Existencia de ondas viajeras con frentes en un sistema parabólico semilineal

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·