Long-time behaviour of solutions of a non-linear diffusion problem with non-local source term

paper is devoted to the long-time behaviour of solutions for the Dirichlet problem of the non-local porous medium equation u(t) = Delta(u(m)) + lambda f (u)/(integral Omega f(u)dx)(q) for the case f(u) = a + u(p) with m > p >= I, lambda, q > 0 and a > 0 We first prove the existence and uniqueness of the solution of the associated steady-state problem Then, for the non-negative initial data u(0)(x) satisfying u(0)(m) is an element of C-1 ((Omega) over bar) and u(0) = 0 on partial derivative Omega, we discuss the asymptotic stability of the unique steady-state solution and the estimate of the convergence rate, at last we give the numerical simulations.