Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 10, pp. 1-18.

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

Authors: Ekaterina T. Kolkovska (Centro de Investigacion en Mat., Mexico)
         Jose Alfredo Lopez-Mimbela (Centro de Investigacion en Mat., Mexico)
	 Aroldo Perez (Univ. Juarez Autonoma de Tabasco, Mexico)

Abstract: 
 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  $0<d\rho\beta /\alpha<1$, 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 $\lambda=0$.
  
Submitted August 24, 2007. Published January 21, 2008.
Math Subject Classifications: 60H30, 35K55, 35K57, 35B35.
Key Words: Semilinear evolution equations; Feynman-Kac representation;
  critical exponent; finite time blowup; nonglobal solution; life span.