Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

Abstract

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i (u i ) may have the property D i (0) = 0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a twocomponent competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.

Quenching problems for a semilinear reaction-diffusion system with singular boundary outflux

“…It is well known that one of the most effective methods to obtain existence and uniqueness results of the solutions of parabolic equations and systems with initial conditions is monotone iterative technique (for details see [1], [5] and [6]). [0, T )) the set of functions that are twice continuously differentiable in x and once continu-…”

Stokes' first problem for a Newtonian fluid in a non‐Darcian porous half‐space using a Laguerre–Galerkin method

“…The well-posedness of (33) in the classical sense can be found in [22]. This flow has discontinuous boundary condition.…”

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

“…[12,28,33,38,39,43]). The recent work in [34] treated a general system of quasilinear parabolic equations with coupled non-linear boundary conditions. In another direction, the study of strongly degenerated equations whose sets of degenerate points contain non-isolated points has attracted increasing attention.…”

“…The global existence and blow-up problem has been extended in [11,12,16,17,22,23,28,45] Recently the authors treated a general coupled system of N equations in the form of (1.1) but with non-linear boundary conditions (cf. [34]). An important difference between the system in [34] and the present system (1.1) is that in the Neumann-Robin type of boundary conditions, including non-linear boundary conditions, it is possible to construct a positive lower bound of the solution for the degenerate case D i (0) = 0.…”

“…[34]). An important difference between the system in [34] and the present system (1.1) is that in the Neumann-Robin type of boundary conditions, including non-linear boundary conditions, it is possible to construct a positive lower bound of the solution for the degenerate case D i (0) = 0. However, for the Dirichlet boundary condition u i (t, x) = 0 (t > 0, x ∈ ∂Ω), i = 1, .…”