Determine the spacial term of a two-dimensional heat source

We consider the problem of determining a pair of functions $(u,f)$ satisfying the heat equation $u_t-\mathrm{\Delta }u=\phi (t)f(x,y)$, where $(x,y)\in \Omega=(0,1)\times (0,1)$andthefunction$\varphi$ is given. The problem is ill-posed. Under a slight condition on $\phi$, we show that the solution is determined uniquely from some boundary data and the initial temperature. Using the interpolation method and the truncated Fourier series, we construct a regularized solution of the source term $f$ from non-smooth data. The error estimate and numerical experiments are given.