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 \begin{equation*} \partial_t u^k = \sum_{i,j=1}^n a_{ij} (t,x,u)\partial^2_{x_i x_j}\!u^k + \sum_{i=1}^n b_i (t,x,u, \partial_x u) \partial_{x_i} u^k +\, c^k(t,x,u,\partial_x u), \\ u^k(0,x) = \varphi^k(x), \\ u^k(t,\,\cdot\,) = 0, \quad \text{on } \partial \mathbb F, k=1,2, \dots, m, \quad x\in\mathbb F, \;\; t>0. \end{equation*} Here, $\mathbb F$ is either a bounded domain or $\mathbb R^n$; 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 $t\to+\infty$, 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.