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Reverse convolution inequalities and applications to inverse heat source problems. (English) Zbl 1029.44002
Authors’ abstract: We introduce reverse convolution inequalities obtained recently and at the same time, we give new type reverse convolution inequalities and their important applications to inverse source problems. We consider the inverse problem of determining \(f(t)\), \(0< t< T\), in the heat source of the heat equation \[ \partial_t u(x, t)=\Delta u(x, t)+ f(t)\varphi(x),\quad x\in\mathbb{R}^n,\quad t> 0 \] from the observation \(u(x_0, t)\), \(0< t< T\), at a remote point \(x_0\) away from the support of \(\varphi\). Under an a priori assumption that \(f\) changes the signs at most \(N\)-times, we give a conditional stability of Hölder type, as an example of applications.

MSC:
44A35 Convolution as an integral transform
26D20 Other analytical inequalities
35K05 Heat equation
35R30 Inverse problems for PDEs
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