The blow-up properties for a degenerate semilinear parabolic equation with nonlocal source

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u t − (x a u x ) x =∫ 0 a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u p , p>1.