Weak solutions to initial-boundary-value problems for quasilinear evolution equations of an odd order

Abstract

Initial-boundary-value problems in three different domains are considered for quasilinear evolution partial differential equations of an odd (not less than third) order with respect to spatial variables in the multidimensional case. The nonlinearity has the divergent form and at most a quadratic rate of growth. Assumptions on the differential operator of odd order provide global estimates on solutions in $L_2$ and a local smoothing effect. Results on existence and uniqueness of global weak solutions are established. The essential part of the study is the construction of special solutions to the corresponding linear equations of the "boundary potential" type, which ensures the results under natural smoothness assumptions on initial and boundary data provided we have certain relations between the dimension and the order of the equations.