Degenerate Hyperbolic Equations Research Articles

The inversion formula for the Volterra integral equation with the Humbert function in the nuclear and its applications to the boundary value problems

Many problems of applied mathematics are reduced to the solution of integral equations with special functions in kernels, therefore the inversion formulas for such equations play an important role in solving boundary value problems for second-order partial differential equations. In this paper, we introduce one degenerate hypergeometric function of two variables through which the solution of the Volterra integral equation of the first kind studing here is expressed. The inversion formula found is applied to finding some relations between the desired solution and its derivative of one boundary value problem for a hyperbolic equation with two degeneration lines of the second kind and with a spectral parameter. Bibliogr.16.Keywords: Volterra integral equation of the first kind, inversion formula, Humbert function, degenerate hypergeometric function of two variables, degenerate hyperbolic equation of the second kind, spectral parameter.

Invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity

In this paper, we will discuss about the invariance of solution set and present the existence and non-existence of the global solutions a class of initial-boundary value problems with dissipative terms is considered for a class of semilinear degenerate hyperbolic equations on the cone Sobolev spaces. First, we will discuss the invariance of some sets corresponding to the problem (1.1) and then, by using a family of potential wells and concavity methods, we obtain existence and non-existence results of global solutions with exponential decay and show the blow-up in finite time of solutions on a manifold with conical singularities.

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.

ОБОБЩЕННЫЕ РЕШЕНИЯ ОДНОГО ВЫРОЖДАЮЩЕГОСЯ ГИПЕРБОЛИЧЕСКОГО УРАВНЕНИЯ ВТОРОГО РОДА СО СПЕКТРАЛЬНЫМ ПАРАМЕТРОМ

For a degenerate hyperbolic equation of the second kind, and with a spectral parameter are studied the Cauchy problem, Cauchy-Goursat and Goursat in a new class of generalized solutions and is given an example that shows the importance of introducing the concept of such a class. Some operators with Bessel functions in the nucleus are introduced and their basic properties are studied. The important identities of these operators are helped to find an explicit representations of the stated problems.

一类非线性退化双曲方程解的整体存在性和爆破的条件

本文研究了一类非线性退化双曲方程的初边值问题。借助Galerkin方法得到了解的整体存在性，与此同时，当初始能量为正且有界时，通过分析的方法可以证明解爆破。 In this paper, the initial-boundary value problem for a class of degenerate hyperbolic equation is studied. The global existence of solution is obtained by introducing Galerkin method. Also the global nonexistence of solution can be verified by using the analysis method while the initial energy is positive and bounded.

Краевая задача для гиперболического уравнения третьего порядка с вырождением порядка внутри области

Исследуется краевая задача для гиперболического уравнения третьего порядка с вырождением типа внутри смешанной области. Рассматриваемое уравнение в положительной части области совпадет с уравнением Аллера, которое является уравнением псевдопараболического типа. А в отрицательной части области - с вырождающимся гиперболическим уравнением первого рода, частным случаем которого является уравнение Бицадзе-Лыкова. Доказана теорема существования и единственности решения. Единственность решения задачи доказана с помощью метода Трикоми. Из функциональных соотношений, принесенных на линию вырождения порядка из положительной и отрицательной частей области, приходим к уравнению Вольтерра второго рода типа свертки относительно следа производной искомого решения. Путем применения метода интегрального преобразования Лапласа решение интегрального уравнения находится в явном виде. Далее решение исследуемой задачи выписывается в явном виде как решение второй краевой задачи для уравнения Аллера в положительной части области и как решение задачи Коши для вырождающегося гиперболического уравнения первого рода в отрицательной части области.

On some hyperbolic type equations and weighted anisotropic Hardy operators

We introduce a version of weighted anisotropic Morrey spaces and anisotropic Hardy operators. We find conditions for boundedness of these operators in such spaces. We also reveal the role of these operators in solving some classes of degenerate hyperbolic partial differential equations. Copyright © 2016 John Wiley & Sons, Ltd.

The motion of closed hypersurfaces in the central force fields

This paper studies the large time existence for the motion of closed hypersurfaces in a radially symmetric potential. Physically, this surface can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The evolution of such surface has been investigated by Schnürer and Smoczyk [20]. To study its motion, we introduce a quasi-linear degenerate hyperbolic equation which describes the motion of the surfaces extrinsically. Our main results show the large time existence of such Cauchy problem and the stability with respect to small initial data. When the radially symmetric potential function v≡1, the local existence and stability results have been obtained by Notz [18]. The proof is based on a new Nash–Moser iteration scheme.

Well-Posedness of Dirichlet and Poincaré Problems in a Cylindrical Domain for Degenerate Multidimensional Hyperbolic Equations with Gellerstedt Operator

We prove that the Dirichlet and Poincar´e problems for degenerate multidimensional hyperbolic equations with Gellerstedt operator in a cylindrical domain are uniquely solvable and establish a criterion of uniqueness for the regular solutions of these problems.

