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

School of Mathematics

Research output: Contribution to journal › Article › peer-review

9 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

Bibliographical note

Publisher Copyright:
© 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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10.1088/1361-6544/aaced3

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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.",

keywords = "Parabolic equations on the real line, convergence, convergent initial data, quasiconvergence",

author = "Antoine Pauthier and Peter Pol{\'a}{\v c}ik",

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AU - Pauthier, Antoine

AU - Poláčik, Peter

PY - 2018/8/7

Y1 - 2018/8/7

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

UR - https://www.scopus.com/pages/publications/85051676840

UR - https://www.scopus.com/pages/publications/85051676840#tab=citedBy

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DO - 10.1088/1361-6544/aaced3

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VL - 31

SP - 4423

EP - 4441

JO - Nonlinearity

JF - Nonlinearity

IS - 9

ER -

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

Abstract

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