# Existencia de ondas viajeras con frentes en un sistema parabólico semilineal

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Esquinas, J. and Herrero, Miguel A. (1989) Existencia de ondas viajeras con frentes en un sistema parabólico semilineal. In Actas de la Reunión Matemática en Honor de A. Dou. Universidad Complutense de Madrid, Madrid, pp. 121-127. ISBN 84-7491-278-4 ==>>> Export to other formats

## Abstract

The authors consider the system, defined for t>0, -∞<x<∞,(1)u t -u xx +v p =0,v t -v xx +u q =0,0<p,q<∞,and their solutions of the form (2)u(x,t)=φ(ct-x),v(x,t)=ψ(ct-x),φ(ξ),ψ(ξ)nonnegative and different from zero, nondecreasing in ξ, φ, ψ∈C 2 (-∞,+∞). If for a certain real ξ 0 φ(ξ)=ψ(ξ)=0 when ξ≤ξ 0 , these solutions (u,v)=(φ,ψ) will be called a finite travelling wave (FTV). In the case here considered, the FTV are unbounded. The main results are:
Theorem 1. There exist FTV of (1) if and only if pq<1. In this case, for every real c there exists a FTV with speed c, and the corresponding profiles φ,ψ are unique up to space and time translations. Definition: f(ξ)≈g(ξ) as ξ→ξ 0 (finite or not) if lim ξ→ξ 0 f(ξ)/g(ξ)=1·
Theorem 2. Let pq<1 and, for every real c, let (φ,ψ) be the FTV with speed c of Theorem 1. Then
i) For every real c, φ(ξ)≈Aξ α , ψ(ξ)≈Bξ β as ξ→0 + . Here α=2(1+p)/(1-pq), β=2(1+q)/(1-pq), A 1-pq =[β(β-1) p α(α-1)] -1 , B=A q (β(β-1)) -1 ·
ii) If c<0, φ(ξ)≈cξ γ , ψ(ξ)≈Dξ δ when ξ≫0, where γ=(1+p)/(1-pq), δ=(1+q)/(1-pq), c 1-pq =[(1-c) 1+p δ p γ] -1 , D=C p [(-c)δ] -1 ·
iii) When c>0, φ (ξ)≈Mexpcξ, ψ (ξ)≈Nexpcξ, where the constants M,N have different dependencies on c, p,q according to p<1, q<1; p<1, q=1; p<1, q>1·

Item Type: Book Section Proceedings of the mathematical meeting in honor of A. Dou held on June, 17th, 1988 in Madrid, Spain Semilinear system; finite travelling wave Sciences > Mathematics > Differential equations 22724 05 Sep 2013 16:04 12 Dec 2018 15:08