Asymptotic behavior of strong solutions for nonlinear parabolic equations with critical Sobolev exponent

Abstract

In this paper, we discuss the asymptotic behavior of solutions of nonlinear parabolic equations in ${\mathbf{R}}^N$ involving critical Sobolev exponent. For the semilinear and subcritical problem, it is well-known that the solution which intersects with the ``stable set'' must be a global one and the solution which enters the "unstable set"feodory should blow up in finite time. But in the critical case, it is not clear that the same result holds or not. In this paper, we show that the same result holds also in the critical case. The proof of our main result requires the method different from that for the subcritical problem and is based on the direct analysis of $L^\infty$-norm of solutions with the aid of the blow up argument and the concentration compactness type argument.