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Acta Mathematica Vietnamica

A Mollification Method for Backward Time-Fractional Heat Equation

icon-email Nguyen Van Duc , Pham Quy Muoi , Nguyen Van Thang

Abstract

In this paper, we study the ill-posed fractional backward heat equation {γutγ=Δu,xRn,t(0,T),u(x,T)=φ(x),xRn, where φ is unknown exact data and only noisy data φε with φε()φ()L2(Rn)ε is available. The problem is regularized by the well-posed mollified problem {γvνtγ=Δvν,xRn,t(0,T),vν(x,T)=Sν(φε(x)),xRn, where ν>0 and Sν(φε(x)), a mollification of φε defined by the convolution of φε(x) with Dirichlet kernel. The error estimates u(,t)vν(,t)Hl(Rn),0l are established for ν chosen a priori and a posteriori.