Large-time behavior of solutions of parabolic equations on the real line with convergent initial data

Antoine Pauthier, Peter Poláčik
2018 Nonlinearity  
We consider the semilinear parabolic equation u_t=u_xx+f(u) on the real line, where f is a locally Lipschitz function on R. We prove that if a solution u of this equation is bounded and its initial value u(x,0) has distinct limits at x=±∞, then the solution is quasiconvergent, that is, all its limit profiles as t→∞ are steady states.
doi:10.1088/1361-6544/aaced3 fatcat:so7iftwy5zgxdlknrv664tbfxa