Existence and attractivity results for a class of degenerate functional- parabolic problems (Q1118755)

We want to study existence, uniqueness and asymptotical behaviour of the solutions of the following problem: \[ \partial_ tu(t,x)=\Delta u^ m(t,x)+a(x)u(t,x)-b(x)u^ 2(t,x)- \] \[ -u(t,x)\int^{t}_{-\infty}ds k(t-s,x)u(s,x)\quad in\quad (0,\infty)\times \Omega, \] \[ u(t,x)=0\quad in\quad (0,\infty)\times \partial \Omega;\quad u(t,x)=u_ 0(t,x)\quad in\quad (-\infty,0]\times \Omega. \]