Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

The Cauchy problem for a nonhomogeneous heat equation with reaction

Pages: 643 - 662, Volume 33, Issue 2, February 2013      doi:10.3934/dcds.2013.33.643

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Arturo de Pablo - Departamento de Matemática Aplicada, Universidad Carlos III de Madrid, 28911 Leganés, Spain (email)
Guillermo Reyes - Departamento de Matemáticas, U. Politécnica de Madrid, 28040 Madrid, Spain (email)
Ariel Sánchez - Departamento de Matemáticas, U. Rey Juan Carlos, 28933 Móstoles, Spain (email)

Abstract: We study the behaviour of the solutions to the Cauchy problem $$ \left\{\begin{array}{ll} \rho(x)u_t=\Delta u+u^p,&\quad x\in\mathbb{R}^N ,\;t\in(0,T),\\ u(x,\, 0)=u_0(x),&\quad x\in\mathbb{R}^N , \end{array}\right. $$ with $p>0$ and a positive density $\rho$ satisfying $\rho(x)\sim|x|^{-\sigma}$ for large $|x|$, $0<\sigma<2< N$. We consider both the cases of a bounded density and the singular density $\rho(x)=|x|^{-\sigma}$. We are interested in describing sharp decay conditions on the data at infinity that guarantee local/global existence of solutions, which are unique in classes of functions with the same decay. We prove that larger data give rise to instantaneous complete blow-up. We also deal with the occurrence of finite-time blow-up. We prove that the global existence exponent is $p_0=1$, while the Fujita exponent depends on $\sigma$, namely $p_c=1+\frac2{N-\sigma}$.
    We show that instantaneous blow-up at space infinity takes place when $p\le1$.
    We also briefly discuss the case $2<\sigma< N$: we prove that the Fujita exponent in this case does not depend on $\sigma$, $\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too small at infinity a phenomenon of instantaneous complete blow-up occurs in the range $1< p < \tilde{p}_c$

Keywords:  Reaction-diffusion equations, initial value problem, well-posedness, blow-up, critical exponents.
Mathematics Subject Classification:  Primary: 35K57, 35A01, 35B44; Secondary: 35A02, 35B33.

Received: July 2011;      Revised: July 2012;      Available Online: September 2012.