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

Abstract

We consider the nonlinear equation $$ frac{partial}{partial t} u (t) = k (t) Delta _{alpha }u (t) + u^{1+eta } (t),quad u(0,x)=lambda varphi (x),; xin mathbb{R} ^{d}, $$ where $Delta _{alpha }:=-(-Delta)^{alpha /2}$ denotes the fractional power of the Laplacian; $00$ 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 $d,alpha, ho,eta$, obeying $0<d hoeta /alpha<1$, 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 $T_{lambdavarphi}sim lambda^{-alpha eta /(alpha -d ho eta )}$ near $lambda=0$.

Keywords

• Semilinear evolution equations
• Feynman-Kac representation
• critical exponent
• finite time blowup
• nonglobal solution
• life span