Global solution of space-fractional diffusion equations with nonlinear reaction source terms

Abstract

In this paper, we study the initial and final value problems (FVPs) of a class of nonlinear diffusion equations including Riesz–Feller derivatives. A natural question on solution continuity with respect to fractional parameters of the Riesz–Feller derivatives is investigated in two problems. For the initial value problem, we firstly study the unique existence of the solution. Secondly, we prove a Lipschitz continuity of the solution with respect to the fractional parameters and the initial condition. Furthermore, we introduce various conditions from which we can predict decay speed of the solution. For the FVP, it is well-known that this problem is ill-posed in the sense of Hadamard. Even so, in our study case, the fractional parameters are assumed to be inexact. This consideration can lead some common regularization strategy to failure. Therefore, a regularization method for the case of noise influencing to the fractional parameters and the final condition is proposed and investigated.

COMMUNICATED BY:

• Michael Klibanov

Keywords:

• Well-posed problem
• ill-posed problem
• regularization scheme
• fractional diffusion problem
• decay estimate

AMS Subject Classifications:

• 35R11
• 35R30
• 35K57
• 65J20

Disclosure statement

No potential conflict of interest was reported by the authors.