The Gellerstedt Equation with Integral Perturbation in the Cauchy Data

We establish the well-posedness of the problem for a degenerate hyperbolic equation of the first kind in the characteristic triangle with integral perturbation in the Cauchy data along the degeneracy line. Bibliography: 16 titles.

Краевая задача для вырождающегося внутри области гиперболического уравнения

Краевая задача для вырождающегося внутри области гиперболического уравнения

Задача Гурса для нагруженного вырождающегося гиперболического уравнения второго порядка с оператором Геллерстедта в главной части

Рассматривается нагруженное вырождающееся гиперболическое уравнение второго порядка с переменными коэффициентами. Главная часть уравнения представляет собой оператор Геллерстедта. Нагруженное слагаемое представляет собой след искомого решения на линии вырождения, которая лежит внутри области. Исследуется задача с данными на одной из характеристик исследуемого уравнения. В модельном случае, когда коэффициенты при младших членах обращаются в ноль, решение задачи Гурса выписано в явном виде. При этом использовалась функция Грина-Адамара для уравнения Эйлера-Дарбу-Пуассона. В общем случае разрешимость задачи Гурса эквивалентным образом редуцирована к вопросу о разрешимости интегрального уравнения Вольтерра второго рода. В этом случае использована схема, реализованная С. Геллерстедтом при доказательстве существования решения второй задачи Дарбу для рассматриваемого уравнения без нагрузки. В обоих случаях существенно использовались известные свойства функции Грина-Адамара.

On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains

In [20,21], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation ∂t2u−t2l−1Δu=f(t,x,u) with an initial data u(0,x)=φ(x)∈Hs(Rn) (0≤s<n2), where (t,x)∈R×Rn, n≥2, l∈N, f(t,x,u) is C1 smooth in its arguments and has compact support with respect to the variable x. Under the assumption of the subcritical growth of f(t,x,u) on u, we will show the existence and regularity of the considered solution in the mixed type domain [−T0,T0]×Rn for some fixed constant T0>0.

Degenerate hyperbolic equations with lower degree degeneracy

Degenerate hyperbolic equations with lower degree degeneracy

Maslov’s canonical operator for degenerate hyperbolic equations

We present a version of Maslov’s canonical operator to be used when constructing asymptotic solutions of hyperbolic equations degenerating in a special way on the boundary of the spatial domain.

Exact Boundary Controllability of 1-D Parabolic and Hyperbolic Degenerate Equations

The boundary controllability of a class of one-dimensional degenerate equations is studied in this paper. The novelty of this work is that the control acts through the part of the boundary where degeneracy occurs. First we consider a class of degenerate hyperbolic equations. Then, we prove sharp observability estimates for these equations using nonharmonic Fourier series. The transmutation method yields a result for the corresponding class of degenerate parabolic equations.

Задача со смещением для вырождающегося внутри области гиперболического уравнения

For a degenerate hyperbolic equation in characteristic region (lune) a boundary-value problem with operators of fractional integro-differentiation is studied. The solution of this equation on the characteristics is related point-to-point to the solution and its derivative on the degeneration line. The uniqueness theorem is proved by the modified Tricomi method with inequality-type constraints on the known functions. Question of the problem solution's existence is reduced to the solvability of a singular integral equation with Cauchy kernel of the normal type.

The existence and singularity structures of low regularity solutions to higher order degenerate hyperbolic equations

This paper is a continuation of our partial work in [21], where we established that the bounded local solution u(t,x) exists and is piecewise smooth for the second order degenerate hyperbolic equation (∂t2−tmΔx)u=f(t,x,u) with the initial data of C1-piecewise smooth u(0,x) and piecewise smooth ∂tu(0,x). In the present paper, we will consider the lower regularity solution of the higher order degenerate hyperbolic equation in the category of discontinuous and even unbounded functions.

A problem with generalized fractional integro-differentiation operators of arbitrary order

For a degenerate hyperbolic equation we study a problem with fractional integro-differentiation operators in the boundary condition on the characteristic part of the boundary. We determine intervals for parameters of generalized operators of arbitrary order with a Gauss hypergeometric function such that the problem either is uniquely solvable or has more than one solution.

Optimal decay–error estimates for the hyperbolic–parabolic singular perturbation of a degenerate nonlinear equation

We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter ε in front of the second-order time-derivative.In a recent paper, under a suitable assumption on initial data, we proved decay–error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem. These estimates show in the same time that the difference tends to zero both as ε→0+, and as t→+∞. In particular, in that case the difference decays faster than the two terms separately.In this paper we consider the complementary assumption on initial data, and we show that now the optimal decay–error estimates involve a decay rate which is slower than the decay rate of the two terms.In both cases, the improvement or deterioration of decay rates depends on the smallest frequency represented in the Fourier components of initial data.