Solutions with positive components to quasilinear parabolic systems

Abstract

We obtain sufficient conditions for the existence and uniqueness of solutions with non-negative components to general quasilinear parabolic problems(P){∂tuk=∑i,j=1naij(t,x,u)∂xixj2uk+∑i=1nbi(t,x,u,∂xu)∂xiuk+ck(t,x,u,∂xu),uk(0,x)=φk(x),uk(t,⋅)=0,on ∂F,k=1,2,…,m,x∈F,t>0. Here, F is either a bounded domain or Rn; in the latter case, we disregard the boundary condition. We apply our results to study the existence and asymptotic behavior of componentwise non-negative solutions to the Lotka-Volterra competition model with diffusion. In particular, we show the convergence, as t→+∞, of the solution for a 2-species Lotka-Volterra model, whose coefficients vary in space and time, to a solution of the associated elliptic problem.