On different types of nonlinear parabolic functional differential equations. (English) Zbl 0928.35183

From the introduction: We consider weak solutions of initial-boundary value problems for the equation \[ \begin{gathered} D_tu(t,x)=\sum_{| \alpha| \leq m}(-1)^{| \alpha| } D^\alpha_x[f_\alpha(t,x,\dots, D^\beta_x u(t,x),\dots)]\tag{1} \\+\sum_{| \alpha| \leq m-1} D^\alpha_x [H_\alpha(u)](t,x)=F(t,x), \quad (t,x)\in Q_T=(0,T)\times \Omega \end{gathered} \] where \(| \beta| \leq m\), \(\Omega \subset \mathbb{R}^n\) is a bounded domain and, denoting by \(V\) a closed linear subspace of the Sobolev space \(W^{m,p}(\Omega)\) \((m\geq 1,p\geq 2)\), \[ H_\alpha:L^p(0,T;V)\to L^q(Q_T) \] is a bounded (nonlinear) operator \((1/p+1/q=1)\). The real valued functions \(f_\alpha\) have polynomial growth in \(D^\beta_x u\).
The main purpose of this paper is to consider operators \(H_\alpha\) of general form such that they contain different types of delay operators (including delay in the boundary condition).


35R10 Partial functional-differential equations
35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations