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Zbl 0631.35041
Friedman, Avner
Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. (English)
[J] Q. Appl. Math. 44, 401-407 (1986). ISSN 0033-569X; ISSN 1552-4485/e

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.
[D.T.Haimo]

MSC 2000:: *35K20 Second order parabolic equations, boundary value problems
35B05 General behavior of solutions of PDE
35B40 Asymptotic behavior of solutions of PDE
35B50 Maximum principles (PDE)
35A05 General existence and uniqueness theorems (PDE)

Keywords: nonlocal boundary conditions; maximum principle; unique solution; monotone decreasing

Cited in: Zbl 0820.35085

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