Quasilinear Parabolic Equation with Power Nonlinearity and Nonlocal Source

Xiaorong Zhang,Zhoujin Cui,Feliz Minhos

doi:10.1155/2021/2208818

Xiaorong Zhang, Zhoujin Cui + Show 1 more

Open Access

https://doi.org/10.1155/2021/2208818

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Abstract

This paper deals with a class of quasilinear parabolic equation with power nonlinearity and nonlocal source under homogeneous Dirichlet boundary condition in a smooth bounded domain; we obtain the blow-up condition and blow-up results under the condition of nonpositive initial energy.

Highlights

In this paper, we consider the following quasilinear parabolic equation with power nonlinearity and nonlocal source term: ð>>>>< ut = Δpu + μup up+1ðy, tÞdy − kjujp−2u, Ω>>>>: uðx, uðx, tÞ 0Þ = =0, u0ðxÞ, ðx, tÞ ∈ Ω × ð0, TÞ, ðx, tÞ ∈ ∂Ω × ð0, TÞ, x ∈ Ω, ð1Þ where Ω ⊂ RN ðN ≥ 1Þ is a bounded domain with smooth boundary ∂Ω and Δpu = div ðj∇ujp−2∇uÞ is the standard p-Laplace operator with p > 2, μ, k u0 ðxÞ ∈ W ðΩÞ \
Where Ω ⊂ RN ðN ≥ 1Þ is a bounded domain with smooth boundary ∂Ω and Δpu = div ðj∇ujp−2∇uÞ is the standard p-Laplace operator with p
Among some other interesting results, they showed that the weak solution uðx, tÞ of problem (5) blows up at finite time under the condition Jðu0Þ ≤ M and Iðu0Þ < 0, where M > 0 is a constant; the energy functional JðuÞ and Nehari functional IðuÞ are defined as follows: JðuÞ =

Summary

Introduction

We consider the following quasilinear parabolic equation with power nonlinearity and nonlocal source term: ð. Blow-up results of parabolic equations with nonlocal sources have been studied as well. 2, f0g: By using the potential well method and a logarithmic Sobolev inequality, the authors obtained results of existence or nonexistence of global weak solution. They provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions. Among some other interesting results, they showed that the weak solution uðx, tÞ of problem (5) blows up at finite time under the condition Jðu0Þ ≤ M and Iðu0Þ < 0, where M > 0 is a constant; the energy functional JðuÞ and Nehari functional IðuÞ are defined as follows: JðuÞ =.

Assume that

Criterions of Blow-Up

Conclusions

Conflicts of Interest