Monotonicity of stable solutions in shadow systems
type: Item
instance of: scholarly article
title: Monotonicity of stable solutions in shadow systems
zbMATH Open document ID: 0981.35018
DOI: 10.1090/S0002-9947-01-02880-X
author: Wei-Ming Ni
author: Peter Poláčik
author: Eiji Yanagida
published in: Transactions of the American Mathematical Society
publication date: 2001-10-21T00:00:00Z
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1663178
zbMATH Keywords: spatially nonhomogeneous solutions are unstable
zbMATH Keywords: evolution operators
MaRDI profile type: Publication
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full work available at URL: s0002-9947-01-02880-x
OpenAlex ID: W1905893663
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review text: The authors study the following reaction-diffusion system NEWLINE\[NEWLINE\begin{aligned} & u_t= u_{xx}+ f(u, v,t),\quad x\in (0,1),\quad t> 0,\quad u_x(0, t)= u_x(1,t)= 0,\\ & v_t= \int^1_c g(u,v,t) dx,\end{aligned}\tag{1}NEWLINE\]NEWLINE where \(u(x,t)\in\mathbb{R}\) and \(v= v(t)\in \mathbb{R}^n\). This system appears as the limit in some sense of the reaction-diffusion system NEWLINE\[NEWLINEu_t= u_{xx}+ f(u, v,t),\quad v_t= \varepsilon^{-2} v_{xx}+ g(u,v,t)NEWLINE\]NEWLINE subject to homogeneous Neumann boundary conditions, as is shown in the cited references. The functions \(f\), \(g\) are subject to certain smoothness and boundedness conditions, labeled as (H1), (H2). The authors now study the stability of bounded solutions of (1). Since (1) is nonautonomous, this notion needs some care. Following \textit{D. Henry} [Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag (1981; Zbl 0456.35001)], they consider the evolution operators \(T(t, s)\), \(t> s\geq 0\) associated with the linearization of (1) at a given bounded solution \((u(t), v(t))\), \(t\geq 0\). This solution is then linearly stable if there are \(\lambda\), \(C>0\) such that \(\|T(t,s)\|\leq Ce^{-\lambda(t- s)}\), \(t> s\geq 0\). There is also a related notion of exponential instability which is given in the paper.NEWLINENEWLINENEWLINETwo conditions are imposed on the bounded conditions under scrutinyNEWLINENEWLINENEWLINE(A1) \((u(x,t), v(t))\) is uniformly bounded on \([0,1]\times [0,\infty)\),NEWLINENEWLINENEWLINE(A2) \(\|u_x(\varepsilon, t)\|_{L^\infty}\geq d\), \(t>0\), for some \(d>0\).NEWLINENEWLINENEWLINEThe main result of the paper states Theorem 1: Assume (H1), (H2) and let \((u(x,t), v(t))\), \(t\geq 0\) be a solution to (1) subject to (A1) and (A2). Then there is \(t_0\) such that \(u_x(x,t)\neq 0\) for all \((x,t)\in (0,1)\times (t_0,\infty)\).NEWLINENEWLINENEWLINEA corollary states that if \(f(u,v,t)\), \(g(u,v,t)\) are \(\tau\)-periodi…
reviewed by: Bruno Scarpellini
description: scientific article; zbMATH DE number 1663178