2020/02/16 by Nguyen Minh Dien, Dien, Nguyen Minh, Erkan Nane +3
Mathematics · #Differential Equations and Numerical Methods #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2002.06747
We consider a class of nonlinear fractional equations having the Caputo\nfractional derivative of the time variable t, the fractional order of the\nself-adjoint positive definite unbounded operator in a Hilbert space and a\nsingular nonlinear source. These equations are generalizations of some\nwell-known fractional equation such as the fractional Cahn-Allen equation, the\nfractional Burger equation, the fractional Cahn-Hilliard equation, the\nfractional Kuramoto-Sivashinsky equation, etc.\n We study both the initial value and the final value problem.\n Under some suitable assumptions, we investigate the existence, uniqueness of\nmaximal solution, and stability of solution of the problems with respect to\nperturbed fractional orders.\n For t=0, we show that the final value problem is instable and deduce that\nthe problem is ill-posed. A regularization method is proposed to recover the\ninitial data from the inexact fractional orders and the final data. By some\nregularity assumptions of the exact solutions of the problems, we obtain an\nerror estimate of H "older type.\n