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Stability of Lipschitz type in determination of initial heat distribution. (English) Zbl 0882.35056
Summary: For the solution \(u(x,t)= u(f)(x,t)\) of the equation \[ u'(x,t)=\Delta u(x,t),\quad x\in\Omega,\;t>0,\quad u(x,0)= f(x),\quad u|_{\partial\Omega}=0, \] where \(\Omega\subset\mathbb{R}^r\), \(2\leq r\leq 3\) is a bounded domain with \(C^2\)-boundary and for an appropriate subboundary \(\Gamma\) of \(\Omega\) we prove a Lipschitz estimate of \(|f|_{L^2(\Omega)}\): For \(\mu\in(1,5/4)\) and for a positive constant \(C\), \[ \begin{split} C^{-1}|f|_{L^2(\Omega)}\leq \int_\Gamma \Biggl\{\sum^\infty_{n=0}{1\over n!\Gamma(n+ 2\mu+1)}\times\\ \int^\infty_0\Biggl|(p\partial^{n+1}_p+ n\partial^n_p)p^{-{3\over 2}}{\partial u(f)\over\partial\nu}\Biggl(x,{1\over 4p}\Biggr)\Biggr|^2 p^{2n+2\mu-1}dp\Biggr\}dS.\end{split} \] .

MSC:
35K05 Heat equation
35R30 Inverse problems for PDEs
93B05 Controllability
35K20 Initial-boundary value problems for second-order parabolic equations
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