On nonlinear Sobolev equations with terminal observations in \(L^p\) spaces

Abstract

In this paper, we investigate the backward problem for the heat equation equipped with the time fractional conformable derivative. This problem is a generalization of the classical heat equation. We consider the problem with a nonlinear source function in a bounded domain. This problem is shown to be ill-posed, so we regularize the solution by the Fourier truncated method and we estimate the error term in the \(L^p(\mathcal{D})\) space. An example to illustrate the theory is given in the final section.