Approximation of mild solutions of a semilinear fractional elliptic equation with random noise

TL;DR

This paper addresses the challenge of approximating solutions to a semilinear fractional elliptic equation with random noise, establishing ill-posedness and proposing a Fourier truncation method for stabilization with proven convergence rates.

Contribution

It is the first to analyze the Cauchy problem for such equations with Gaussian white noise, introducing a regularization technique and deriving convergence rates.

Findings

Ill-posedness of the problem established

Fourier truncation method effectively stabilizes the solution

Convergence rates in $L^2$ and $H^q$ norms demonstrated

Abstract

We study for the first time the Cauchy problem for semilinear fractional elliptic equation. This paper is concerned with the Gaussian white noise model for the initial Cauchy data. We establish the ill-posedness of the problem. Then, under some assumption on the exact solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Some convergence rates between the exact solution and the regularized solution is established in $L^2$ and $H^q$ norms.