Bulletin of the
Korean Mathematical Society BKMS

This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system $$ \left\{\!\!\!\! \begin{array}{cc} u_{t}\! =\! (|(u^{m_1})_x|^{p_1-1}(u^{m_1})_x)_x \!+\! u^{l_{11}}\!\!\int_0^av^{l_{12}}(\xi,t)\DXI, \!(x, t)\text{ in } [0, a] \!\times\! (0,T),\\[0.5em] \!v_{t}\!=\! (|(v^{m_2})_x|^{p_2-1}(v^{m_2})_x)_x \!+\! v^{l_{22}}\!\! \int_0^au^{l_{21}}(\xi,t)\DXI, \!(x, t)\text{ in } [0, a] \!\times\! (0,T) \end{array} \right. $$ 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 $m_1, m_2\geq1$, $p_1, p_2 > 1$, $l_{11}$, $l_{12}$, $l_{21}$, $l_{22}$, $q_{11}$, $q_{12}$, $q_{21}$, $q_{22} > 0$. 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

MSC numbers: 35K55, 35K57, 35K60, 35K65