Existence of the fundamental solution of a second order evolution equation

Jan Bochenek

Displaying similar documents to “Existence of the fundamental solution of a second order evolution equation”

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.

Forms, functional calculus, cosine functions and perturbation

Wolfgang Arendt, Charles J. K. Batty (2007)

Banach Center Publications

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In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded $H^{\infty}$ -calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function...

Universal stability of Banach spaces for ε -isometries

Lixin Cheng, Duanxu Dai, Yunbai Dong, Yu Zhou (2014)

Studia Mathematica

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Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T : L (f) \equiv \bar{s p a n} f (X) \to X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally...

On stable currents in positively pinched curved hypersurfaces

Jintang Li (2003)

Colloquium Mathematicae

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Let Mⁿ (n ≥ 3) be an n-dimensional complete hypersurface in a real space form N(c) (c ≥ 0). We prove that if the sectional curvature $K_{M}$ of M satisfies the following pinching condition: $c + δ < K_{M} \leq c + 1$ , where δ = 1/5 for n ≥ 4 and δ = 1/4 for n = 3, then there are no stable currents (or stable varifolds) in M. This is a positive answer to the well-known conjecture of Lawson and Simons.

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.

On the eigenfunction expansion method for semilinear dissipative equations in bounded domains and the Kuramoto-Sivashinsky equation in a ball

V. V. Varlamov (2001)

Studia Mathematica

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Presented herein is a method of constructing solutions of semilinear dissipative evolution equations in bounded domains. For small initial data this approach permits one to represent the solution in the form of an eigenfunction expansion series and to calculate the higher-order long-time asymptotics. It is applied to the spatially 3D Kuramoto-Sivashinsky equation in the unit ball B in the linearly stable case. A global-in-time mild solution is constructed in the space $C ⁰ ([0, \infty), H ₀^{s} (B))$ , s < 2, and...

Stable invariant subspaces for operators on Hilbert space

John B. Conway, Don Hadwin (1997)

Annales Polonici Mathematici

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If T is a bounded operator on a separable complex Hilbert space ℋ, an invariant subspace ℳ for T is stable provided that whenever $T_{n}$ is a sequence of operators such that $‖ T_{n} - T ‖ \to 0$ , there is a sequence of subspaces $ℳ_{n}$ , with $ℳ_{n}$ in $L a t T_{n}$ for all n, such that $P_{ℳ_{n}} \to P_{ℳ}$ in the strong operator topology. If the projections converge in norm, ℳ is called a norm stable invariant subspace. This paper characterizes the stable invariant subspaces of the unilateral shift of finite multiplicity and normal operators. It also...

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 ......................................................................................................................................................................................

Extreme operators on 2-dimensional $l_{p}$ -spaces

Ryszard Grząślewicz (1981)

Colloquium Mathematicae

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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.

On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

Sameer Chavan (2011)

Studia Mathematica

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The Cauchy dual operator T’, given by $T {(T * T)}^{- 1}$ , provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1...

The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Cristinel Mortici (2006)

Czechoslovak Mathematical Journal

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Let $T$ be a $γ$ -contraction on a Banach space $Y$ and let $S$ be an almost $γ$ -contraction, i.e. sum of an $(ε, γ)$ -contraction with a continuous, bounded function which is less than $ε$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u = T u .$ If moreover there exists $v$ in $Y$ with $v = S v$ , then we will give estimates for $∥ u - v ∥ .$ Finally, we establish some inequalities related to the Cauchy problem.

Stable solutions of $- Δ u = f (u)$ in $ℝ^{N}$

Louis Dupaigne, Alberto Farina (2010)

Journal of the European Mathematical Society

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Several Liouville-type theorems are presented for stable solutions of the equation $- Δ u = f (u)$ in $ℝ^{N}$ , where $f > 0$ is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

Admissibly integral manifolds for semilinear evolution equations

Nguyen Thieu Huy, Vu Thi Ngoc Ha (2014)

Annales Polonici Mathematici

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We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u (t) = U (t, s) u (s) + \int_{s}^{t} U (t, ξ) f (ξ, u (ξ)) d ξ$ when the evolution family ${(U (t, s))}_{t \geq s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the...

