THE EXISTENCE OF SOLUTIONS OF EVOLUTION AND ELLIPTIC EQUATIONS WITH SINGULAR COEFFICIENTS
YAREMENKO MIKOLA IVANOVICH *
Department of Mathematics, Kyiv Polytechnic Institute, National Technical University of Ukraine, Ukraine.
*Author to whom correspondence should be addressed.
In this paper we study existence of weak solutions of quasi-linear evolution differential equations in space. To prove the existence of the solution of quasi-linear evolution equation with singular coefficients we consider the form, that is associated with non-linear operator and studying the properties this associated operator by means of form, then applying Galerkin method and showing that a given equation has a solution in the Sobolev space. We introduced a new type of nonlinear elliptic operators that are associated with left side of elliptic equation and studied their properties. To prove necessity of existence of solution we proved some a priori estimates which are theorems about properties of solutions under certain conditions on the functional coefficients of this equation. The estimates of solutions are the key point for proving theorem of existence, in case such estimates are known we can use different methods of proving the solvability of the equation.
Keywords: Differential form, parabolic equations, evolution equations, a priori estimate, weak solution, singular coefficients, Banach space