О внутренней регулярности решений двумерного уравнения Захарова-Кузнецова

In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov-Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a semistrip. In this problem, different types of initial conditions (e. g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of areas: a plane, a half-plane, and a strip.В статье рассматриваются вопросы внутренней регулярности слабых решений начально-краевых задач для уравнения Захарова-Кузнецова с двумя пространственными переменными. Начальная функция предполагается нерегулярной, а основным параметром, влияющим на регулярность, является скорость убывания начальной функции на бесконечности. Основные результаты работы относятся к случаю задачи, поставленной на полуполосе. При этом различные типы краевых условий (например, Дирихле или Неймана) влияют на характер внутренней регулярности. Приводится также обзор ранее полученных результатов для других типов областей: всей плоскости, полуплоскости и полосы

Weak solutions to one initial-boundary value problem with three boundary conditions for quasilinear evolution equations of the third order

Global well-posedness in a class of weak solutions is established to one initial-boundary value problem with three boundary conditions for a wide class of quasilinear dispersive evolution equations of the third order in the multidimensional case. The considered class of equations generalizes the Korteweg–de Vries, the Korteweg–de Vries–Burgers and the Zakharov–Kuznetsov equations

О начально-краевой задаче на полуоси для обобщенного уравнения Кавахары

In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.В статье рассматривается начально-краевая задача на полуоси для обобщенного уравнения Кавахары с нелинейностью высокого порядка. Получен результат о существовании и единственности глобального решения. Также в случае наличия в уравнении абсорбирующего слагаемого, исчезающего на бесконечности, устанавливается затухание решения при больших временах

An Initial-Boundary Value Problem in a Strip for Two-Dimensional Equations of Zakharov-Kuznetsov Type

An initial-boundary value problem in a strip with homogeneous Dirichlet boundary conditions for the two-dimensional generalized Zakharov-Kuznetsov equation is considered. In particular, dissipative and absorbing degenerate terms can be supplemented to the original Zakharov-Kuznetsov equation. Results on global existence, uniqueness and long-time decay of weak solutions are established

An initial-boundary value problem in a strip for two-dimensional Zakharov-Kuznetsov-Burgers equation

An initial-boundary value problem in a strip with homogeneous Dirichlet boundary conditions for two-dimensional Zakharov-Kuznetsov-Burgers equation is considered. Results on global well-posedness and large-time decay of solutions in the spaces Hs for sε[0,2] are established. © 2014 Elsevier Ltd. All rights reserved

On mixed problems for the Korteweg-de Vries equation under irregular boundary data

Results are established concerning the non-local solubility and well posedness in various function spaces of the mixed problem for the Korteweg-de Vries equation u(t) + u(xxx) + au(x) + uu(x) = f(t, x) in the half-strip (0,T) x (-infinity,0). Some a priori estimates of the solutions are obtained using a special solution J(t, x) of the linearized Kdv equation of boundary potential type. Properties of J are studied which differ essentially as x --> +infinity or x --> -infinity. Application of this boundary potential enables us in particular to prove the existence of generalized solutions with non-regular boundary values

Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function

In the strip Π = (-1, 0) × ℝ, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u t + u xxx + uu x = 0 with initial condition either 1) u(-1, x) = -xθ(x), or 2) u(-1, x) = -xθ(-x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (-1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity. © 2008 Pleiades Publishing, Ltd

Initial-boundary value problems in a rectangle for two-dimensional Zakharov–Kuznetsov equation

Initial-boundary value problems in a bounded rectangle with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global well-posedness in the classes of weak and regular solution are established. As applications of the developed technique results on boundary controllability and long-time decay of weak solutions are also obtained. © 2018 Elsevier Inc