Blowup and life span bounds for a reaction-diffusion equation with a time-dependent generator

Kolkovska, Ekaterina T. ; Lopez-Mimbela, Jose Alfredo ; Perez, Aroldo

Electronic journal of differential equations, Tome 2008 (2008)

Résumé

We consider the nonlinear equation

$ \frac{\partial}{\partial t} u (t) = k (t) \Delta _{\alpha }u (t) + u^{1+\beta } (t),\quad u(0,x)=\lambda \varphi (x),\; x\in \mathbb{R} ^{d}, $

where $\Delta _{\alpha }:=-(-\Delta)^{\alpha /2}$ denotes the fractional power of the Laplacian; $0\alpha \leq 2, \lambda, \beta >0$ are constants; $ \varphi$ is bounded, continuous, nonnegative function that does not vanish identically; and $k$ is a locally integrable function. We prove that any combination of positive parameters $d,\alpha,\rho,\beta$, obeying $0$, yields finite time blow up of any nontrivial positive solution. Also we obtain upper and lower bounds for the life span of the solution, and show that the life span satisfies $T_{\lambda\varphi}\sim \lambda^{-\alpha \beta /(\alpha -d\rho \beta )}$ near $\lambda=0$.

Détail

Classification : 60H30, 35K55, 35K57, 35B35
Keywords: semilinear evolution equations, Feynman-Kac representation, critical exponent, finite time blowup, nonglobal solution, life span