The Blow-Up Rate for a Non-Scaling Invariant Semilinear Heat Equation

Abstract

We consider the semilinear heat equation ∂tu-Δu=f(u),(x,t)∈RN×[0,T),(1)with f(u) = | u| p-1ulog a(2 + u2) , where p> 1 is Sobolev subcritical and a∈ R. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely u′= | u| p-1ulog a(2 + u2). In other words, all blow-up solutions in the Sobolev subcritical range are Type I solutions. To the best of our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.