Abstract
We study traveling waves bifurcating from stable standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. We prove the existence of weekly decaying traveling fronts that emerge in the presence of a weakly stable direction on a center manifold. Moreover, we show the existence of bifurcating traveling waves of constant mass. The main difficulty is to prove the smoothness of the ansatz in exponentially weighted spaces required to apply the Lyapunov-Schmidt methods.
Highlights
In this paper we prove the existence traveling fronts bifurcating from layers in a class of parabolic systems that couple a scalar conservation law with a scalar reaction–diffusion equation
From a theoretical point of view, (1.1) is interesting as a system just slightly more complex than a scalar equation: the steady-state problem can be readily seen to reduce to a scalar equation after integrating the first equation for u as a function of v and substituting the result into the second equation
We point out that there exists a special class of traveling waves bifurcating from the standing layer with constant mass u∞ L, under the additional, generic assumption that b′(u∞ L ) ̸= 0
Summary
In this paper we prove the existence traveling fronts bifurcating from (standing) layers in a class of parabolic systems that couple a scalar conservation law with a scalar reaction–diffusion equation. In different circumstances, crossing of a zero eigenvalue of the linearization at a standing layer induces bifurcation of traveling fronts; see [3, 4, 7, 11, 16]. Assuming Hypothesis 1, there exists a locally unique family of traveling fronts with weak decay, parameterized by ±s ∈ [0, s0) (speed) and ε ∈ (−ε0, ε0) (bifurcation parameter), bifurcating from the standing layer. We point out that there exists a special class of traveling waves bifurcating from the standing layer with constant mass u∞ L , under the additional, generic assumption that b′(u∞ L ) ̸= 0. One is usually interested in the stability of solutions While this question is of interest for the full two-parameter family found in Theorem 1.1, the analysis in this general setting is quite intricate.
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