Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system

Abstract

The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$ u_t = v^p \left( {\Delta u + a\int_\Omega {udx} } \right),v_t = u^q \left( {\Delta v + b\int_\Omega {vdx} } \right) $$ with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution (u, v) to this problem. Moreover, a necessary and sufficient condition for the non- global existence of the solution is obtained. Under some further conditions on the initial data, we get criteria for the finite time blow-up of the solution.