Elliptic Problems with Nonlocal Potential Arising in Models of Nonlinear Optics

The Dirichlet problem in the half-plane for strong elliptic differential-difference equations with nonlocal potentials is considered. The classical solvability of this problem is proved, and the integral representation of this classical solution by a Poisson-type relation is constructed

Decay of Nonnegative Solutions of Singular Parabolic Equations with KPZ-Nonlinearities

The Cauchy problem for quasilinear parabolic equations with KPZ-nonlinearities is considered. It is proved that the behavior of the solution as $t \to \infty$ can change substantially as compared with the homogeneous case if the equation involves zero-order terms. More specifically, the solution decays at infinity irrespective of the behavior of the initial function and the rate and character of this decay depend on the conditions imposed on the lower order coefficients of the equation

Elliptic Equations with Arbitrarily Directed Translations in Half-Spaces

In this paper, we investigate the half-space Dirichlet problem for elliptic differential-difference equations with superpositions of differential operators and translation operators acting in arbitrary directions parallel to the boundary hyperplane. The summability assumption is imposed on the boundary-value function of the problem. The specified equations, substantially generalizing classical elliptic partial differential equations, arise in various models of mathematical physics with nonlocal and (or) heterogeneous properties or the process or medium: multi-layer plates and envelopes theory, theory of diffusion processes, biomathematical applications, models of nonlinear optics, etc. The theoretical interest to such equations is caused by their nonlocal nature: they connect values of the desired function (and its derivatives) at different points (instead of the same one), which makes many classical methods unapplicable. For the considered problem, we establish the solvability in the sense of generalized functions, construct Poisson-like integral representations of solutions, and prove the infinite smoothness of the solution outside the boundary hyperplane and its uniform convergence to zero (together with all its derivatives) as the timelike variable tends to infinity. We find a power estimate of the velocity of the specified extinction of the solution and each its derivative