Differential and Integral Equations

Global existence and blow up results for a heat equation with nonlinear nonlocal term

Miguel Loayza

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Abstract

We study the nonlocal parabolic equation $$ u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s) \, ds $$ with $p>1$. We assume that $k$ is continuous and there exists $\gamma \in {\mathbb{R}}$ such that $k(\lambda t, \lambda s)=\lambda^{-\gamma}k(t,s)$ for all $\lambda>0$, $0<s<t$. We consider the problem in ${\mathbb{R}}^N$ and a Dirichlet problem in a bounded smooth domain $\Omega$. We analyze the conditions for either blow up or global existence of solutions.

Article information

Source
Differential Integral Equations, Volume 25, Number 7/8 (2012), 665-683.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012657

Mathematical Reviews number (MathSciNet)
MR2975689

Zentralblatt MATH identifier
1265.35122

Subjects
Primary: 35K154 35B33: Critical exponents 35B44: Blow-up

Citation

Loayza, Miguel. Global existence and blow up results for a heat equation with nonlinear nonlocal term. Differential Integral Equations 25 (2012), no. 7/8, 665--683. https://projecteuclid.org/euclid.die/1356012657


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