Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

Cyrill B. Muratov; Xing Zhong

doi:10.3934/dcds.2017038

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2017, Volume 37, Issue 2: 915-944. Doi: 10.3934/dcds.2017038

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Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

Cyrill B. Muratov and
Xing Zhong¹

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA

Author Bio: E-mail address: muratov@njit.edu

¹The author is deceased.

Received: May 31, 2015

Revised: December 31, 2015

Published: May 2017

This work was supported, in part, by NSF via grants DMS-0908279, DMS-1119724 and DMS-1313687. CBM wishes to express his gratitude to V. Moroz for many valuable discussions.

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Abstract

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $\mathbb{R}^N$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $L^2$ initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.

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Mathematics Subject Classification: Primary:35K57, 35K15, 35A15.

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Table 1. List of critical exponents.

Name	Exponent	Validity	$N = 3$
Fujita	$p_F = (N + 2)/N$	$N \geq 1$	5/3
Serrin	$p_{sg} = N / (N - 2)$	$N \geq 3$	3
Sobolev	$p_S = (N + 2) / (N - 2)$	$N \geq 3$	5
Joseph-Lundgren	$p_{JL} = 1 + 4/ \left( N - 4 - 2 \sqrt{N - 1} \, \right)$	$N \geq 11$	-

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