Cauchy problem for a semilinear elliptic equation with contaminated coefficients

In this paper, we assume that Omega is a bounded domain of R-N with smooth boundary. Let A : D(A) -> L-2(Omega) be a self-adjoint operator defined on a dense subspace D(A) subset of L-2(Omega) with an orthonormal basis of eigenfunctions in L-2(Omega). For a constant Y > 0, giving the function F : Omega x [0, Y] x R -> R, we consider the problem of finding a function u : Omega x [0, Y] -> R such that -tau(y)Au(x, y) + u(yy) (x, y) = F (x, y, u(x, y)), x is an element of Omega, 0 < y < Y, u(x, 0) = f (x), u(y)(x, 0) = g(x) x is an element of Omega, where f, g is an element of L-2(Omega) are given. In many practical cases, the function tau : [0, Y] -> (0, infinity) is contaminated with noise and it can only be approximated by an experimentally observable function mu : [0, Y] -> (0, infinity) such that sup(0 <= y <= Y) (Sigma(2)(m=0) vertical bar d(m)tau/dy(m) (y) - d(m)mu/ d y(m) (y)vertical bar) <= delta for a delta > 0. Similarly, for epsilon > 0, the initial data f, g are often perturbed by f(epsilon), g(epsilon) as follows. parallel to f - f(epsilon)parallel to(L2(Omega)) + parallel to g - g(epsilon)parallel to(L2(Omega)) <= epsilon. 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.