  In this work we study partial differential equations defined in a domain that
moves in time according to the flow of a given ordinary differential equation,
starting out of a given initial domain. We first derive a formulation for a
particular case of partial differential equations known as balance equations.
For this kind of equations we find the equivalent partial differential
equations in the initial domain and later we study some particular cases with
and without diffusion. We also analyze general second order differential
equations, not necessarily of balance type. The equations without diffusion are
solved using the characteristics method. We also prove that the diffusion
equations, endowed with Dirichlet boundary conditions and initial data, are
well posed in the moving domain. For this we show that the principal part of
the equivalent equation in the initial domain is uniformly elliptic. We then
prove a version of the weak maximum principle for an equation in a moving
domain. Finally we perform suitable energy estimates in the moving domain and
give sufficient conditions for the solution to converge to zero as time goes to
infinity.
