Regularized Wavelet Solutions for Ill-posed Nonhomogeneous Parabolic Equations

We consider the nonhomogeneous problem u_xx(x, t) = u_t(x, t) + f(x, t), 0 ≤ x <; 1, t ≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). In this paper, we shall define a Meyer wavelet solution to obtain well-posed solution in the scaling space V_j. We shall also show that under certain conditions this regularized solution is convergent to the exact solution. In the previous papers, most of the theoretical results concerning the error estimate are about the homogeneous equation, i.e., f(x, t) ≡ 0.