Transition fronts in unbounded domains with multiple branches
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Calculus of Variations
Transition fronts in unbounded domains with multiple branches Hongjun Guo1 Received: 5 March 2020 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper is concerned with the existence and uniqueness of transition fronts of a general reaction-diffusion-advection equation in domains with multiple branches. In this paper, every branch in the domain is not necessary to be straight and we use the notions of almost-planar fronts to generalize the standard planar fronts. Under some assumptions of existence and uniqueness of almost-planar fronts with positive propagating speeds in extended branches, we prove the existence of entire solutions emanating from some almost-planar fronts in some branches. Then, we get that these entire solutions converge to almost-planar fronts in some of the rest branches as time increases if no blocking occurs in these branches. Finally, provided by the complete propagation of every front-like solution emanating from one almost-planar front in every branch, we prove that there is only one type of transition fronts, that is, the entire solutions emanating from some almost-planar fronts in some branches and converging to almost-planar fronts in the rest branches. Mathematics Subject Classification 35B51 · 35J61 · 35K57
1 Introduction In this paper, we consider the following reaction-diffusion-advection equation in unbounded domains u t − div(A(x)∇u) + q(x) · ∇u = f (x, u), t ∈ R, x ∈ , (1.1) ν A(x)∇u = 0, t ∈ R, x ∈ ∂, where is a smooth non-empty open connected subset of R N with N ≥ 2 and ν(x) denotes the outward unit normal to ∂. More precise assumptions on will be given later. Such equations arise in various models in combustion, population dynamics and ecology (see [14,22,24,29,37]), where u typically stands for the temperature or the concentration of a species.
Communicated by P. Rabinowitz.
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Hongjun Guo [email protected] Department of Mathematics and Statistics, University of Wyoming, Laramie, WY, USA 0123456789().: V,-vol
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H. Guo
Throughout the paper, A(x) = (Ai j (x))1≤i, j≤N denotes a globally C 1,α (with α > 0) matrix field defined in and there exist 0 < β1 ≤ β2 such that Ai, j (x)ξi ξ j ≤ β2 |ξ |2 for all x ∈ and ξ ∈ R N . (1.2) β1 |ξ |2 ≤ 1≤i, j≤N
The vector field q(x) = (qi (x))1≤i≤N is bounded and of class C 0,α (). The term q(x) · ∇u is understood as a transport term, or a driving flow. In some sense, the flow is driven by some exogeneously given flow represented by q(x). The reaction term f (x, u) : R N × [0, 1] → R is assumed to be of class C 0,α in x ∈ R N uniformly in u ∈ [0, 1], and of C 1,1 in u uniformly in x ∈ R N . Assume that f (x, u) is Lipschitz-continuous in u uniformly for x ∈ R N . One also assumes that 0 and 1 are uniformly (in x) stable zeroes of f (x, ·) in the sense that there exist γ > 0 and σ ∈ (0, 1/2) such that f (x, u) is decreasing in u for (x, u) ∈ R N × [0, σ ] and (x, u) ∈ R N × [1 − σ, 1] and f (x, u) ≤ −γ u, for al
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