Asymptotic symmetry of positive solutions of parabolic equations

This note is primarily concerned with symmetry properties of solutions of parabolic equations. To put the questions in broader perspective, we first recall several symmetry results on elliptic equations (on bounded and unbounded domains). We then proceed by summarizing earlier theorems on asymptotic symmetry for parabolic equations on bounded domains. Finally, we announce a new theorem on nonautonomous equations on R-N which asserts that positive solutions decaying at spatial infinity are asymptotically radially symmetric about some center. As our discussion of the symmetry problem suggests, dealing with parabolic equations on R-N one is faced with interesting extra difficulties not present in elliptic equations or in parabolic equations on bounded domains.