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Constant solutions, groundstate solutions and radial terrace solutions Du, Yihong
Description
In many applications, one is interested in the nonlinear parabolic problem ut −∆u=f(u)(x∈RN,t>0), u(x,0)=u0(x)(x∈RN), where u0 ∈ L∞(RN) is nonnegative and has compact support, f is a smooth function satisfying f(0) = 0. One wants to know how much of the longtime dynamics of this problem is determined by the corre sponding elliptic problem −∆u=f(u), u≥0(x∈RN). We show that, if u(·,t) stays bounded in L∞(RN) for all t > 0, then as t → ∞, u(·,t) converges to a stationary solution in L∞loc(RN), provided that all the zeros of f(u) in [0,∞) are nondegenerate (i.e., f(u) = 0 and u ≥ 0 imply f′(u) ̸= 0). Moreover, this stationary solution is either a stable constant solution (hence a stable zero of f ), or a groundstate solution based on a stable zero of f (namely a solution v(x) of the elliptic problem which is radially symmetric about some point x0 ∈ RN , decreases in x−x0, and limx→∞ v(x) is a stable zero of f). Thus, viewed in the space L∞loc(RN), the longtime behavior of u(·,t) is determined by two simplest types of solutions of the corresponding elliptic problem. Furthermore, we show that, viewed in the space L∞(RN), the long time behavior of u(·, t) resembles that of a radial terrace solution v(x, t), whose limit as t → ∞ is determined completely by a propagating ter race of 1space dimension, which by definition is a set of traveling wave solutions {wi}ki=1, with each wi solving the elliptic equation −wzz + ciwz = f(w)(z ∈ R1), where ci is a certain constant, called the wave speed of wi. This talk is based on joint works with Peter Polacik and with Hiroshi Matano.
Item Metadata
Title 
Constant solutions, groundstate solutions and radial terrace solutions

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20150901T09:54

Description 
In many applications, one is interested in the nonlinear parabolic problem
ut −∆u=f(u)(x∈RN,t>0), u(x,0)=u0(x)(x∈RN),
where u0 ∈ L∞(RN) is nonnegative and has compact support, f is a smooth function satisfying f(0) = 0. One wants to know how much of the longtime dynamics of this problem is determined by the corre sponding elliptic problem
−∆u=f(u), u≥0(x∈RN).
We show that, if u(·,t) stays bounded in L∞(RN) for all t > 0, then as t → ∞, u(·,t) converges to a stationary solution in L∞loc(RN), provided that all the zeros of f(u) in [0,∞) are nondegenerate (i.e., f(u) = 0 and u ≥ 0 imply f′(u) ̸= 0). Moreover, this stationary solution is either a stable constant solution (hence a stable zero of f ), or a groundstate solution based on a stable zero of f (namely a solution v(x) of the elliptic problem which is radially symmetric about some point x0 ∈ RN , decreases in x−x0, and limx→∞ v(x) is a stable zero of f). Thus, viewed in the space L∞loc(RN), the longtime behavior of u(·,t) is determined by two simplest types of solutions of the corresponding elliptic problem.
Furthermore, we show that, viewed in the space L∞(RN), the long time behavior of u(·, t) resembles that of a radial terrace solution v(x, t), whose limit as t → ∞ is determined completely by a propagating ter race of 1space dimension, which by definition is a set of traveling wave solutions {wi}ki=1, with each wi solving the elliptic equation
−wzz + ciwz = f(w)(z ∈ R1),
where ci is a certain constant, called the wave speed of wi.
This talk is based on joint works with Peter Polacik and with Hiroshi Matano.

Extent 
38 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: University of New England

Series  
Date Available 
20160311

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0300080

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Faculty

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International