Solutions with positive components to quasilinear parabolic systems

We obtain sufficient conditions for the existence and uniqueness of solutions with non-negative components to general quasilinear parabolic problems {partial derivative(t)u(k )= & sum;(n )(i,j=1)a(ij)(t, x, u)partial derivative(2)(xixj)u(k )+ & sum;(n )(i=1)b(i)(t, x, u, partial derivative(x)u)partial derivative(xi )u(k )+ c(k)(t, x, u, partial derivative(x)u), u(k) (0,x) = phi(k)(x), u(k)(t, center dot) = 0, on partial derivative F, k=1, 2, ...,m, x is an element of F, t > 0. Here, is either a bounded domain or R-n; 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 -> infinity, 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. (c) 2024 Elsevier Inc. All rights reserved.