Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity

Jerzy August Gawinecki

Displaying similar documents to “Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity”

Sur la classification des hexagones hyperboliques à angles droits en dimension 5

François Delgove, Nicolas Retailleau (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The aim of this paper is to give a classification of the right-angled hyperbolic hexagons in the real hyperbolic space $ℍ_{ℝ}^{5}$ , by using a quaternionic distance between geodesics in $ℍ_{ℝ}^{5}$ .

On the Picard problem for hyperbolic differential equations in Banach spaces

Antoni Sadowski (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation $z_{x y} = f (x, y, z, z_{x y})$ on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation $z_{x y} = f (x, y, z, z_{x}, z_{x y})$ using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].

$L^{p} - L^{q}$ time decay estimates for the solution of the linear partial differential equations of thermodiffusion

Arkadiusz Szymaniec (2010)

Applicationes Mathematicae

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We consider the initial-value problem for a linear hyperbolic parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove $L^{p} - L^{q}$ time decay estimates for the solution of the associated linear Cauchy problem.

Computing the determinantal representations of hyperbolic forms

Mao-Ting Chien, Hiroshi Nakazato (2016)

Czechoslovak Mathematical Journal

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The numerical range of an $n \times n$ matrix is determined by an $n$ degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an $n$ degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus $g = 1$ . We reformulate the Fiedler-Helton-Vinnikov formulae for the genus $g = 0, 1$ , and present an elementary...

An invariance method for constructing fundamental solutions for $P (□_{m} n)$

Zofia Szmydt, Bogdan Ziemian (1985)

Annales Polonici Mathematici

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Hyperbolic spaces in Teichmüller spaces

Christopher J. Leininger, Saul Schleimer (2014)

Journal of the European Mathematical Society

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We prove, for any $n$ , that there is a closed connected orientable surface $S$ so that the hyperbolic space $ℍ^{n}$ almost-isometrically embeds into the Teichmüller space of $S$ , with quasi-convex image lying in the thick part. As a consequence, $ℍ^{n}$ quasi-isometrically embeds in the curve complex of $S$ .

Smooth Gevrey normal forms of vector fields near a fixed point

Laurent Stolovitch (2013)

Annales de l’institut Fourier

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We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $α$ -Gevrey vector field with an hyperbolic linear part admits a smooth $β$ -Gevrey transformation to a smooth $β$ -Gevrey normal form. The Gevrey order...

Fundamental solution $E_{n}$ of the operator $\partial^{2} / \partial t^{2} - Δ_{n}$ for n>3

Zofia Szmydt, Bogdan Ziemian (1979)

Annales Polonici Mathematici

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Front propagation for nonlinear diffusion equations on the hyperbolic space

Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)

Journal of the European Mathematical Society

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We study the Cauchy problem in the hyperbolic space $ℍ^{n} (n \geq 2)$ for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space $ℝ^{n}$ new phenomena arise, which depend on the properties of the diffusion process in $ℍ^{n}$ . We also investigate a family of travelling wave solutions,...

The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the following Darboux problem for the functional differential equation $\partial ² u / \partial x \partial y (x, y) = f (x, y, u_{(x, y)}, \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] $0, a] \times (0, b],$ where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

Systole growth for finite area hyperbolic surfaces

Florent Balacheff, Eran Makover, Hugo Parlier (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this note, we observe that the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g, n)$ is greater than a function that grows logarithmically in terms of the ratio $g / n$ .

Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

David Borthwick, Colin Guillarmou (2016)

Journal of the European Mathematical Society

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On geometrically finite hyperbolic manifolds $Γ ∖ ℍ^{d}$ , including those with non-maximal rank cusps, we give upper bounds on the number $N (R)$ of resonances of the Laplacian in disks of size $R$ as $R \to \infty$ . In particular, if the parabolic subgroups of $Γ$ satisfy a certain Diophantine condition, the bound is $N (R) = 𝒪 (R^{d} {(\log R)}^{d + 1})$ .

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila, Mikhail Lyubich, Weixiao Shen (2011)

Journal of the European Mathematical Society

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We study the parameter space of unicritical polynomials $f_{c} : z \mapsto z^{d} + c$ . For complex parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives

Adrian Karpowicz (2008)

Annales Polonici Mathematici

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We consider the Darboux problem for a functional differential equation: $(\partial ² u) / (\partial x \partial y) (x, y) = f (x, y, u_{(x, y)}, u (x, y), \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]∖(0,a]×(0,b], where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.

Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Victor Gerasimov, Leonid Potyagailo (2013)

Journal of the European Mathematical Society

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We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using the Floyd completion we further prove that the property of relative hyperbolicity is invariant under quasi-isometric maps. If a finitely generated group $H$ admits a quasi-isometric map $ϕ$ into a relatively hyperbolic group $G$ then $H$ is itself relatively hyperbolic with respect to a system of subgroups whose image under $ϕ$ is situated within a uniformly bounded...

Hyperideal polyhedra in hyperbolic 3-space

Xiliang Bao, Francis Bonahon (2002)

Bulletin de la Société Mathématique de France

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A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic $3$ -space $ℍ^{3}$ which, in the projective model for $ℍ^{3} \subset {ℝℙ}^{3}$ , is just the intersection of $ℍ^{3}$ with a projective polyhedron whose vertices are all outside $ℍ^{3}$ and whose edges all meet $ℍ^{3}$ . We classify hyperideal polyhedra, up to isometries of $ℍ^{3}$ , in terms of their combinatorial type and of their dihedral angles.

Noncharacteristic mixed problems for hyperbolic systems of the first order

Ewa Zadrzyńska

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CONTENTS1. Introduction....................................................................................................................................................52. Notations and preliminaries .........................................................................................................................11 2.1. Function spaces and spaces of distributions............................................................................................11 2.2. Perturbations...

Nonuniform center bunching and the genericity of ergodicity among $C^{1}$ partially hyperbolic symplectomorphisms

Artur Avila, Jairo Bochi, Amie Wilkinson (2009)

Annales scientifiques de l'École Normale Supérieure

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We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns–Wilkinson and Avila–Santamaria–Viana. Combining this new technique with other constructions we prove that $C^{1}$ -generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

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We prove the central limit theorem for the multisequence $\sum_{1 \leq n ₁ \leq N ₁} \dots \sum_{1 \leq n_{d} \leq N_{d}} a_{n ₁, . . ., n_{d}} c o s (⟨ 2 π m, A ₁^{n ₁} . . . A_{d}^{n_{d}} x ⟩)$ where $m \in ℤ^{s}$ , $a_{n ₁, . . ., n_{d}}$ are reals, $A ₁, . . ., A_{d}$ are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in ${[0, 1]}^{s}$ . The main tool is the S-unit theorem.

Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps

Martine Babillot, Marc Peigné (2006)

Bulletin de la Société Mathématique de France

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We consider a large class of non compact hyperbolic manifolds $M = ℍ^{n} / Γ$ with cusps and we prove that the winding process $(Y_{t})$ generated by a closed $1$ -form supported on a neighborhood of a cusp $𝒞$ , satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp $𝒞$ and the Poincaré exponent $δ$ of $Γ$ . No assumption on the value of $δ$ is required and this theorem generalises previous results due to Y. Guivarc’h, Y. Le Jan, J. Franchi and N. Enriquez. ...

New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino, Henrique F. de Lima (2015)

Archivum Mathematicum

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In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $ℍ^{n + 1}$ , that is, complete hypersurfaces of $ℍ^{n + 1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R = a H + b$ for some $a$ , $b \in ℝ$ . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $ℍ^{n + 1}$ . Furthermore,...