The ANZIAM Journal

Research Article

GLOBAL EXISTENCE AND BLOW-UP FOR A NON-NEWTON POLYTROPIC FILTRATION SYSTEM WITH NONLOCAL SOURCE

JUN ZHOUa1 c1 and CHUNLAI MUa1

a1 School of Mathematics and Physics, Chongqing University, Chongqing, 400044, People’s Republic of China (email: zhoujun_math@hotmail.com)

Abstract

This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system with nonlocal source,

\[ u_t-\Delta _{m,p}u=a\int _{\Omega }v^\alpha (x,t)\,dx,\quad v_t-\Delta _{n,q}v=b\int _{\Omega }u^\beta (x,t)\,dx. \]

Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depending on the initial data and the relations between αβ and mn(p−1)(q−1). In the special case, α=n(q−1), β=m(p−1), we also give a criteria for the solution to exist globally or blow up in finite time, which depends on a,b and ζ(x),xs03D1(x) as defined in our main results.

(Received April 17 2007)

(Revised September 19 2008)

2000 Mathematics subject classification

  • primary 35K50; secondary 35K55;
  • 35K65;
  • 35B33

Keywords and phrases

  • non-Newtonian polytropic system;
  • nonlocal source;
  • global existence;
  • blow-up

Correspondence:

c1 For correspondence; e-mail: zhoujun_math@hotmail.com