Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent. (English) Zbl 1170.35456

Summary: 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.


35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35B33 Critical exponents in context of PDEs
35K15 Initial value problems for second-order parabolic equations