On the Cauchy problem for hyperbolic functional-differential equations

Adrian Karpowicz; Henryk Leszczyński

Displaying similar documents to “On the Cauchy problem for hyperbolic functional-differential equations”

$C^{\infty}$ -well posedness of the Cauchy problem for quasi-linear hyperbolic equations with coefficients non-Lipschitz in time and smooth in space

Akisato Kubo, Michael Reissig (2003)

Banach Center Publications

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In this paper we prove the $C^{\infty}$ -well posedness of the Cauchy problem for quasi-linear hyperbolic equations of second order with coefficients non-Lipschitz in t ∈ [0,T] and smooth in x ∈ ℝⁿ.

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.

Evolution equations with parameter in the hyperbolic case

Jan Bochenek, Teresa Winiarska (1996)

Annales Polonici Mathematici

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The purpose of this paper is to give theorems on continuity and differentiability with respect to (h,t) of the solution of the initial value problem du/dt = A(h,t)u + f(h,t), u(0) = u₀(h) with parameter $h \in Ω \subset ℝ^{m}$ in the “hyperbolic” case.

Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity

Jerzy August Gawinecki

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CONTENTS1. Introduction..................................................................................................................................... 5 1.1. Main Theorem 1.1................................................................................................................. 8 1.2. Main Theorem 1.2................................................................................................................. 92. Radon transform.......................................................................................................................................

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.

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,...

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 $ℍ_{ℝ}^{5}$ .

Bounded Solutions for the Degasperis-Procesi Equation

Giuseppe Maria Coclite, Kenneth H. Karlsen (2008)

Bollettino dell'Unione Matematica Italiana

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This paper deals with the well-posedness in $L^{1}\cap L^{\infty}$ of the Cauchy problem for the Degasperis-Procesi equation. This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves.

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.

Comparison of solutions and successive approximations in the theory of the equation $\partial^{2} z / \partial x \partial y = f (x, y, z, \partial z / \partial x, \partial z / \partial y)$

J. Kisyński, A. Pelczar

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CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^{1}' * (Δ_{a, b}; E)$ AND $L_{1}^{1, *} (Δ_{a, b}; E)$ ......................... 71. Basic function classes ......................................................................................................................................................................................

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...

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation $\partial_{t} u - \partial ³_{t x x} u + 2 κ \partial_{x} u + \partial_{x} [g (u) / 2] = γ (2 \partial_{x} u \partial ²_{x x} u + u \partial ³_{x x x} u)$ for the initial data u₀(x) in the Besov space $B_{p, r}^{s} (ℝ)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^{m}$ -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

Tomasz Człapiński (1997)

Annales Polonici Mathematici

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We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order (1) $D ₓ z (x, y) = f (x, y, z (x, y), (W z) (x, y), D_{y} z (x, y))$ on E, (2) z(x,y) = ϕ(x,y) on [-τ₀,0]×[-b,b], where E is the Haar pyramid and τ₀ ∈ ℝ₊, b = (b₁,...,bₙ) ∈ ℝⁿ₊. Using the method of bicharacteristics and the method of successive approximations for a certain functional integral system we prove, under suitable assumptions, a theorem on the local existence of weak solutions of the problem (1),(2).

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...

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

Zofia Szmydt, Bogdan Ziemian (1985)

Annales Polonici Mathematici

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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...

Regularized cosine existence and uniqueness families for second order abstract Cauchy problems

Jizhou Zhang (2002)

Studia Mathematica

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Let $C_{i}$ (i = 1,2) be two arbitrary bounded operators on a Banach space. We study (C₁,C₂)-regularized cosine existence and uniqueness families and their relationship to second order abstract Cauchy problems. We also prove some of their basic properties. In addition, Hille-Yosida type sufficient conditions are given for the exponentially bounded case.

An inequality for spherical Cauchy dual tuples

Sameer Chavan (2013)

Colloquium Mathematicae

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Let T be a spherical 2-expansive m-tuple and let $T^{}$ denote its spherical Cauchy dual. If $T^{}$ is commuting then the inequality $\binom{\sum_{| β | = k} {(β!)}^{- 1} {(T^{})}^{β} (T^{}) *^{β} \leq (k + m - 1}{k) \sum_{| β | = k} {(β!)}^{- 1} (T^{}) *^{β} {(T^{})}^{β}}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.

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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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.

The Cauchy kernel for cones

Sławomir Michalik (2004)

Studia Mathematica

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A new representation of the Cauchy kernel $_{Γ}$ for an arbitrary acute convex cone Γ in ℝⁿ is found. The domain of holomorphy of $_{Γ}$ is described. An estimation of the growth of $_{Γ}$ near the singularities is given.

Convergence results for unbounded solutions of first order non-linear differential-functional equations

Henryk Leszczyński (1996)

Annales Polonici Mathematici

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We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t} z (t, x) = f (t, x, z (t, x), z_{(t, x)}, D_{x} z (t, x))$ or $D_{t} z (t, x) = f (t, x, z (t, x), z, D_{x} z (t, x))$ . We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.

On a functional equation with derivative and symmetrization

Adam Bobrowski, Małgorzata Kubalińska (2006)

Annales Polonici Mathematici

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We study existence, uniqueness and form of solutions to the equation $α g - β g^{'} + γ g_{e} = f$ where α, β, γ and f are given, and $g_{e}$ stands for the even part of a searched-for differentiable function g. This equation emerged naturally as a result of the analysis of the distribution of a certain random process modelling a population genetics phenomenon.