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Hiroyuki Takamura - One of the best experts on this subject based on the ideXlab platform.
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nonexistence of global solutions for a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case
Mediterranean Journal of Mathematics, 2020Co-Authors: Alessandro Palmieri, Hiroyuki TakamuraAbstract:In this paper, we consider the blow-up for solutions to a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case. The assumption on the time-dependent coefficients for the damping terms means that these coefficients are summable and nonnegative. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single Semilinear Equation, we employ Kato’s lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case, an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p - q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of Semilinear not-damped wave Equations with the same kind of nonlinearities.
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nonexistence of global solutions for a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case
arXiv: Analysis of PDEs, 2018Co-Authors: Alessandro Palmieri, Hiroyuki TakamuraAbstract:In this paper we consider the blow-up for solutions to a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single Semilinear Equation, we employ Kato's lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p-q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of Semilinear not-damped wave Equations with the same kind of nonlinearities.
Hong-wei Zhang - One of the best experts on this subject based on the ideXlab platform.
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Wave Equation on certain noncompact symmetric spaces
2020Co-Authors: Hong-wei ZhangAbstract:In this paper, we prove sharp pointwise kernel estimates and dis-persive properties for the linear wave Equation on noncompact Riemannian symmetric spaces G/K of any rank with G complex. As a consequence, we deduce Strichartz inequalities for a large family of admissible pairs and prove global well-posedness results for the corresponding Semilinear Equation with low regularity data as on hyperbolic spaces.
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Wave and Klein-Gordon Equations on certain locally symmetric spaces
The Journal of Geometric Analysis, 2020Co-Authors: Hong-wei ZhangAbstract:This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave Equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global well-posedness results for the corresponding Semilinear Equation with low regularity data as on real hyperbolic spaces.
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Wave Equation on general noncompact symmetric spaces
2020Co-Authors: Jean-philippe Anker, Hong-wei ZhangAbstract:We establish sharp pointwise kernel estimates and dispersive properties for the wave Equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. As consequences, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding Semilinear Equation with low regularity data as on hyperbolic spaces.
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Wave and Klein–Gordon Equations on Certain Locally Symmetric Spaces
The Journal of Geometric Analysis, 2019Co-Authors: Hong-wei ZhangAbstract:This paper is devoted to study the dispersive properties of the linear Klein–Gordon and wave Equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global well-posedness results for the corresponding Semilinear Equation with low regularity data as on real hyperbolic spaces.
Jianqing Chen - One of the best experts on this subject based on the ideXlab platform.
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Multiple positive solutions for a Semilinear Equation with critical exponent and prescribed singularity
Nonlinear Analysis, 2016Co-Authors: Yaoping Chen, Jianqing ChenAbstract:Abstract This paper is devoted to the study of a Semilinear Equation of the form: − Δ u − μ V ( x ) u = | u | 2 ∗ − 2 u + θ h ( x ) , u ∈ H 0 1 ( Ω ) , where Ω ⊂ R N ( N ≥ 3 ) is an open bounded domain with smooth boundary ∂ Ω , 0 μ μ = ( N − 2 2 ) 2 , 2 ∗ = 2 N N − 2 , h ( x ) > 0 and V ( x ) has prescribed finitely many singular points. Using variational methods, we establish some existence and multiplicity of positive solutions for the problem.
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Multiple non-negative solutions to a Semilinear Equation on Heisenberg group with indefinite nonlinearity
Boundary Value Problems, 2015Co-Authors: Lirong Huang, Jianqing Chen, Eugénio M. RochaAbstract:This paper is concerned with the existence and multiplicity of non-negative solutions to the Semilinear Equation $-\Delta_{H} u = K(\xi)\vert u\vert ^{2^{\sharp}-2}u + \mu \vert \xi \vert _{H}^{\alpha}u$ in a bounded domain $\Omega\subset\mathbb{H}^{N}$ with Dirichlet boundary conditions. Here $\mathbb{H}^{N}$ is the Heisenberg group and $2^{\sharp}= 2q/(q-2)$ is the critical exponent of the Sobolev embedding on the Heisenberg group. The function $K(\xi)$ may be sign changing on Ω. Using the variational method, we prove that this problem has at least two non-negative solutions provided μ, α, and $K(\xi)$ satisfy some conditions.
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Multiple positive solutions for a Semilinear Equation with prescribed singularity
Journal of Mathematical Analysis and Applications, 2005Co-Authors: Jianqing ChenAbstract:Variational methods are used to prove the existence of multiple positive solutions for a Semilinear Equation with prescribed finitely many singular points. Some exact local behavior for positive solutions are also given.
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Multiple solutions for a Semilinear Equation involving singular potential and critical exponent
Zeitschrift für angewandte Mathematik und Physik, 2005Co-Authors: Jianqing ChenAbstract:Variational methods are used to prove the existence of positive and sign-changing solutions for a Semilinear Equation involving singular potential and critical exponent in any bounded domain.
Alessandro Palmieri - One of the best experts on this subject based on the ideXlab platform.
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nonexistence of global solutions for a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case
Mediterranean Journal of Mathematics, 2020Co-Authors: Alessandro Palmieri, Hiroyuki TakamuraAbstract:In this paper, we consider the blow-up for solutions to a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case. The assumption on the time-dependent coefficients for the damping terms means that these coefficients are summable and nonnegative. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single Semilinear Equation, we employ Kato’s lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case, an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p - q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of Semilinear not-damped wave Equations with the same kind of nonlinearities.
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nonexistence of global solutions for a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case
arXiv: Analysis of PDEs, 2018Co-Authors: Alessandro Palmieri, Hiroyuki TakamuraAbstract:In this paper we consider the blow-up for solutions to a weakly coupled system of Semilinear damped wave Equations of derivative type in the scattering case. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single Semilinear Equation, we employ Kato's lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p-q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of Semilinear not-damped wave Equations with the same kind of nonlinearities.
Joaquim Serra - One of the best experts on this subject based on the ideXlab platform.
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Radial symmetry of solutions to diffusion Equations with discontinuous nonlinearities
Journal of Differential Equations, 2013Co-Authors: Joaquim SerraAbstract:We prove a radial symmetry result for bounded nonnegative solutions to the p-Laplacian Semilinear Equation −Δpu=f(u) posed in a ball of Rn and involving discontinuous nonlinearities f. When p=2 we obtain a new result which holds in every dimension n for certain positive discontinuous f. When p⩾n we prove radial symmetry for every locally bounded nonnegative f. Our approach is an extension of a method of P.L. Lions for the case p=n=2. It leads to radial symmetry combining the isoperimetric inequality and the Pohozaev identity.
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Radial symmetry of solutions to diffusion Equations with discontinuous nonlinearities
arXiv: Analysis of PDEs, 2011Co-Authors: Joaquim SerraAbstract:We prove a radial symmetry result for bounded nonnegative solutions to the $p$-Laplacian Semilinear Equation $-\Delta_p u=f(u)$ posed in a ball of $\mathbb R^n$ and involving discontinuous nonlinearities $f$. When $p=2$ we obtain a new result which holds in every dimension $n$ for certain positive discontinuous $f$. When $p\ge n$ we prove radial symmetry for every locally bounded nonnegative $f$. Our approach is an extension of a method of P. L. Lions for the case $p=n=2$. It leads to radial symmetry combining the isoperimetric inequality and the Pohozaev identity.
