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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 ${\bbfR}\sp N$ and let A be a uniformly elliptic operator on $\Omega$. The author considers the parabolic problem $u\sb t-Au=0$ in $\Omega\sb{\infty}=\Omega \times (0,\infty)$, $u(x,0)=u\sb 0(x)$ for $x\in \Omega$ where $u\sb 0\not\equiv 0$ for all $u\sb 0$ in C(${\bar \Omega}$), and $u(x,t)=\int\sb{\Omega} f(x,y)u(y,t) dy$, $0<t<\infty$, where f is a continuous function defined for $x\in \partial \Omega$, $y\in {\bar \Omega}$ and such that, for every $x\in \partial \Omega$, $\int\sb{\Omega} \vert f(x,y)\vert dy\le \rho <1$. He appeals to the maximum principle to prove that the problem has a unique solution u in C(${\bar \Omega}\sb{\infty})$; that $U(t)=\max\sb{x\in {\bar \Omega}} \vert u(x,t)\vert$ is monotone decreasing in t; that there are constants C, $\gamma$ such that for all $t>0$, U(t)$\le Ce\sp{-\gamma t}$; and that there is a $T\sb*$ with $0<T\sb*\le \infty$ such that U(t) is strictly decreasing for $0<t<T\sb*$ whereas U(t)$\equiv 0$ for $t>T\sb*$. 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

35K20Second order parabolic equations, initial boundary value problems
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B40Asymptotic behavior of solutions of PDE
35B50Maximum principles (PDE)
35A05General existence and uniqueness theorems (PDE) (MSC2000)