Nonexistence of Nontrivial Solutions for Supercritical Equations of Mixed Elliptic-Hyperbolic Type

Lupo, Daniela; Payne, Kevin R.; Popivanov, Nedyu

doi:10.1007/3-7643-7401-2_25

Cited by 25 publications

(25 citation statements)

References 22 publications

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“…The key ingredient for this nonexistence result is a Pohožaev type identity for the operator G γ in the domain . For further details see for instance [21], [23] and [19].…”

Section: Two Applications Of the Moving Planes To The Grushin Operatormentioning

confidence: 99%

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Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Monticelli¹

2010

J. Eur. Math. Soc.

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Abstract. We deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation. A good example of such an operator is the Grushin operator on R d+k , to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains, which is a generalization of the result of Gidas-Ni-Nirenberg [12], [13], and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space, which is a generalization of the result of Gidas-Spruck [14] and Chen-Li [6]. We use the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.

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“…The key ingredient for this nonexistence result is a Pohožaev type identity for the operator G γ in the domain . For further details see for instance [21], [23] and [19].…”

Section: Two Applications Of the Moving Planes To The Grushin Operatormentioning

confidence: 99%

“…Let ε 0 := inf ∂B 1 (0) v. Then ε 0 > 0 since v is strictly positive on R d+k \ {0}. Now, recalling definition (19), for every m ∈ N one has in m…”

Section: (24)mentioning

confidence: 99%

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Monticelli¹

2010

J. Eur. Math. Soc.

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“…Concerning nonexistence principle for nontrivial solution of semilinear mixed-type equations in multidimensional case, we refer to [29].…”

Section: Historical Remarks On the Main Resultsmentioning

confidence: 99%

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

Popivanov

Popov

Tesdall

2014

Abstract and Applied Analysis

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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 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 . Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

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“…Different aspects of Protter problems and several their generalizations (including some applications in the industrial explosion process) are studied by many authors (see Aldashev and Kim [18], Choi and Park [19], Aldashev [20], and references therein). For different statements of other related problems for mixed-type equations of the first kind, including nonlinear equations, see [21][22][23][24][25][26][27].…”

Section: History Of the Problem And Motivationmentioning

confidence: 99%

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

Popivanov

Hristov

Nikolov

et al. 2017

Advances in Mathematical Physics

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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.

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Nonexistence of Nontrivial Solutions for Supercritical Equations of Mixed Elliptic-Hyperbolic Type

Cited by 25 publications

References 22 publications

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

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

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

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