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On the backward heat equation. (English) Zbl 0726.35135

Summary: Let u(t) be a solution of the problem \[ u'(t)-\Delta u(t)=0\text{ in } \Omega,\quad t>0;\quad u(t)=0\text{ on } \partial \Omega,\quad t>0, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\) with a smooth boundary \(\partial \Omega\). Suppose \[ \sum^{\infty}_{n=1}\lambda^ s_ n | (u(0),\phi_ n)|^ 2 \leq E^ 2\text{ for } an\quad s\geq 0\text{ and } \| g-u(1)\| <\epsilon, \] where \((\phi_ n)\) are the orthonormal eigenfunctions of -\(\Delta\) in \(H^ 1_ 0(\Omega)\cap H^ 2(\Omega)\) and \((\lambda_ n)\) are the corresponding eigenvalues. Here \(\| \|\) is the \(L_ 2\)-norm. In the paper we construct, by truncated eigenfunction expansion, an approximate solution \(u_{\epsilon}(t)\), stable with respect to variations in g, such that \[ \| u_{\epsilon}(t)-u(t)\| \leq (1+2^ s)^{1/2} E^{1-t} \epsilon^ t(\log (E/\epsilon))^{-s(1-t)/2},\quad 0\leq t<1, \] for small \(\epsilon >0\) if \(s>0\) and for all \(\epsilon >0\) if \(s=0\). The paper also shows how error estimates can be further improved by strengthening regularity conditions on u(0).

MSC:

35R25 Ill-posed problems for PDEs
35K05 Heat equation
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