On the Generalization of Burgers Equation to the Case of Linebreak Bounded Dissipation Flux

It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux u_t+f(u)_x=Q(u_x)_x, (t,x)∈ R₊ × R, where Q' >0, |Q(s)|<+ \infty . Such equation degenerates to hyperbolic one as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted. The two close definitions of the generalized solution are given in the preprint in the spirit of definitions of A.I.Vol'pert and S.N.Kruzkov. For the former definition the existence theorem is established while for the latter one the uniqueness theorem is proven in the classes of functions of bounded variation. The main feature of used apriori estimates is the fact that one needs to estimate only Q(u_x) which allows to have in fact it arbitrary local growth of the velocity gradient.