On asymptotically symmetric parabolic equations
Juraj Földes, Peter Poláčik · Networks and Heterogeneous Media · 2012
We consider global bounded solutions offully nonlinear parabolic equations on boundedreflectionally symmetric domains, under nonhomogeneous Dirichletboundary condition. We assume that, as $t→∞$,the equation is asymptotically symmetric, the boundary condition isasymptotically homogeneous, and the solution is asymptoticallystrictly positive in the sense that all its limit profiles arestrictly positive. Our main theorem states that all the limit profilesare reflectionally symmetric and decreasing on one side of thesymmetry hyperplane in the direction perpendicular to the hyperplane. We also illustrate by example that, unlike forequations which are symmetric at all finitetimes, the result does not hold under arelaxed positivity condition requiring merely that atleast one limit profile ofthe solution be strictly positive.