Radial equivalence and applications to the qualitative theory for a class of nonhomogeneous reaction-diffusion equations

Abstract

Some transformations acting on radially symmetric solutions to the followingclass of nonhomogeneous reaction-diffusion equations|x|𝜎1𝜕tu=Δum+|x|𝜎2up,(x,t)∈RN×(0,∞),which has been proposed in a number of previous mathematical works as wellas in several physical models, are introduced. We consider herem≥1,p≥1,N≥1, and𝜎1,𝜎2real exponents. We apply these transformations in connec-tion to previous results on the one hand to deduce general qualitative propertiesof radially symmetric solutions and on the other hand to construct self-similarsolutions, which are expected to be patterns for the dynamics of the equations,strongly improving the existing theory. We also introduce mappings betweensolutions which work in the semilinear casem=1