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

We consider the semilinear heat equation partial derivative(t)u - Delta u = f (u), (x, t) is an element of R-N x [0, T), with f (u) = vertical bar u vertical bar(p-1)u log(a)(2+ u(2)), where p > 1 is Sobolev subcritical and a is an element of 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' = vertical bar u vertical bar(p-1)u log(a)(2 + u(2)). 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.