Local and global existence and blow-up of solutions to a polytropic filtration system with nonlinear memory and nonlinear boundary conditions

摘要

This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system $$ left{!!!! egin{array}{cc} u_{t}! =! (|(u^{m_1})_x|^{p_1-1}(u^{m_1})_x)_x !+! u^{l_{11}}!!int_0^av^{l_{12}}(xi,t)DXI, !(x, t)ext{ in } [0, a] !imes! (0,T),[0.5em] !v_{t}!=! (|(v^{m_2})_x|^{p_2-1}(v^{m_2})_x)_x !+! v^{l_{22}}!! int_0^au^{l_{21}}(xi,t)DXI, !(x, t)ext{ in } [0, a] !imes! (0,T) end{array} ight. $$ with nonlinear boundary conditions $u_x|_{x = 0} = 0$, $u_x|_{x = a} = u^{q_{11}}v^{q_{12}}|_{x = a}$, $v_x|_{x = 0} = 0$, $v_x|_{x = a} = u^{q_{21}}v^{q_{22}}|_{x = a}$ and the initial data ($u_{0}$, $v_{0}$), where $m_1, m_2geq1$, $p_1, p_2 > 1$, $l_{11}$, $l_{12}$, $l_{21}$, $l_{22}$, $q_{11}$, $q_{12}$, $q_{21}$, $q_{22} > 0$. 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.