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Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. (English) Zbl 0631.35041

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ N\) and let A be a uniformly elliptic operator on \(\Omega\). The author considers the parabolic problem \(u_ t-Au=0\) in \(\Omega_{\infty}=\Omega \times (0,\infty)\), \(u(x,0)=u_ 0(x)\) for \(x\in \Omega\) where \(u_ 0\not\equiv 0\) for all \(u_ 0\) in C(\({\bar \Omega}\)), and \(u(x,t)=\int_{\Omega} f(x,y)u(y,t) dy\), \(0<t<\infty\), where f is a continuous function defined for \(x\in \partial \Omega\), \(y\in {\bar \Omega}\) and such that, for every \(x\in \partial \Omega\), \(\int_{\Omega} | f(x,y)| dy\leq \rho <1\). He appeals to the maximum principle to prove that the problem has a unique solution u in C(\({\bar \Omega}_{\infty})\); that \(U(t)=\max_{x\in {\bar \Omega}} | u(x,t)|\) is monotone decreasing in t; that there are constants C, \(\gamma\) such that for all \(t>0\), U(t)\(\leq Ce^{-\gamma t}\); and that there is a \(T_*\) with \(0<T_*\leq \infty\) such that U(t) is strictly decreasing for \(0<t<T_*\) whereas U(t)\(\equiv 0\) for \(t>T_*\). Imposing added conditions on the coefficients of A, on the boundary \(\partial \Omega\), and on an extension of the function f, the author then shows that for any \(x\in \Omega\), u(x,t) is analytic in t, \(0<t<\infty\), and that U(t) is strictly decreasing in t for all \(t>0\) so that U(t) does not vanish in finite time.
Reviewer: D.T.Haimo

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B50 Maximum principles in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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