On parabolic functional differential equations in unbounded domains. (English) Zbl 1090.35562

Simon, László

Arch. Math., Brno 36, Suppl., 603-616 (2000).

The author considers an initial-boundary value problem for parabolic functional differential equations of the form \(D_tu(x,t) - \sum _{j=1}^n D_j[f_j(t,x,u(t,x),\nabla u(t,x))] + f_0(t,x,u(t,x), \nabla u(t,x)) + g(t,x,u(t,x)) + h(t,x,[H(u)](t,x))= F(t,x),\) where \(g,h\) are discontinuous in the unknown function. The existence of weak solutions is proved under certain growth conditions on the functions \(f_i, g, h\) and conditions on the operator \(H\). Further, boundedness of solutions and stability as \(t \to \infty \) is established. The results are new in the case of unbounded domain, the bounded case was considered by the author earlier.

Reviewer: Martin Kolář (Brno)

MSC:

35R10

Partial functional-differential equations

Keywords:

partial functional differential equations; differential equations with discontinuous terms