Large-time behavior of solutions of parabolic equations on the real line with convergent initial data II: Equal limits at infinity

We continue our study of bounded solutions of the semilinear parabolic equation u_t=u_xx+f(u) on the real line, where f is a locally Lipschitz function on R. Assuming that the initial value u₀=u(⋅,0) of the solution has finite limits θ^± as x→±∞, our goal is to describe the asymptotic behavior of u(x,t) as t→∞. In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as t→∞ are steady states. It is known that this result is not valid in general if the limits are equal: θ^±=θ₀. In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either f(θ₀)≠0, or f(θ₀)=0 and θ₀ is a stable equilibrium of the equation ξ˙=f(ξ). If f(θ₀)=0 and θ₀ is an unstable equilibrium of the equation ξ˙=f(ξ), we also prove some quasiconvergence theorem making (necessarily) additional assumptions on u₀. A major ingredient of our proofs of the quasiconvergence theorems—and a result of independent interest—is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.