On qualitative behavior of multiple solutions of quasilinear parabolic functional equations

We shall consider weak solutions of initial-boundary value problems for semilinear and nonlinear parabolic differential equations for $t\in (0,\infty )$ with certain nonlocal terms. We shall prove theorems on the number of solutions and certain qualitative properties of the solutions. These statements are based on arguments for fixed points of some real functions and operators, respectively, and theorems on the existence, uniqueness and qualitative properties of the solutions of partial differential equations (without functional terms).