An inverse problem of determining unknown source parameter in a parabolic equation is considered. The variational iteration method (VIM) is presented to solve inverse problems. The solution gives good approximations by VIM. A numerical example shows that the VIM works effectively for an inverse problem.

Let D be a domain in ℝm+1 and E be a domain in ℝn+1. A pair of a smooth mapping f : D → E and a smooth positive function ϕ on D is called a caloric morphism if ϕ ˙u o f is a solution of the heat equation in D whenever u is a solution of the heat equation in E. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case of m < n, there are no caloric morphisms. In the case of m = n, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case of m > n, under some assumption on f, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ℝn+1.

In this paper, we consider the parabolic equation with anisotropic growth conditions, and obtain some criteria on boundedness of solutions, which generalize the corresponding results for the isotropic case.

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].