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

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