Asymptotic Analysis - Volume 78, issue 3

Authors: Felmer, Patricio | Quaas, Alexander

Article Type: Research Article

Abstract: In this article we study existence of boundary blow up solutions for some fractional elliptic equations including (−Δ)α u+up =f in Ω, u=g on Ωc , lim x∈Ω,x→∂Ω u(x)=∞, where Ω is a bounded domain of class C2 , α∈(0,1) and the functions f :Ω→R and g :RN \Ω− →R are continuous. We obtain existence of a solution u when the boundary value g blows up at the boundary and we get explosion rate for u under an additional assumption on the rate of explosion of g. Our results are extended for an ample class of elliptic fractional nonlinear operators of …Isaacs type. Show more

Keywords: fractional Laplacian, boundary blow up, Isaacs operators, supersolution, subsolution

DOI: 10.3233/ASY-2011-1081

Citation: Asymptotic Analysis, vol. 78, no. 3, pp. 123-144, 2012

Authors: Arrieta, Jose M. | López-Fernández, María | Zuazua, Enrique

Article Type: Research Article

Abstract: We consider an evolution equation of parabolic type in R having a travelling wave solution. We study the effects on the dynamics of an appropriate change of variables which transforms the equation into a non-local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. This procedure allows us to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples. We analyze the relation of the new equation with the original one in the entire real line. We also …analyze the behavior of the non-local problem in a bounded interval with appropriate boundary conditions. We show that it has a unique stationary solution which approaches the traveling wave as the interval gets larger and larger and that is asymptotically stable for large enough intervals. Show more

Keywords: travelling waves, reaction–diffusion equations, implicit coordinate-change, non-local equation, asymptotic stability, numerical approximation

DOI: 10.3233/ASY-2011-1088

Citation: Asymptotic Analysis, vol. 78, no. 3, pp. 145-186, 2012