Existence for reaction-diffusion systems - a compactness method approach

Abstract

The authors study the existence of weak solutions for the following system: ut−Δφ(u)∈F(u,v), vt−Δψ(v)∈G(u,v) in (0,T)×Ω, φ(u)=ψ(v)=0 on (0,T)×∂Ω, u(0,x)=u0(x), v(0,x)=v0(x) in the region Ω⊂⊂Rn with smooth boundary ∂Ω. The functions ψ,φ:R→R are assumed to be continuous and nondecreasing with ψ(0)=φ(0)=0, u0,v0∈L∞(Ω), F,G:R2→2R with F an upper semicontinuous mapping (u.s.c.). The following local existence results are shown: (1) for the diffusive case, i.e. when both ψ and φ are strictly increasing with u.s.c. G; (2) for the semi-diffusive case (only one function φ is strictly increasing) with G being either with separated variables (i.e. having the form of the product or of the sum of two functions g(u) and H(v)) or globally Lipschitz with respect to its second variable (i.e. |G(u,v)−G(u,v′)|≤L|v−v′| for each u∈B⊂⊂R, v,v′∈R and some L=L(B)). Additional conditions (of linear form) on the growth of F and G are indicated to guarantee global existence results.