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LOCAL AND GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO A POLYTROPIC FILTRATION SYSTEM WITH NONLINEAR MEMORY AND NONLINEAR BOUNDARY CONDITIONS

  • Wang, Jian ;
  • Su, Meng-Long ;
  • Fang, Zhong-Bo
  • Received : 2010.11.26
  • Published : 2013.01.31

Abstract

This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system $$\{u_t=(\mid(u^{m_1})_x{\mid}^{{p_1}^{-1}}(u^{m_1})_x)_x+u^{l_{11}}{{\int_0}^a}v^{l_{12}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T),\\{v_t=(\mid(v^{m_2})_x{\mid}^{{p_2}^{-1}}(v^{m_2})_x)_x+v^{l_{22}}{{\int_0}^a}u^{l_{21}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T)}$$ with nonlinear boundary conditions $u_x{\mid}{_{x=0}}=0$, $u_x{\mid}{_{x=a}}=u^{q_{11}}u^{q_{12}}{\mid}{_{x=a}}$, $v_x{\mid}{_{x=0}}=0$, $v_x|{_{x=a}}=u^{q21}v^{q22}|{_{x=a}}$ and the initial data ($u_0$, $v_0$), where $m_1$, $m_2{\geq}1$, $p_1$, $p_2$ > 1, $l_{11}$, $l_{12}$, $l_{21}$, $l_{22}$, $q_{11}$, $q_{12}$, $q_{21}$, $q_{22}$ > 0. Under appropriate hypotheses, the authors establish local theory of the solutions by a regularization method and prove that the solution either exists globally or blows up in finite time by using a comparison principle.

Keywords

nonlinear boundary value problem;nonlinear memory;polytropic filtration system;global existence;blow-up

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