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

Title & Authors
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;

Abstract
This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system $\small{\{u_t=(\mid(u^{m_1})_x{\mid}^{{p_1}^{-1}}(u^{m_1})_x)_x+u^{l_{11}}{{\int_0}^a}v^{l_{12}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T),\\{v_t=(\mid(v^{m_2})_x{\mid}^{{p_2}^{-1}}(v^{m_2})_x)_x+v^{l_{22}}{{\int_0}^a}u^{l_{21}}({\xi},t)d{\xi},\;(x,t)\;in\;[0,a]{\times}(0,T)}}$ with nonlinear boundary conditions $\small{u_x{\mid}{_{x=0}}=0}$, $\small{u_x{\mid}{_{x=a}}=u^{q_{11}}u^{q_{12}}{\mid}{_{x=a}}}$, $\small{v_x{\mid}{_{x=0}}=0}$, $\small{v_x|{_{x=a}}=u^{q21}v^{q22}|{_{x=a}}}$ and the initial data ($\small{u_0}$, $\small{v_0}$), where $\small{m_1}$, $\small{m_2{\geq}1}$, $\small{p_1}$, $\small{p_2}$ > 1, $\small{l_{11}}$, $\small{l_{12}}$, $\small{l_{21}}$, $\small{l_{22}}$, $\small{q_{11}}$, $\small{q_{12}}$, $\small{q_{21}}$, $\small{q_{22}}$ > 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;
Language
English
Cited by
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