Local and global existence and blow-up of solutions to a polytropic filtration system with nonlinear memory and nonlinear boundary conditions
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This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system ( ut=(|(um1 )x|p1−1(um1 )x)x+ul11 R a 0 vl12 (, t)d,(x, t) in [0, a]×(0, T), vt=(|(vm2 )x|p2−1(vm2 )x)x+vl22 R a 0 ul21 (, t)d,(x, t) in [0, a]×(0, T) with nonlinear boundary conditions ux|x=0 = 0, ux|x=a = uq11 vq12 |x=a, vx|x=0 = 0, vx|x=a = uq21 vq22 |x=a and the initial data (u0, v0), where m1,m2 1, p1, p2 > 1, l11, l12, l21, l22, q11, q12, q21, q22 > 0. Under appropriate hypotheses, the authors establish local theory of the solu- tions by a regularization method and prove that the solution either exists globally or blows up in finite time by using a comparison principle.
Abstract
This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system ( ut=(|(um1 )x|p1−1(um1 )x)x+ul11 R a 0 vl12 (, t)d,(x, t) in [0, a]×(0, T), vt=(|(vm2 )x|p2−1(vm2 )x)x+vl22 R a 0 ul21 (, t)d,(x, t) in [0, a]×(0, T) with nonlinear boundary conditions ux|x=0 = 0, ux|x=a = uq11 vq12 |x=a, vx|x=0 = 0, vx|x=a = uq21 vq22 |x=a and the initial data (u0, v0), where m1,m2 1, p1, p2 > 1, l11, l12, l21, l22, q11, q12, q21, q22 > 0. Under appropriate hypotheses, the authors establish local theory of the solu- tions by a regularization method and prove that the solution either exists globally or blows up in finite time by using a comparison principle.
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