Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity

Abstract

This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation u_t=u_{xx}+f(u) on the real line whose initial data u_0=u(cdot ,0) have finite limits theta ^pm as xrightarrow pm infty . We assume that f is a locally Lipschitz function on mathbb {R} satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u(x, t) as trightarrow infty . In the first two parts of this series we mainly considered the cases where either theta ^-ne theta ^+; or theta ^pm =theta _0 and f(theta _0)ne 0; or else theta ^pm =theta _0, f(theta _0)=0, and theta _0 is a stable equilibrium of the equation {{dot{xi }}}=f(xi ). In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of u(cdot ,t) as trightarrow infty are steady states. The limit profiles, or accumulation points, are taken in L^infty _{loc}(mathbb {R}). In the present paper, we take on the case that theta ^pm =theta _0, f(theta _0)=0, and theta _0 is an unstable equilibrium of the equation {{dot{xi }}}=f(xi ). Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on u(cdot ,t) is that it is nonoscillatory (has only finitely many critical points) at some tge 0. Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal.