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

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= u(· , 0) have finite limits θ^± as x→ ± ∞. We assume that f is a locally Lipschitz function on R satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u(x, t) as t→ ∞. In the first two parts of this series we mainly considered the cases where either θ^-≠ θ⁺; or θ^±= θ and f(θ) ≠ 0 ; or else θ^±= θ, f(θ) = 0 , and θ is a stable equilibrium of the equation ξ˙ = f(ξ). 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(· , t) as t→ ∞ are steady states. The limit profiles, or accumulation points, are taken in Lloc∞(R). In the present paper, we take on the case that θ^±= θ, f(θ) = 0 , and θ is an unstable equilibrium of the equation ξ˙ = f(ξ). Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on u(· , t) is that it is nonoscillatory (has only finitely many critical points) at some t≥ 0. Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal.