Existence and uniqueness of solutions to semilinear nonlocal parabolic equations. (English) Zbl 0814.35060

Jackson, Dennis

J. Math. Anal. Appl. 172, No. 1, 256-265 (1993).

The author establishes existence and uniqueness of mild solutions for a class of second order parabolic problems of the form \[ u_ t + Au = f(u) \qquad \text{on } \Omega \times (0,T) \]
\[ u | \partial \Omega \equiv 0,\quad \partial \Omega \times (0,T),\qquad u(x,0) = \psi (x) - g(t_ 1, \dots, t_ k,u), \qquad x \in \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(k \in \mathbb{N}\) and \(f\) and \(g\) are nonlinear operators allowing for nonlocal terms. E.g., \(g = g(T,u) : = - u(T, \cdot)\), \(\psi \equiv 0\) correspond to in time \(T\)-periodic solutions of the boundary value problem, and \(f\) may stand for a Volterra integral term. The proof makes use of the contraction mapping principle and analytic semigroups.

Reviewer: G.Hetzer (Auburn)

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K65	Degenerate parabolic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

nonlocal reaction and boundary terms; mild solutions; contraction mapping principle; analytic semigroups