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

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)
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