Large-time behavior of solutions of parabolic equations on the real line with convergent initial data

Antoine Pauthier; Peter Poláčik

doi:10.1088/1361-6544/aaced3

Large-time behavior of solutions of parabolic equations on the real line with convergent initial data

Antoine Pauthier, Peter Poláčik

Mathematics

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8 Scopus citations

Abstract

We consider the semilinear parabolic equation ut = u_xx + 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.

Original language	English (US)
Pages (from-to)	4423-4441
Number of pages	19
Journal	Nonlinearity
Volume	31
Issue number	9
DOIs	https://doi.org/10.1088/1361-6544/aaced3
State	Published - Aug 7 2018

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© 2018 IOP Publishing Ltd & London Mathematical Society Printed in the UK.

Keywords

Parabolic equations on the real line
convergence
convergent initial data
quasiconvergence

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abstract = "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.",

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N2 - 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.

AB - 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.

KW - Parabolic equations on the real line

KW - convergence

KW - convergent initial data

KW - quasiconvergence

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Large-time behavior of solutions of parabolic equations on the real line with convergent initial data

Abstract

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Keywords

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