We consider the semilinear parabolic equation ut = uxx + 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 profles as t → ∞ are steady states.
Bibliographical noteFunding Information:
1 Supported in part by the NSF Grant DMS-1565388.
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- Parabolic equations on the real line
- convergent initial data