Existence and regularity in inverse source problem for fractional reaction-subdiffusion equation perturbed by locally Lipschitz sources

Abstract

<p style='text-indent:20px;'>In this paper, we consider an inverse problem of determining a space-dependent source in the time fractional reaction-subdiffusion equation involving locally Lipschitz perturbations, where the additional measurements take place at the terminal time which are allowed to be nonlinearly dependent on the state. By providing regularity estimates on both time and space of resolvent operator and using local estimates on Hilbert scales, we establish some results on the existence and uniqueness of solutions and the Lipschitz type stability of solution map of the problem under consideration. In addition, when the input data take more regular values, we obtain results on regularity in time of solution for both the direct linear problem and the inverse problem above.</p>

Regularity theory for fractional reaction–subdiffusion equation and application to inverse problem

“…$$ It is useful to notice that for each $$$ \mu >0 $$$, the function $$$ {s}_{\alpha}\left(t,\mu \right) $$$ is completely monotonic on $$$ \left(0,\infty \right) $$$, that is, $$$$ {\left(-1\right)}^n\frac{\partial^n}{\partial {t}^n}{s}_{\alpha}\left(t,\mu \right)\ge 0,\kern0.4em \mathrm{for}\ \mathrm{all}\kern0.4em n=0,1,2,\dots, t>0, $$$$ thanks to Gorenflo et al 42, Proposition 3.23, p. 47 (see also Pollard 43 ). We collect some useful other properties of $$$ {s}_{\alpha}\left(\cdotp, \mu \right),{r}_{\alpha}\left(\cdotp, \mu \right) $$$ in the following proposition whose proof can be borrowed from Tuan 44, Propositions 1 and 2 …”

“…It is apparent that $$$ {\mathcal{S}}_{\alpha }(t) $$$ given by () is a bounded linear operator on $$$ {L}^2\left(\Omega \right) $$$ for all $$$ t\ge 0 $$$. Some other interesting properties of $$$ {\mathcal{S}}_{\alpha }(t) $$$ are stated in the following lemma, whose proof can be found in Tuan 44, Lemma 2.1 …”