Blowup for degenerate and singular parabolic system with nonlocal source

Jun Zhou,Chunlai Mu,Zhongping Li

doi:10.1155/bvp/2006/21830

Abstract

We deal with the blowup properties of the solution to the degenerate and singular parabolic system with nonlocal source and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained. Furthermore, under certain conditions it is proved that the blowup set of the solution is the whole domain.

Highlights

Since |q1| + r1 = 0, |q2| + r2 = 0, the coefficients of ut, ux, uxx and vt, vx, vxx may tend to 0 or ∞ as x tends to 0, we can regard the equations as degenerate and singular
Floater [9] and Chan and Liu [4] investigated the blowup properties of the following degenerate parabolic problem: xqut − uxx = up, (x, t) ∈ (0, a) × (0, T), u(0, t) = u(a, t) = 0, t ∈ (0, T), u(x, 0) = u0(x), x ∈ [0, a], (1.2)
Under certain conditions on the initial datum u0(x), Floater [9] proved that the solution u(x,t) of (1.2) blows up at the boundary x = 0 for the case 1 < p ≤ q + 1

Summary

Introduction

In this paper, we consider the following degenerate and singular nonlinear reactiondiffusion equations with nonlocal source: a xq1 ut − xr1 ux x = vp1 dx, (x, t) ∈ (0, a) × (0, T), xq2 vt − xr2 vx x = up2 dx, (x, t) ∈ (0, a) × (0, T), u(0, t) = u(a, t) = v(0, t) = v(a, t) = 0, t ∈ (0, T), u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ [0, a], where u0(x), v0(x) ∈ C2+α(D) for some α ∈ (0, 1) are nonnegative nontrivial functions. u0(0) = u0(a) = v0(0) = v0(a) = 0, u0(x) ≥ 0, v0(x) ≥ 0, u0, v0 satisfy the compatibility condition, T > 0, a > 0, r1, r2 ∈ [0, 1), |q1| + r1 = 0, |q2| + r2 = 0, and p1 > 1, p2 > 1. Let D = (0, a) and Ωt = D × (0, t], D and Ωt are their closures, respectively. Since |q1| + r1 = 0, |q2| + r2 = 0, the coefficients of ut, ux, uxx and vt, vx, vxx may tend to 0 or ∞ as x tends to 0, we can regard the equations as degenerate and singular. Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 21830, Pages 1–19 DOI 10.1155/BVP/2006/21830

Blowup for degenerate and singular parabolic system

Local existence

Blowup of solution