periodic solutions

About uniqueness for periodic-parabolic problems

Autores: 
V. N. Schuchman & M. P. Vishnevskii
Resumo: 
In this paper we study the semilinear periodic-parabolic boundary value problem of the form:
\[
(1)\;\;\;\;\;\;\;\;\;\;u_t = L(x, t)u + \lambda f(x,t,u)\;\;\;\;in\;\;\;\;Q = \Omega \times (0, +\infty)
\]
\[
(2)\;\;\;\;\;\;\;\;\;\;B(x,t)u = \beta\frac{\partial u}{\partial \nu}+ b(x,t)u = 0,\;\;\;\;on\;\;\;\;\Gamma = \partial\Omega \times (0, + \infty)
\]
\[
(3)\;\;\;\;\;\;\;\;\;\;u(x,0) = u_0 (x)
\]

Let $u(x,t;u_0)$ be a solution of problem (1) - (3). We also consider the
periodic solution of the problem (1), (2) with periodic conditions:
\[
(4)\;\;\;\;\;\;\;\;\;\;u(x,t,) = u(x, t + \omega)
\]

Here $\Omega \subset R^n$, $n \geq 1$ is a bounded domain with boundary of the class
$C^{2+\alpha}$, $0 < \alpha < 1$, $L(x, t)$ is a strongly uniformly elliptic operator,
$B(x,t)$ is of Dirichlet, Neumann or Robin type of boundary conditions,
operators $L(x,t)$, $B(x,t)$ and positive function $b(x,t)$ are $\omega$-periodic of
time.
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