Fractional evolution equation with Cauchy data in L p $L^{p}$ spaces

Abstract

Abstract In this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in L 2 $L^{2}$ and H s $H^{s}$ . However, there have not been any papers dealing with this problem with observed data in L p $L^{p}$ with p ≠ 2 $p \neq 2$ . We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in L p $L^{p}$ . To our knowledge, L p $L^{p}$ evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.

Keywords

• Fractional evolution equation
• Caputo derivative
• Cauchy problem
• Fourier truncation regularization
• Sobolev embeddings