Blow-up analysis for a doubly nonlinear parabolic system with multi-coupled nonlinearities

Jian Wang,Yanyan Ge

doi:10.14232/ejqtde.2012.1.1

Abstract

This paper deals with the global existence and the global nonexistence of a doubly nonlinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux. The authors first establish a weak comparison principle, then by constructing various upper and lower solutions, some appropriate conditions for global existence and global nonexistence of solutions are determined respectively.

Highlights

|∇u|k−1uxN ), Ω is a bounded domain in RN with smooth boundary ∂Ω, mi > 1, ni, αi, βi > 0, pi, qi ≥ 0, i = 1, 2. ν denotes the outer unit normal on the boundary, u0(x), v0(x) ∈ C1(Ω ) are positive and satisfy the compatibility conditions
In porous media, the theory of non-Newtonian fluids perturbed by nonlinear terms and forced by rather irregular period in time excitations, the flow of a gas through a porous medium in a turbulent regime or the spread of biological
A connection has been revealed with soil science, with flows in reservoirs exhibiting fractured media

Summary

Introduction

We consider the following problem:. (vn2 )t = ∆m2 v + up vβ1 , ∇m2 v · ν = uq vβ2 , v(x, 0) = v0(x), x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,. In [12], Li et al considered the following system with nonlinear boundary conditions (uk1 )t = ∆mu, (vk2 )t = ∆nv, ∇mu · ν = uαvp, ∇nv · ν = uqvβ, u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω They obtained necessary and sufficient conditions on the global existence of all positive (weak) solutions. A function (u(x, t), v(x, t)) is called a weak upper(or lower) solution of Problem (1.1)-(1.3) in QT if all of the following hold: (i) u, v ∈ L∞(0, T ; W 1,∞(Ω)) ∩ W 1,2(0, T ; L2(Ω)) ∩ C(QT ); (ii) (u(x, 0), v(x, 0)) ≥ (≤)(u0(x), v0(x)); (iii) For any positive two functions ψ1(x, t), ψ2(x, t) ∈ L1(0, T ; W 1,2(Ω))∩L2(QT ), one has [(un1 )tψ1 + ∇m1 u · ∇ψ1]dxdt. Since λ < 1, (0, 0) < (δ, δ) ≤ (u(x, 0), v(x, 0)) ≤ (u0(x), v0(x)), it follows from the continuity of u, v, u and v that there exists a τ > 0 sufficiently small such that λuα2 ≤ uα , vp1 ≤ vp for (x, t) ∈ Qτ

It follows that

We know that