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

Victoria Clark; John Christopher Meyer

doi:10.1016/j.jde.2020.01.007

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

Victoria Clark, John Christopher Meyer

Mathematics

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Abstract

In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as η→∞.

Original language	English
Pages (from-to)	1401-1431
Number of pages	31
Journal	Journal of Differential Equations
Volume	269
Issue number	2
Early online date	15 Jan 2020
DOIs	https://doi.org/10.1016/j.jde.2020.01.007
Publication status	Published - 5 Jul 2020

Keywords

semi-linear parabolic PDE
well-posedness
oscillation
non-Lipschitz
Self-similar solutions
Oscillation
Semi-linear parabolic PDE
Non-Lipschitz
Well-posedness

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2020.01.007Licence: None: All rights reserved

ClarkV2020signedAccepted author manuscript, 512 KBLicence: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND)

1 Article

The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data
Meyer, J. C. & Needham, D. J., 5 Feb 2017, In: Journal of Differential Equations. 262, 3, p. 1747-1776 31 p.
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title = "On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity",

abstract = "In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as η→∞.",

keywords = "semi-linear parabolic PDE, well-posedness, oscillation, non-Lipschitz, Self-similar solutions, Oscillation, Semi-linear parabolic PDE, Non-Lipschitz, Well-posedness",

author = "Victoria Clark and Meyer, {John Christopher}",

year = "2020",

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doi = "10.1016/j.jde.2020.01.007",

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AU - Clark, Victoria

AU - Meyer, John Christopher

PY - 2020/7/5

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N2 - In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as η→∞.

AB - In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as η→∞.

KW - semi-linear parabolic PDE

KW - well-posedness

KW - oscillation

KW - non-Lipschitz

KW - Self-similar solutions

KW - Oscillation

KW - Semi-linear parabolic PDE

KW - Non-Lipschitz

KW - Well-posedness

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UR - https://arxiv.org/abs/1906.09216

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DO - 10.1016/j.jde.2020.01.007

M3 - Article

SN - 0022-0396

VL - 269

SP - 1401

EP - 1431

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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Research output

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

Cite this