Search

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.

Four-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.