Abstract
We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.
Highlights
We study the existence and qualitative properties of traveling-wave solutions to the scalar diffusion-convection-reaction equation ρt + f (ρ)x = (D(ρ)ρx)x + g(ρ), t ≥ 0, x ∈ R
About the diffusivity D and the reaction term g we consider two different scenarios, where the assumptions are made on the pair D, g; we assume either (D1) D ∈ C1[0, 1], D > 0 in (0, 1) and D(1) = 0, Corresponding author
We prove that in both cases there is a threshold c∗ such that profiles only exists for c ≥ c∗; we study their regularity and strict monotonicity, namely whether they are classical (i.e., C1) or sharp
Summary
The former profiles are called wavefronts, the latter are semi-wavefronts; precise definitions are provided in Definition 2.1 Notice that in both cases the equilibria may be reached for a finite value of the variable ξ as a consequence of the degeneracy of D at those points. These solutions represent single-shape smooth transitions between the two constant densities 0 and 1. The main results of those papers is that there is a critical threshold c∗, depending on both f and the product Dg, such that traveling waves satisfying (1.4) exist if and only if c ≥ c∗.
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