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On some model diffusion problems with a nonlocal lower order term. (English) Zbl 1039.35056
The authors consider a mixed initial boundary value problem for a semilinear parabolic equation of the type $u_t-\Delta u+ a(l(u(t)))u=f,\quad l(u(t))=\int_\Omega g(x)u(x,t)\,dx\quad \text{ in}\quad \Omega\times{\mathbb R}^+$ The function $$a:{\mathbb R}\to{\mathbb R}$$ is continuous, satisfying $$0<a(\xi)<M$$ $$\forall \xi\in{\mathbb R}$$ and $$g\in L^2(\Omega).$$ It is studied existence of unique weak solution of the parabolic problem, existence of stationary solutions and in some cases of asymptotic behaviour.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs 35J25 Boundary value problems for second-order elliptic equations
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##### References:
 [8] doi:10.1007/978-3-642-61798-0 · Zbl 0361.35003
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