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Keywords:
strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up
Summary:
The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$ u_t = v^p\biggl (\Delta u + a\int _\Omega u \,{\rm d} x\biggr ),\quad v_t =u^q\biggl (\Delta v + b\int _\Omega v \,{\rm d} x\biggr ) $$ 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.
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