Boundary Control and Calder\'on type Inverse Problems in Non-local heat equation

Abstract

We examine various density results related to the solutions of the non-local heat equation at a specific time slice, focusing on two distinct models: one with homogeneous Dirichlet boundary condition and the other with singular boundary data. In both the cases, we assume the non-local exponent $a\in(\frac{1}{2},1)$. We explore both the qualitative and quantitative aspects of the approximations. Additionally, we address Calder\'on-type inverse problems for these parabolic models, where we recover the potentials by analyzing the solutions either on the boundary or at a particular time slice. In both the density results and the Calder\'on type inverse problems, the Pohozaev identity plays a crucial role. Finally, in the last section, we apply the Pohozaev identity to a specific elliptic eigenvalue problem and demonstrate that the eigenfunctions, when divided by an appropriate power of the distance function, can not vanish on any non-empty open subset of the boundary. This particular eigenvalue problem does not need any restriction on the non-local exponent.