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 u(0) = u(., 0) of the solution has finite limits theta(+/-) as x -> +/-infinity, our goal is to describe the asymptotic behavior of u(x, t) as t -> infinity. 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 -> infinity are steady states. It is known that this result is not valid in general if the limits are equal: theta(+/-) = theta(0). 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(theta(0)) not equal 0, or f(theta(0)) = 0 and theta(0) is a stable equilibrium of the equation (xi)over dot = f(xi). If f(theta(0)) = 0 and theta(0) is an unstable equilibrium of the equation (xi)over dot = f(xi), we also prove some quasiconvergence theorem making (necessarily) additional assumptions on u(0). 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. (C) 2021 Elsevier Masson SAS. All rights reserved.