The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data

John Meyer; David Needham

doi:10.1016/j.jde.2016.10.027

The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data

John Meyer, David Needham

Mathematics

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

2 Citations (Scopus)

218 Downloads (Pure)

Abstract

In this paper, we establish the existence of spatially inhomogeneous classical self-similar solutions to a non-Lipschitz semi-linear parabolic Cauchy problem with trivial initial data. Specifically we consider bounded solutions to an associated two-dimensional non-Lipschitz non-autonomous dynamical system, for which, we establish the existence of a two-parameter family of homoclinic connections on the origin, and a heteroclinic connection between two equilibrium points. Additionally, we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin.

Original language	English
Pages (from-to)	1747-1776
Number of pages	31
Journal	Journal of Differential Equations
Volume	262
Issue number	3
Early online date	17 Nov 2016
DOIs	https://doi.org/10.1016/j.jde.2016.10.027
Publication status	Published - 5 Feb 2017

Keywords

semi-linear parabolic PDE
heteroclinic connection
homoclinic connection
non-Lipschitz
self-similar solutions

ASJC Scopus subject areas

Analysis

Access to Document

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Meyer_&_Needham_The_evolution_to_localized_and_front_solutions_Journal_of_differential_equations
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http://www.sciencedirect.com/science/article/pii/S0022039616303618Licence: None: All rights reserved

2 Citations
3 Article
1 Book

On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity
Clark, V. & Meyer, J., 5 Jul 2020, In: Journal of Differential Equations. 269, 2, p. 1401-1431 31 p.
Research output: Contribution to journal › Article › peer-review

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109 Downloads (Pure)
On a L^∞ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity
Meyer, J. & Needham, D., 15 Oct 2018, In: Journal of Differential Equations. 265, 8, p. 3345-3362
Research output: Contribution to journal › Article › peer-review

Open Access
File
1 Citation (Scopus)

153 Downloads (Pure)
Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations
Meyer, J. C. & Needham, D. J., 1 Aug 2016, In: Proceedings of the Royal Society of Edinburgh: Section A (Mathematics). 146, 4, p. 777-832
Research output: Contribution to journal › Article › peer-review

Open Access
File
1 Citation (Scopus)

194 Downloads (Pure)
The cauchy problem for non-lipschitz semi-linear parabolic partial differential equations
Meyer, J. & Needham, D., Nov 2015, Cambridge, UK: Cambridge University Press. 173 p. (London Mathematical Society Lecture Note Series; no. 419)
Research output: Book/Report › Book
2 Citations (Scopus)

Cite this

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title = "The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data",

abstract = "In this paper, we establish the existence of spatially inhomogeneous classical self-similar solutions to a non-Lipschitz semi-linear parabolic Cauchy problem with trivial initial data. Specifically we consider bounded solutions to an associated two-dimensional non-Lipschitz non-autonomous dynamical system, for which, we establish the existence of a two-parameter family of homoclinic connections on the origin, and a heteroclinic connection between two equilibrium points. Additionally, we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin.",

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T1 - The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data

AU - Meyer, John

AU - Needham, David

PY - 2017/2/5

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N2 - In this paper, we establish the existence of spatially inhomogeneous classical self-similar solutions to a non-Lipschitz semi-linear parabolic Cauchy problem with trivial initial data. Specifically we consider bounded solutions to an associated two-dimensional non-Lipschitz non-autonomous dynamical system, for which, we establish the existence of a two-parameter family of homoclinic connections on the origin, and a heteroclinic connection between two equilibrium points. Additionally, we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin.

AB - In this paper, we establish the existence of spatially inhomogeneous classical self-similar solutions to a non-Lipschitz semi-linear parabolic Cauchy problem with trivial initial data. Specifically we consider bounded solutions to an associated two-dimensional non-Lipschitz non-autonomous dynamical system, for which, we establish the existence of a two-parameter family of homoclinic connections on the origin, and a heteroclinic connection between two equilibrium points. Additionally, we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin.

KW - semi-linear parabolic PDE

KW - heteroclinic connection

KW - homoclinic connection

KW - non-Lipschitz

KW - self-similar solutions

UR - https://arxiv.org/abs/1607.08423

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DO - 10.1016/j.jde.2016.10.027

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JF - Journal of Differential Equations

IS - 3

ER -

The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

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Research output

On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity

On a L∞ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity

Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

The cauchy problem for non-lipschitz semi-linear parabolic partial differential equations

Cite this

On a L^∞ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity