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On the asymptotic behaviour of some nonlocal mixed boundary value problems. (English) Zbl 1052.35100
Agarwal, Ravi P. (ed.) et al., Nonlinear analysis and applications: To V. Lakshmikantham on his 80th birthday. Vol. 1. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1711-1/hbk). 431-449 (2003).
The authors study the following parabolic problem \[ \begin{aligned} u_t-a(l(u(t)))Au&=f \quad \text{in}\;\Omega\times {\mathbb R}^+,\\ u&=0\quad\text{on}\;\Gamma\times{\mathbb R}^+,\\ \partial_{\nu_A}&= 0\quad\text{on}\;\partial\Omega \setminus \Gamma\times \mathbb{R}^+,\\ u(\cdot,0)&=u_0\;\;\text{in}\;\Omega \end{aligned} \] from the point of view of existence, uniqueness and asymptotic behaviour. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), and \(\Gamma\) is a part of \(\partial\Omega\). Moreover, \(A\) is a second-order elliptic operator in divergence form, and \(\partial_{\nu_A}\) is the corresponding conormal derivative. Finally, \(a: \mathbb{R}\to \mathbb{R}\) is a function, and \(\ell(u):= \int_\Omega g(x) u(x)\,dx\) with \(g\in L^2(\Omega)\).
For the entire collection see [Zbl 1030.00016].

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
35K20 Initial-boundary value problems for second-order parabolic equations