The nonlinear fractional diffusion equations with Nagumo-type sources and perturbed orders

We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable $t$, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular nonlinear source. These equations are generalizations of some well-known fractional equation such as the fractional Cahn-Allen equation, the fractional Burger equation, the fractional Cahn-Hilliard equation, the fractional Kuramoto-Sivashinsky equation, etc. We study both the initial value and the final value problem. Under some suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. For $t=0$, we show that the final value problem is instable and deduce that the problem is ill-posed. A regularization method is proposed to recover the initial data from the inexact fractional orders and the final data. By some regularity assumptions of the exact solutions of the problems, we obtain an error estimate of Hölder type.

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