Monotonicity and convergence results in order-preserving systems in the presence of symmetry

Abstract

This paper deals with various applications of two basic theorems in orderpreserving systems under a group action -monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equations and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudotraveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.2000 Mathematics Subject Classification. 35K40, 35K57. Key words and phrases. order-preserving systems, semilinear elliptic equations, traveling waves and pseudo-traveling waves, quasilinear diffusion equations.

Speed Selection and Stability of Wavefronts for Delayed Monostable Reaction-Diffusion Equations

“…The studies of wavefront stability in monostable monotone delayed model (1) (including its non-local and discrete Laplacian versions) were initiated in 2004-2005 by Mei et al [29] and Ma and Zou [21]. Their research was influenced by a series of previous results about a) the existence of monotone wavefronts [20,46]; b) the stability of wavefronts in delayed bistable equations [31,38] and discrete monostable equations [5]. Over the last decade, the wave stability problem for equation (1) has attracted attention of many other mathematicians so that it would be difficult to mention all interesting findings in this area.…”

Dynamics of time-periodic reaction-diffusion equations with compact initial support on R

“…This lemma follows directly from [4, Lemma 4.3] by a sliding method. Let us mention that it can also be proved by a dynamical system approach used in an earlier work [20] (see Corollary 8.3 and Proposition B.2 in Appendix 2 of [20]).…”

Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction

“…Hence, the main aim of the present paper is to study the stability properties of monostable pushed fronts to the monotone delayed model (1). We are going to achieve this goal by developing several ideas and methods from [10,30,33,42]. We also will establish the asymptotic convergence of solutions for the initial value problem (1), (2) to an appropriate pushed wavefront when, in addition to (H), g is monotone and when w 0 satisfies, for some A, B > 0, σ ∈ (0, κ) and µ > λ 1 (c * ) the following conditions (IC): At first glance, if additionally we assume the monotonicity of g, Proposition 1.2 seems to follow from quite general results on spreading speeds to continuous-time semiflows established in [22,23].…”

“…In order to prove the above theorem, instead of looking for an appropriate Lyapunov functional (as it was done in [10,33]) for functional differential equation (10), we will use the Berestycki and Nirenberg method of the sliding solutions as well as some ideas of the approach developed by Ogiwara and Matano in [30]. Proof.…”