On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent

Abstract

In this paper, we are concerned with the existence of $L^\infty$-global bounds for global-in-time solutions of some semilinear parabolic problems. It is well known that every global-in-time solution for the subcritical problem is globally bounded in $L^\infty$, while there exists a global solution which is not bounded in $L^\infty$ globally in time in the critical case. In this paper, we discuss the necessary and sufficient condition for the existence of $L^\infty$-global bounds, which is valid for the subcritical and the critical case in a unified way. Moreover, using our main results, we provide various examples with the critical exponent in which every global-in-time solution has an $L^\infty$-global bound.