Showing papers by "Thierry Gallay published in 1998"
TL;DR: In this article, the authors studied a one-dimensional reaction-diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic (A+B --> 2B) and a cubic (A + 2B --> 3B) auto-catalysis.
35 citations
TL;DR: In this article, the Ginzburg-Landau equation on the real line has spatially periodic steady states of the form "with" and "without" with the boundary conditions, and the existence of the limiting profile is established as an application of monotone operators.
Abstract: The Ginzburg–Landau equation \(\) on the real line has spatially periodic steady states of the form \(\), with \(\) and \(\). For \(\), \(\), we construct solutions which converge for all t>0 to the limiting pattern \(\) as \(\). These solutions are stable with respect to sufficiently small \(\) perturbations, and behave asymptotically in time like \(\), where \(\) is uniquely determined by the boundary conditions \(\). This extends a previous result of [BrK92] by removing the assumption that \(\) should be close to zero. The existence of the limiting profile \(\) is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.
29 citations
Posted Content•
TL;DR: It is proved that the traveling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space and approach a universal self-similar profile, which is independent of $\epsilon$, F, and the initial data.
Abstract: We consider the damped hyperbolic equation (1) \epsilon u_{tt} + u_t = u_{xx} + F(u), x \in R, t \ge 0, where \epsilon is a positive, not necessarily small parameter. We assume that F(0) = F(1) = 0 and that F is concave on the interval [0,1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This family is indexed by a parameter c \ge c_* related to the speed of the front. In the critical case c=c_*, we prove that the travelling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space. In addition, we show that the perturbations decay to zero like t^{-3/2} as t \to +\infty and approach a universal self-similar profile, which is independent of \epsilon, F and of the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting \epsilon = 0 in Eq.(1). The proof of our results relies on careful energy estimates for the equation (1) rewritten in self-similar variables x/\sqrt{t}, \log t.
1 citations