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where u0 , v0 are known functions. A physical derivation of the above equations, with A(v) — constant-v, appears in a forthcoming report by J. Fleck [5]. The purpose of this paper is to determine a priori bounds for norms of u and v. Under natural assumptions on the given functions A, u0 , v0 , we suppose that smooth solutions u, v exist to the mixed initialand boundary-value problem posed by the preceding equations. We then derive numerical bounds, depending only on the given functions, for the maximum-norm of v and for an integral-norm of u. These bounds will hold in the infinite domain 0<z<a, 0 < £ < °°. Our results are found by the elementary sort of argument used by E. Hopf [1] and by L. Nirenberg [2] in their justifications for the maximum principles for elliptic and for parabolic equations. To prove a preliminary result, we shall use the following form of the weak maximum principle for parabolic equations. Let U(x, t) be a solution of the equation

A priori bounds for temperature in circulating fuel reactors