Electron. J. Differential Equations, Vol. 2018 (2018), No. 32, pp. 1-11.

Blow-up for a semilinear heat equation with moving nonlinear reaction

Raul Ferreira

Abstract:
We study the behavior of solutions of the semilinear problem
$$\displaylines{ u_t=u_{xx}+(1+(T-t)^{-\alpha}\chi_{\{|x|<(T-t)^{1/2}\}}) u^p,\quad x\in\mathbb R,\; t\in(0,T),\cr u(x,0)=u_0(x)\ge 0, \quad x\in\mathbb R, }$$
with $\alpha>0$ and p>0 We describe, in terms of the parameters when the solution is bounded and when it blows up. For blowing up solutions we find the blow-up rate and the blow-up set.

Submitted July 13, 2017. Published January 22, 2018.
Math Subject Classifications: 35K58, 35B44, 35B40.
Key Words: Semilinear parabolic equation; blow-up; asymptotic behaviour.

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Raul Ferreira
Departamento de Matemática Aplicada
Univ. Complutense de Madrid
28040 Madrid, Spain
email: raul_ferreira@mat.ucm.es

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