We establish here the convergence (thereby proving the existence) of a semi-discrete scheme for the quasilinear hyperbolic equation u(0, .) = 4
where x E R", t E [O,T], and d E L"(R"). It is well known that the above problem does not necessarily have global classical solutions and the usual concepts of weak solutions do not lead to a unique solution. The existence of a unique solution to the above problem in a suitable sense was established in 131, where a parabolic problem obtained by introducing the term -E A U was studied and then the behavior as E ---f 0 was discussed. A difference scheme approach to a problem of the above type where CP, does not depend on x and t and $ does not depend on u was also studied in 121. The aim of this paper is to present a proof for the case when @ depends on x , $ depends on u, and the technical complications in this case are nontrivial. The discussions in this paper may be considered as a continuation of the ideas in the above papers.
I. Assumptions
Consider the equation where x = (xI, . . . ,x,,) E R" and t E 10, T I , T > 0. Denotz by R, = (0, T ) x R". We will study the Cauchy problem for (1.1) given an initial condition of the type ul,=o = 4
(1.2)
We make the following assumptions:
( H -I ) @!:ar X R + R is continuous and V , @ , ( t , x , u ) = {(d@,/dx,) x *This work was done when the second author was visiting the University of Texas at Arlington.