Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a similar one proposed by M. Protter in 1952, but for Tricomi-type equations which, in part, are closely connected with transonic fluid dynamics.We consider a properly defined, in a special function space, generalized solution to problem PK for which existence and uniqueness theorems hold. It is known that it may have a strong power-type singularity at one boundary point even for very smooth right-hand sides of the equation. In the present paper we study the asymptotic behavior of the generalized solutions of problem PK at the singular point. There are given orthogonality conditions on the right-hand side of the equation, which are necessary and sufficient for the existence of a generalized solution with fixed order of singularity

Asymptotic expansions of singular solutions for (3+1)-D Protter problems

AbstractFour-dimensional boundary value problems for the nonhomogeneous wave equation are studied, which are analogues of Darboux problems in the plane. The smoothness of the right-hand side function of the wave equation is decisive for the behavior of the solution of the boundary value problem. It is shown that for each n∈N there exists such a right-hand side function from Cn, for which the uniquely determined generalized solution has a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. The present article describes asymptotic expansions of the generalized solutions in negative powers of the distance to this singularity point. Some necessary and sufficient conditions for existence of regular solutions, or solutions with fixed order of singularity, are derived and additionally some a priori estimates for the singular solutions are given

Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a similar one proposed by M. Protter in 1952, but for Tricomi-type equations which, in part, are closely connected with transonic fluid dynamics. We consider a properly defined, in a special function space, generalized solution to problem PK for which existence and uniqueness theorems hold. It is known that it may have a strong power-type singularity at one boundary point even for very smooth right-hand sides of the equation. In the present paper we study the asymptotic behavior of the generalized solutions of problem PK at the singular point. There are given orthogonality conditions on the right-hand side of the equation, which are necessary and sufficient for the existence of a generalized solution with fixed order of singularity

Сингулярни решения на задачата на Протър за клас от тримерни хиперболични уравнения

Недю Иванов Попиванов, Алексей Йорданов Николов - През 1952 г. М. Протър формулира нови гранични задачи за вълновото уравнение, които са тримерни аналози на задачите на Дарбу в равнината. Задачите са разгледани в тримерна област, ограничена от две характеристични конуса и равнина. Сега, след като са минали повече от 50 години, е добре известно, че за безброй гладки функции в дясната страна на уравнението тези задачи нямат класически решения, а обобщеното решение има силна степенна особеност във върха на характеристичния конус, която е изолирана и не се разпространява по конуса. Тук ние разглеждаме трета гранична задача за вълновото уравнение с младши членове и дясна страна във формата на тригонометричен полином. Дадена е по-нова от досега известната априорна оценка за максимално възможната особеност на решенията на тази задача. Оказва се, че при по-общото уравнение с младши членове възможната сингулярност е от същия ред като при чисто вълновото уравнение.For 3-D wave equation M. Protter formulated (1952) some boundary value problems which are three-dimensional analogues of the Darboux problems on the plane. Protter studied these problems in a 3-D domain, bounded by two characteristic cones and by a plane region. Now, more than 50 years later, it is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions and the generalized solution have a strong power type singularity at the vertex of the characteristic cone, which is isolated and does not propagate along the cone. In the present paper we consider the third boundary value problem for the wave equation involving lower order terms with a right-hand function of the form of trigonometric polynomial and give a new upper estimate of possible singularity of the solutions. It is interesting that the solutions of the considered problem have the same order of possible singularity as the solutions of the wave equation without lower order terms. *2000 Mathematics Subject Classification: 35L05, 35L20, 35D05, 35A20.This research was partially supported by the Bulgarian NSF under Grant DO 02-75/2008 and Grant DO 02-115/2008

