Abstract
We study the global existence and the global nonexistence of a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux. We first establish a weak comparison principle, then discuss the large time behavior of solutions by using modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained.
Highlights
RN is a bounded domain with smooth boundary ∂Ω, ν is the outward normal vector on the boundary ∂Ω, k1, k2, m, n > 0, α, β ≥ 0, p, q > 0, and u0 x, v0 x ∈ C1 Ω are positive and satisfy the compatibility conditions
The main purpose of this paper is to study the influence of nonlinear power exponents on the existence and nonexistence of global solutions of 1.1 – 1.3
If we extend the solution to 1.6 to the interval −1, 1 by symmetry, we get a solution to the same problem 1.6 with the condition at x 0, substituted by a condition at x −1, −ux −1, t λuαvp −1, t, −vx −1, t λuqvβ −1, t, t > 0
Summary
We study the global existence and the global nonexistence of a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux. We first establish a weak comparison principle, discuss the large time behavior of solutions by using modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive weak solutions are obtained.
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