Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent

Abstract

We consider the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the Joseph-Lundgren sense. We find the grow-up rate of solutions that approach a singular steady state from below as $t\to\infty$. The grow-up rate in the critical case contains a logarithmic term which does not appear in the Joseph-Lundgren supercritical case, making the calculations more delicate.