Задача на дарбу за клас тримерни слабо хиперболични уравнения

Недю Попиванов, Цветан Христов - Изследвани са някои тримерни аналози на задачата на Дарбу в равнината. През 1952 М. Протер формулира нови тримерни гранични задачи както за клас слабо хиперболични уравнения, така и за някои хиперболично-елиптични уравнения. За разлика от коректността на двумерната задача на Дарбу, новите задачи са некоректни. За слабо хиперболични уравнения, съдържащи младши членове, ние намираме достатъчни условия както за съществуване и единственост на обобщени решения с изолирана степенна особеност, така и за единственост на квази-регулярни решения на задачата на Протер.Some three-dimensional analogues of the plane Darboux problems for weakly hyperbolic equations are studied. In 1952 M. Protter formulated new 3-D boundary value problems for a class of weakly hyperbolic equations, as well as for some hyperbolic- elliptic equations. In the contrast of the well-posedness of the Darboux problem in 2-D case, the new problems are strongly ill-posed. For weakly hyperbolic equation, involving lower order terms, we find sufficient conditions for existence and uniqueness of generalized solutions with isolated power-type singularities as well as for uniqueness of quasi-regular solutions to the Protter problem. *2000 Mathematics Subject Classification: 35L20, 35A20.This work was partially supported by the Bulgarian NSF under Grant DO–02–75/2008 and Grant DO–02–115/2008, and by Sofia University Grant 184/2010

Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex O of the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance to O. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained

Квазирегулярни решения на тримерни уравнения от типа на Трикоми и типа на Келдиш

Цветан Д. Христов, Недю Ив. Попиванов, Манфред Шнайдер - Изучени са някои тримерни гранични задачи за уравнения от смесен тип. За уравнения от типа на Трикоми те са формулирани от М. Протер през 1952, като тримерни аналози на задачите на Дарбу или Коши–Гурса в равнината. Добре известно е, че новите задачи са некоректни. Ние формулираме нова гранична задача за уравнения от типа на Келдиш и даваме понятие за квазиругулярно решение на тази задача и на eдна от задачите на Протер. Намерени са достатъчни условия за единственост на такива решения.Some 3D boundary value problems for equations of mixed type are studied. For equations of Tricomi type they are formulated by M. Protter in 1952 as three-dimensional analogues of the plane Darboux or Cauchy-Goursat problems. It is well-known that the new problems are strongly ill-posed. We formulate a new boundary value problem for equations of Keldish type and give a notion for quasi-regular solutions to this problem and to one of Protter problems. Sufficient conditions for uniqueness of such solution are found.This work was partially supported by the Bulgarian NSF under Grant DO–02–75/2008 and Grant DO–02–115/2008, and by Sofia University Grant 153/2011

Сингулярни решения с експоненциален ръст за четиримерното вълново уравнение

Недю И. Попиванов, Тодор П. Попов, Рудолф Шерер - Разглеждат се четиримерни гранични задачи за нехомогенното вълново уравнение. Те са предложени от М. Протер като многомерни аналози на задачата на Дарбу в равнината. Известно е, че единственото обобщено решение може да има силна степенна особеност само в една гранична точка. Тази сингулярност е изолирана във върха на характеристичния конус и не се разпространява по конуса. Друг аспект на проблема е, че задачата не е фредхолмова, тъй като има безкрайномерно коядро. Предишни резултати сочат, че решението може да има най-много експоненциален ръст, но оставят открит въпроса дали наистина съществуват такива решения. Показваме, че отговора на този въпрос е положителен и строим обобщено решение на задачата на Протер с експоноциална особеност.Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied. They were proposed by M. Protter as multidimensional analogues of Darboux problems in the plane. It is known that the unique generalized solution may have a strong power-type singularity at only one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Another aspect is that the problem is not Fredholm, since it has infinite-dimensional cokernel. Some known results suggest that the solution may have at most exponential growth, but the question whether such solutions really exist was still open. We show that the answer is positive and construct generalized solution of Protter problem with exponential singularity. *2000 Mathematics Subject Classification: 35L05, 35L20, 35D05, 35D10, 35C10.The research was partially supported by the Bulgarian Sofia University Grant 184/2010 and Bulgarian NSF under Grants DO 02-115/2008 and DO 02-75/2008