Let Q be a heat conduction body and let ' = '(t) be given. We consider the problem of nding a two-dimensional heat source having the form '(t)f(x; y) in Q. The problem is ill-posed. Assuming @Q is insulated and ' 6 0, we show that the heat source is de ned uniquely by the temperature history on @Q and the temperature distribution in Q at the initial time t = 0 and at the nal time t = 1. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate.

We study the nonhomogeneous heat equation under the form u t − u xx = ϕ(t)f (x), where the unknown is the pair of functions (u, f ). Under various assumptions about the function ϕ and the final value in t = 1, i.e., g(x), we propose different regularizations on this ill-posed problem based on the Fourier transform associated with a Lebesgue measure. For ϕ / ≡ 0 the solution is unique.

Communicated by W. TornigWe study the following initial and boundary value problem:In section 1. with ug in L2(12), fcontinuous such thatf(u) + EU non-decreasing for E positive, we prove the existence of a unique solution on (0, T), for each T > 0. in section 2 it is proved that the unique solution u belongs to Lz(O, T; HA n H') n L"(0, T; HA) if we assume uo in HA andfin C ' ( R . W). Numerical results are given for these two cases. IntroductionIn this paper, we study the following initial and boundary value problem:

We consider the two-dimensional problem of recovering globally in time the heat distribution on the surface of a layer inside of a heat conducting body from two interior temperature measurements. The problem is ill-posed. The approximation function is represented by a two-dimensional Sinc series and the error estimate is given. ᭧