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

Popivanov, Nedyu; Hristov, Tsvetan; Nikolov, Aleksey; Schneider, Manfred

doi:10.1155/2017/1571959

Cited by 18 publications

(7 citation statements)

References 31 publications

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“…In fact, if set ( ) = 1+ , then we can choose such that < for Dirichlet problem (6) in , and then (51) holds which is independent of (4).…”

Section: (50)mentioning

confidence: 99%

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Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term

Zhang

2018

Advances in Mathematical Physics

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“…In fact, if set ( ) = 1+ , then we can choose such that < for Dirichlet problem (6) in , and then (51) holds which is independent of (4).…”

Section: (50)mentioning

confidence: 99%

“…[5] established the fundamental solutions to the corresponding differential operator for < 1/2 by using the finite part of divergence integrals in theory of distribution. For = 3, the existence of singular solution to Protter problem and Protter-Morawetz problem was considered in [6,7].…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term

Zhang

2018

Advances in Mathematical Physics

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“…Currently, we have constructed a singular solution for the same linear Protter problem for wave equation (m=0) with exponential growth of singularity at the point O(see [13] [28], [29].…”

Section: Remark 10mentioning

confidence: 99%

“…Here we are looking for different kinds of solutions of the homogeneous Protter weakly hyperbolic problem (28) - (29).…”

Section: Remark 10mentioning

confidence: 99%

Nonexistence of nontrivial generalized solutions for 2-D and 3-D BVPs with nonlinear mixed type equations

Dechevski¹,

Payne

Popivanov

2017

AIP Conference Proceedings

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Abstract. A brief survey of known results, open problems and new contributions to the understanding of the nonexistence of nontrivial solutions to nonlinear boundary value problems (BVPs) whose linear part is of mixed elliptic-hyperbolic type is given. Crucial issues discussed include: the role of so-called critical growth of the nonlinear terms in the equation (often related to threshold values of continuous and compact embedding for Sobolev spaces in Lebesgue spaces), the role that hyperbolicity in the principal part plays in over-determining solutions with classical regularity if data is prescribed everywhere on the boundary, the relative lack of regularity that solutions to such problems possess and the subsequent importance to address nonexistence of generalized solutions.

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“…For instance, such representations have been found for solutions of Protter problems for equations of Keldysh type in (3 + 1) dimensions [24,25]. In [24], the representations have also been applied to derive asymptotic formulas for the solution. The Euler-Darboux equation and integral representations of its solution have also arisen in the context of hierarchies of integrable systems such as the KdV hierarchy [20].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics to all orders of the Euler–Darboux equation in a triangle

Mauersberger

2019

Journal of Mathematical Analysis and Applications

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In Einstein's theory of relativity, the interaction of two collinearly polarized plane gravitational waves can be described by a Goursat problem for the Euler-Darboux equation in a triangular domain. In this paper, using a representation of the solution in terms of Abel integrals, we give a full asymptotic expansion of the solution near the diagonal of the triangle. The expansion is related to the formation of a curvature singularity of the spacetime. In particular, our framework allows for boundary data with derivatives which are singular at the corners. This level of generality is crucial for the application to gravitational waves.PDEs in the case of colliding electromagnetic plane waves [3,7]. In the linear limit, both of these coupled equations reduce to the Euler-Darboux equation (1.1).Of particular interest is the behavior of the solution of (1.1) near the triangle's diagonal edge x+y = 1. Indeed, this behavior is related to the formation of a curvature singularity of the spacetime by the mutual focusing of the colliding waves. In [26], it was observed that the solution must behave like ln(1 − x − y) as x + y → 1. In this paper, using an Abel integral representation of the solution, we derive an asymptotic expansion to all orders as x + y → 1.Our first result (Theorem 1) establishes a mathematically precise version of the classical Abel integral representation of the solution of the Goursat problem for (1.1) with given boundary data. This type of representation is well known (cf. e.g. [11,14,15]) and the main purpose of Theorem 1 is to provide a formulation that is suitable for our needs. We discuss regularity, the behavior at the boundary of D, and uniqueness of the solution under reasonable assumptions on the boundary data. In this context, reasonable means that our results can be applied to the common examples of collinear solutions (cf. e.g. [9,19,26]). In particular, we allow for boundary data with singular derivatives at the corners of D. As a consequence, our representation of the solution of (1.1) contains singular integrands, which is the main challenge in the analysis of the Goursat problem.Our second result (Theorem 2) describes the asymptotic behavior of the solution near the diagonal of D. We show that V (x, 1 − x − ) admits an asymptotic expansion to all orders of the formwhere the coefficients f j , g j are given explicitly in terms of the boundary data V 0 (x) = V (x, 0) and V 1 (y) = V (0, y), and where the error term is uniform on compact subsets

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

Cited by 18 publications

References 31 publications

Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term

Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term

Nonexistence of nontrivial generalized solutions for 2-D and 3-D BVPs with nonlinear mixed type equations

Asymptotics to all orders of the Euler–Darboux equation in a triangle

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