, More recently the paper of L. Ambrosio [1] studies the same subject However this context don't permits to de?ne and solve (7) when is a distribution in (t; x)- variables. In a simpli?ed case where f = 0 and = a t 1 x where a t 2 D 0 (R), the problem is posed an solved in [4]. When 2 D 0 R 2 we turn back to the regular case as the starting point of our generalized methods. It is well known that when , f and u 0 are of class C 1 , the problem (7) admits a unique solution of class C 1 given by integrating along the characteristics (8) u(t; x)
, function n and f n and an Egorov theorem She gives a result similar to (8) when the irregularities are controled by: u 0 2 Lip loc (R; R) (R 2 ; R) and for some > 0; 1 (t; x) ; f 2 L 1 (R 2 ; R) R; (:; x) and f (:; x) are locally Lipschitz uniformly in x: But as in the previous case, this context don't permit to de?ne and solve (7) when is a distribution in (t; x)-variables
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