RT Book, Section
T1 Existencia de ondas viajeras con frentes en un sistema parabólico semilineal
A1 Esquinas, J.
A1 Herrero, Miguel A.
A2 Díaz Díaz, Jesús Ildefonso
A2 Vegas Montaner, José Manuel
AB 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. Theni) 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·
PB Universidad Complutense de Madrid
SN 84-7491-278-4
YR 1989
FD 1989
LK https://hdl.handle.net/20.500.14352/60768
UL https://hdl.handle.net/20.500.14352/60768
NO Proceedings of the mathematical meeting in honor of A. Dou held on June, 17th, 1988 in Madrid, Spain
DS Docta Complutense
RD 4 abr 2025