Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 837–850
We consider nonlinear equations of parabolic type in reflexive Banach spaces. We present sufficient conditions for the existence of solutions of these equations. We use methods for the investigation of problems with operators of pseudomonotone (on a subspace) type. In addition, a sufficient criterion in the Sobolev space L p (0, T; W p 1 (Ω)∩L 2 (0, T; L2(Ω)) is considered for the case where an operator introduced with the use of functional coefficients belongs to a given class. We also show that it is possible to weaken the classical condition of coerciveness.
Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1250-1264
We propose a method for the construction of generalized solutions for some nondivergent partial differential systems using set-valued analogs of the generalized statement of the problem based on subdifferential calculus. We establish new sufficient conditions for the existence of solutions of a variational inequality with set-valued operator under weakened coerciveness conditions. We consider examples of a weighted p-Laplacian in the Sobolev spaces \(W_p^1 \left( \Omega \right)\) , p ≥ 2.