On higher order parabolic functional differential equations (Q1915643)

The author proves existence of weak solutions of the higher-order parabolic functional differential equation \[ D_tu+\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^\alpha_x[f_\alpha(t,x,u,\dots, D^\beta_xu,\dots)]+ \sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^\alpha_x[g_\alpha(t,x,u,\dots, D^\gamma_xu,\dots)]+ \] \[ \sum_{|\alpha|\leq m}(-1)^{|\alpha|} \int^t_{t-r}D^\alpha_x[h_\alpha(t,\tau,x,u,\dots, D^\gamma_xu,\dots)]d\tau= F,\;t>0,\;x\in \Omega\subset\mathbb{R}^n \] with initial condition \(u(t,x)=u_0(t,x)\), \(t\in(-r,0]\), and some homogeneous boundary condition on \(\partial\Omega\). Also, certain asymptotic properties of the solution for \(t\to\infty\) are investigated.