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.

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.

On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations

P. Besala (1971)

Annales Polonici Mathematici

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Non-autonomous stochastic Cauchy problems in Banach spaces

Mark Veraar, Jan Zimmerschied (2008)

Studia Mathematica

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We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $d U (t) = A (t) U (t) d t + B (t) d W_{H} (t)$ , t ∈ [0,T], U(0) = u₀. Here, $W_{H}$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${(B (t))}_{t \in [0, T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${(A (t))}_{t \in [0, T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting...

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.

Stochastic differential equations with Sobolev drifts and driven by $α$ -stable processes

Xicheng Zhang (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this article we prove the pathwise uniqueness for stochastic differential equations in $ℝ^{d}$ with time-dependent Sobolev drifts, and driven by symmetric $α$ -stable processes provided that $α \in (1, 2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $α \in (\frac{2 d}{d + 1}, 2)$ . Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

Stable cohomotopy groups of compact spaces

Sławomir Nowak (2003)

Fundamenta Mathematicae

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We show that one can reduce the study of global (in particular cohomological) properties of a compact Hausdorff space X to the study of its stable cohomotopy groups $π_{s}^{k} (X)$ . Any cohomology functor on the homotopy category of compact spaces factorizes via the stable shape category ShStab. This is the main reason why the language and technique of stable shape theory can be used to describe and analyze the global structure of compact spaces. For a given Hausdorff compact space X, there exists...

Convergence to stable laws and a local limit theorem for stochastic recursions

Mariusz Mirek (2010)

Colloquium Mathematicae

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We consider the random recursion $X ₙ^{x} = M ₙ X_{n - 1}^{x} + Q ₙ + N ₙ (X_{n - 1}^{x})$ , where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and $N ₙ (X_{n - 1}^{x})$ is smaller at infinity, than $M ₙ X_{n - 1}^{x}$ . Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $S ₙ^{x} = \sum_{k = 0}^{n} X_{k}^{x}$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

Divergent solutions to the 5D Hartree equations

Daomin Cao, Qing Guo (2011)

Colloquium Mathematicae

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We consider the Cauchy problem for the focusing Hartree equation $i u_{t} + Δ u + {(| \cdot |}^{- 3} * | u | ²) u = 0$ in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of $- Q + Δ Q + (| \cdot |^{- 3} * | Q | ²) Q = 0$ in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂...

On vertex stability with regard to complete bipartite subgraphs

Aneta Dudek, Andrzej Żak (2010)

Discussiones Mathematicae Graph Theory

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A graph G is called (H;k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its k vertices. Q(H;k) denotes the minimum size among the sizes of all (H;k)-vertex stable graphs. In this paper we complete the characterization of $(K_{m, n}; 1)$ -vertex stable graphs with minimum size. Namely, we prove that for m ≥ 2 and n ≥ m+2, $Q (K_{m, n}; 1) = m n + m + n$ and $K_{m, n} * K ₁$ as well as $K_{m + 1, n + 1} - e$ are the only $(K_{m, n}; 1)$ -vertex stable graphs with minimum size, confirming the conjecture of Dudek and Zwonek.

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $α$ with drift and diffusion coefficients $b$ , $σ$ . When $α \in (1, 2)$ , we investigate pathwise uniqueness for this equation. When $α \in (0, 1)$ , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $α \in (0, 1)$ or $α \in (1, 2)$ and on whether the driving stable process is symmetric or not. Our...

Semilinear perturbations of Hille-Yosida operators

Horst R. Thieme, Hauke Vosseler (2003)

Banach Center Publications

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The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), $u (0) = x \in \bar{D (A)}$ , with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, $B {(λ - A)}^{- 1} x = λ \int_{0}^{\infty} e^{- λ t} V (s) x d s$ , where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0, \infty) \times \bar{D (A)}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤...