×

Remarks on a backward parabolic problem. (English) Zbl 1085.35026

From the introduction: Let \(A(x, t)\) be an \(n\times n\) matrix-valued measurable function on \(\mathbb{R}^n\times\mathbb{R}^+\) such that (i) \(A\) is periodic in both \(x\) and \(t\) with period 1, (ii) \(A\) is bounded and positive definite for all \((x, t)\in\mathbb{R}^n\times \mathbb{R}^+\). We consider the following problem: \[ \begin{aligned} {\partial u^\varepsilon\over\partial t}- \text{div}(A({x\over\varepsilon}, {t\over\varepsilon^2})\nabla u^\varepsilon)= 0\quad &\text{in }\Omega\times (0,T),\\ u^\varepsilon(t)\equiv 0\quad &\text{on }\partial\Omega\times [0, T].\tag{\(*\)}\end{aligned} \] Here \(\varepsilon> 0\) is assumed to be very small, and \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\). Suppose \(f(x)\) is an observed data at the time \(t= T\), i.e., \(f(x)\) is close to some true solution \(u^\varepsilon\) of \((*)\) at the time \(t= T\). Then our problem is how to recover (one of) such \(u^\varepsilon\), a solution of \((*)\), so that \(\| u^\varepsilon(\cdot, T)- f(\cdot)\|\) is small for a suitable given norm \(\|\cdot\|\).

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
35R25 Ill-posed problems for PDEs

Keywords:

uniqueness
PDFBibTeX XMLCite
Full Text: DOI Euclid