We study the solvability in Sobolev spaces of boundary value problems for elliptic and parabolic equations with variable coefficients presence an involution (involutive deviation) at higher derivatives, both nondegenerate degenerate cases. For under study, we prove existence theorems as well uniqueness regular solutions, i.e., those that have all weak derivatives equation.

In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to the well-known thin film equation, but the Laplace operator is replaced by a Dirichlet-toNeumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann bo...

Abstract We study the existence of weak solutions to a Newtonian fluid?non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). u t = <m...