Protter Problem Research Articles

New singular solutions for the (3+1)-D Protter problem

New singular solutions for the (3+1)-D Protter problem

On the existence and uniqueness of a generalized solution of the Protter problem for (3+1)-D Keldysh-type equations

A (3+1)-dimensional boundary value problem for equations of Keldysh type (the second kind) is studied. Such problems for equations of Tricomi type (the first kind) or for the wave equation were formulated by M.H. Protter (1954) as multidimensional analogues of Darboux or Cauchy-Goursat plane problems. Now, it is well known that Protter problems are not correctly set, and they have singular generalized solutions, even for smooth right-hand sides. In this paper an analogue of the Protter problem for equations of Keldysh type is given. An appropriate generalized solution with possible singularity is defined. Results for uniqueness and existence of such a generalized solution are obtained. Some a priori estimates are stated.

Protter-Morawetz multidimensional problems

About 50 years ago M.H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter-Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain—the nonhomogeneous wave equation is studied in a (3+1)-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in ℝ2. In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex O of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.

Exact Asymptotic Expansion of Singular Solutions for the (2 + 1)‐D Protter Problem

We study three‐dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ2. In contrast to the planar Darboux problem the three‐dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right‐hand side functions there is a uniquely determined generalized solution that may have a strong power‐type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions.

Singular solutions of protter's problem for the 3 + 1;-D wave equation

For the wave equation we study boundary value problems, which are four-dimensional analogues of Darboux problems on the plane. It is shown that for n in ℕ there exists a right hand side smooth function from C n , for which the corresponding unique generalized solution has a strong power-type singularity at the point O. This singularity is isolated at the vertex O of the characteristic cone and does not propagate along the cone. The present article describes the exact behavior of the singular solutions at the point O. We give some necessary and sufficient conditions for existence of solutions with fixed order of singularity. It states some exact a priori estimates for the singular solutions

Criterion for the Uniqueness of a Solution of the Darboux–Protter Problem for Degenerate Multidimensional Hyperbolic Equations

We obtain a criterion for the uniqueness of a regular solution of the Darboux–Protter problem for degenerate multidimensional hyperbolic equations.