Abstract

We consider the problem of finding, from the final data \(u(x,T)= \varphi(x)\), the temperature \(u\) satisfying \[ u_t-u_{xx}= f(x,t,u(x,t)), \quad (x,t)\in\mathbb R\times(0,T). \] The problem is nonlinearly ill-posed. We use the Fourier transforms to get a nonlinear integral equation and give a regularized solution by perturbing directly the integral equation. Error estimates are given.