Evolution equations with monotone operator and functional non-linearity atthe time derivative

Conditions for the solubility of the so-called doubly non-linear equations Au + partial derivative/partial derivativet Bu = f, u(0) = u(0) are investigated. Here A is a monotone operator induced by a differential e xpression containing higher-order partial derivatives and B is an operator induced by a monotone function. A theorem on the existence of a solution is proved. The method of monotone operators is used in combination with the m ethod of compact operators. Examples of applications to parabolic different ial equations are presented.