THE CAUCHY-PROBLEM FOR ODD-ORDER QUASI-LINEAR EQUATIONS

A nonlocal Cauchy problem for multidimensional quasilinear evolution equations containing a linear differential operator L(t, x, D(x)) with leading derivatives of odd order is considered. The conditions on the nonlinear terms are chosen so that they are subordinate to the operator L. The Korteweg-de Vries equation is a special case of such equations. No smoothness conditions are imposed on the initial function u0(x) (u0(x) is-an-element-of L2(R(n))). Theorems on the existence, uniqueness, and continuous dependence on the initial data of generalized solutions are established.