Vol 235, No 3 (2018)

We study the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in two dimensions. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

We find necessary and sufficient conditions for the generalized smoothness of the coordinate functions obtained from the approximate relations. We show that for the coordinate functions the smoothness on their supports is equivalent to that on the boundaries of supports. We obtain conditions for the continuity of Courant type finite element approximations and conditions for the uniqueness of a linear space of such approximations.

We study the continuity in the sense of the strong topology for the flux function υ → l(υ) = |υ|^{p(⋅) − 2}υ acting from the Lebesgue–Orlicz space L^p(⋅)(Ω, ℝ^m) to the dual L^{p ′ (⋅)}(Ω, ℝ^m), where p^′(⋅) is the Hölder-conjugate exponent, under the assumption that p(·) is an L^∞(Ω)-function such that 1 < α ≤ p(·) ≤ β < ∞. We obtain estimates for the convergence \( {\left\Vert l\left({\upsilon}_n\right)-l\left(\upsilon \right)\right\Vert}_{p^{\prime}\left(\cdot \right)}\to 0 \) with respect to the smallness order as ‖υ_n − υ‖_p(⋅) → 0. The strong continuity of the energy functional \( \underset{\varOmega }{\int }{\left|\upsilon \right|}^{p\left(\cdot \right)} dx \) is a consequence of the strong continuity of the flux function.

We establish the integrability for some classes of dynamic systems on the tangent bundles of multidimensional manifolds. We consider the case where the force fields possess variable dissipation. An example of a four-dimensional manifold is discussed in detail.