Recovering the initial wave amplitude for nonlinear elliptic equation with locally Lipschitz source in multiple-dimensional domain

Abstract

The backward problems for the elliptic equation (BPEE) are widely used for modelling problems in many fields such as physics, geometry or engineering. This paper is the first investigation of BPEE in unbounded multiple-dimensional domain associated with locally Lipschitz source as follows ∂2u∂x12+∂2u∂x22+...∂2u∂xn2+∂2u∂y2=S(x1,x2,…,xn,y,u(x1,x2,…,xn,y)),in which (x1,x2,…,xn,y)∈Rn×0,L, where L>0. Based on the idea of the truncation approach, we construct the regularized solution and obtain the H ölder type of its convergence rate under some assumptions on regularity of the exact solution. Eventually, a numerical experiment for the elliptic Allen–Cahn equation is proposed to illustrate the effectiveness and applicability of our method.