On a potential-well type result and global bounds of solutions for semilinear parabolic equation involving critical Sobolev exponent (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

In this note, we are concerned with the asymptotics of (mainly nonnegative) solutions for a semilinear parabolic equation involving critical Sobolev exponent. The existence of a potential-well structure is discussed. In addition, the existence of the global bounds for the Sobolev norm of time-global solutions and long-time asymptotics of them are given.