In §1 we formulate a differential game when the dynamics is the inhomogeneous heat equation. In §2 we state the basic theory of differential games when the controls must choose uniformly Lipschitz control functions. We then prove some general theorems for the case when the controls may choose any measurable control functions. These theorems hold for games with any dynamics. In §3 we apply our theory developed to our particular example and in §4 we prove the existence of value for games with partial differential equations.