On a class of nonlocal porous medium equations of Kirchhoff type

We study the Dirichlet problem for the degenerate parabolic equation of the Kirchhoff type u(t) - a (parallel to u parallel to(p)(Lp)(Omega) Sigma D-n(i=1)i(vertical bar u vertical bar(p-2)D(i)u) + b(x, t, u) - f(x, t) in Q(T) - Omega x (0, T), where p >= 2, T > 0, Omega subset of R-n, n >= 2, is a smooth bounded domain. The coefficient a(.) is real-valued function defined on R+ and b(., ., tau) is a measurable function with variable nonlinearity in tau. We prove existence of weak solutions of the considered problem under appropriate and general conditions on a and b. Sufficient conditions for uniqueness are found and in the case f equivalent to 0 the decay rates for parallel to u parallel to(L2(Omega)) are obtained.