Cauchy problem for a semilinear elliptic equation with contaminated coefficients

Abstract

In this paper, we assume that Ω is a bounded domain of RN with smooth boundary. Let A:D(A)→L2(Ω) be a self-adjoint operator defined on a dense subspace D(A)⊂L2(Ω) with an orthonormal basis of eigenfunctions in L2(Ω). For a constant Y>0, giving the function F:Ω×[0,Y]×R→R, we consider the problem of finding a function u:Ω×[0,Y]→R such that −τ(y)Au(x,y)+uyy(x,y)=F(x,y,u(x,y)),x∈Ω,0<y<Y,u(x,0)=f(x),uy(x,0)=g(x),x∈Ω, where f, g∈L2(Ω) are given. In many practical cases, the function τ:[0,Y]→(0,∞) is contaminated with noise and it can only be approximated by an experimentally observable function μ:[0,Y]→(0,∞) such that sup0⩽y⩽Y∑m=02dmτdym(y)−dmμdym(y)⩽δfor a δ>0. Similarly, for ɛ>0, the initial data f,g are often perturbed by fɛ,gɛ as follows. ‖f−fɛ‖L2(Ω)+‖g−gɛ‖L2(Ω)⩽ɛ.The Cauchy problem with these contaminated data is nonlinear and ill-posed. Due to the usefulness in physics and other fields, we consider a regularization for the problem.