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in the region t>0. Here x =(Xl, x2 . . . . , xn), u =u(t, x), the f j , j = 1, 2, . . . , n, are continuously differentiable functions of a single real variable, and u0 (x) is in the class F; i.e., Uo(X) is a bounded function having locally bounded variation in the sense of TOt~LLI-CESARI. Moreover for each fixed t, u(t, x) is in the same class F. A function f is said to have bounded variation in the sense of TONeLLI-CEsld~I over a compact set (2 if there exists a set Z of measure zero in Q such that the functions

archive for rational mechanics and analysis

Uniqueness and stability theorem for the generalized solution of the initial-value problem for a class of quasi-linear equations in several space variables