Please use this identifier to cite or link to this item: https://hdl.handle.net/1959.11/18391
Title: Locally Uniform Convergence to an Equilibrium for Nonlinear Parabolic Equations on ℝN
Contributor(s): Du, Yihong  (author)orcid ; Polacik, Peter (author)
Publication Date: 2015
DOI: 10.1512/iumj.2015.64.5535
Handle Link: https://hdl.handle.net/1959.11/18391
Abstract: 

We consider bounded solutions of the Cauchy problem

{Ut - Δu = ƒu, x ∈ℝN, t > 0,
u(O, x) = uo (x), x ∈ℝN,
where u0 is a non-negative function with compact support and ƒ is a C1 function on ℝ with ƒ (O) = 0. Assuming that ƒ' is locally Hölder continuous, and that ƒ satisfies minor nondegeneracy condition we prove that as t → ∞ the solution u(∙, t) converges to an equilibriun φ locally uniformly in ℝN. Moreover, either the limit function φ is a constant equilibrium, or there is a point xo ∈ ℝN such that φ is radically symmetric and radially decreasing about xo, and it approaches a constant equilibrium as |x - xo| → ∞. The nondegeneracy condition only concerns a specific set of zeros of ƒ and we make no assumption whatsoever on the nonconstant equilibria. The set of such equilibria can be very complicated and indeed a complete understanding of this set is usually beyond reach in dimension N ⋝ 2. Moreover, because of the symmetries of the equation there are always continua of such equilibria. Our result shows that the assumption "uo has compact support" is powerful enough to guarantee that, first, the equilibria that can possibly be observed in the w-limit set of u have a rather simple structure; and, second, exactly one of them is selected. Our convergence result remains valid if Δu is replaced by a general elliptic operator of the form ∑i7jaijuxixj with constant coefficients aij.
Publication Type: Journal Article
Grant Details: ARC/DP130102773
Source of Publication: Indiana University Mathematics Journal, 64(3), p. 787-824
Publisher: Indiana University, Department of Mathematics
Place of Publication: United States of America
ISSN: 1943-5258
0022-2518
1943-5266
Fields of Research (FoR) 2008: 010110 Partial Differential Equations
Fields of Research (FoR) 2020: 490410 Partial differential equations
Socio-Economic Objective (SEO) 2008: 970101 Expanding Knowledge in the Mathematical Sciences
Socio-Economic Objective (SEO) 2020: 280118 Expanding knowledge in the mathematical sciences
Peer Reviewed: Yes
HERDC Category Description: C1 Refereed Article in a Scholarly Journal
Appears in Collections:Journal Article
School of Science and Technology

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