The Cauchy problem for odd-order quasilinear 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 $u_0(x)$ $(u_0(x)\in L_2(\mathbf R^n))$. Theorems on the existence, uniqueness, and continuous dependence on the initial data of generalized solutions are established.
Bibliography: 20 titles.