Some functional differential equations

A. Pelczar

Displaying similar documents to “Some functional differential equations”

Global existence and $L_{p}$ decay estimate of solution for Cahn-Hilliard equation with inertial term

Hongmei Xu, Qi Li (2021)

Applications of Mathematics

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The Cauchy problem of the Cahn-Hilliard equation with inertial term in multi space dimension is considered. Based on detailed analysis of Green’s function, using fixed-point theorem, we get the global existence in time of classical solution with large initial data. Furthermore, we get $L_{p}$ decay rate of the solution.

Bounds on the global offensive k-alliance number in graphs

Mustapha Chellali, Teresa W. Haynes, Bert Randerath, Lutz Volkmann (2009)

Discussiones Mathematicae Graph Theory

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Let G = (V(G),E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆ V(G) is called a global offensive k-alliance if |N(v)∩S| ≥ |N(v)-S|+k for every v ∈ V(G)-S, where N(v) is the neighborhood of v. The global offensive k-alliance number $γ ₒ^{k} (G)$ is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on $γ ₒ^{k} (G)$ in terms of order, maximum degree, independence number, chromatic number and minimum degree.

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).

Comparison theorems for infinite systems of parabolic functional-differential equations

Danuta Jaruszewska-Walczak (2001)

Annales Polonici Mathematici

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The paper deals with a weakly coupled system of functional-differential equations $\partial_{t} u_{i} (t, x) = f_{i} (t, x, u (t, x), u, \partial_{x} u_{i} (t, x), \partial_{x x} u_{i} (t, x))$ , i ∈ S, where (t,x) = (t,x₁,...,xₙ) ∈ (0,a) × G, $u = {u_{i}}_{i \in S}$ and S is an arbitrary set of indices. Initial boundary conditions are considered and the following questions are discussed: estimates of solutions, criteria of uniqueness, continuous dependence of solutions on given functions. The right hand sides of the equations satisfy nonlinear estimates of the Perron type with respect to the unknown functions. 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.

Large data local solutions for the derivative NLS equation

Ioan Bejenaru, Daniel Tataru (2008)

Journal of the European Mathematical Society

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We consider the derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension $n = 2$ . Here we prove a similar result for large initial data in all dimensions $n \geq 2$ .

Local-global principle for certain biquadratic normic bundles

Yang Cao, Yongqi Liang (2014)

Acta Arithmetica

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Let X be a proper smooth variety having an affine open subset defined by the normic equation $N_{k (\sqrt a, \sqrt b) / k} (x) = Q (t ₁, . . ., t ₘ) ²$ over a number field k. We prove that: (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of X; (2) the analogue for rational points is also valid assuming Schinzel’s hypothesis.

Global strong solutions of a 2-D new magnetohydrodynamic system

Ruikuan Liu, Jiayan Yang (2020)

Applications of Mathematics

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The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic model in the two dimensional space. Based on Agmon, Douglis, and Nirenberg’s estimates for the stationary Stokes equation and Solonnikov’s theorem on $L^{p}$ - $L^{q}$ -estimates for the evolution Stokes equation, it is shown that this coupled magnetohydrodynamic equations possesses a global strong solution. In addition, the uniqueness of the global strong solution...

A generalization of the Keller-Segel system to higher dimensions from a structural viewpoint

Fujie, Kentarou, Senba, Takasi

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We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than ${(8 π)}^{2}$ , whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known $8 π$ problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel...

A new look at an old comparison theorem

Jaroslav Jaroš (2021)

Archivum Mathematicum

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We present an integral comparison theorem which guarantees the global existence of a solution of the generalized Riccati equation on the given interval $[a, b)$ when it is known that certain majorant Riccati equation has a global solution on $[a, b)$ .

Dynamics of a modified Davey-Stewartson system in ℝ³

Jing Lu (2016)

Colloquium Mathematicae

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We study the Cauchy problem in ℝ³ for the modified Davey-Stewartson system $i \partial ₜ u + Δ u = λ ₁ | u | ⁴ u + λ ₂ b ₁ u v_{x ₁}$ , $- Δ v = b ₂ {(| u | ²)}_{x ₁}$ . Under certain conditions on λ₁ and λ₂, we provide a complete picture of the local and global well-posedness, scattering and blow-up of the solutions in the energy space. Methods used in the paper are based upon the perturbation theory from [Tao et al., Comm. Partial Differential Equations 32 (2007), 1281-1343] and the convexity method from [Glassey, J. Math. Phys. 18 (1977), 1794-1797].

Local analytic solutions of the functional equation $Φ (z) = H (z; Φ [f_{1} (z)], . . ., Φ [f_{m} (z)])$

Wilhelmina Smajdor (1970)

Annales Polonici Mathematici

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Non-local Gel'fand problem in higher dimensions

Tosiya Miyasita, Takashi Suzuki (2004)

Banach Center Publications

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The non-local Gel’fand problem, $Δ v + λ e^{v} / \int_{Ω} e^{v} d x = 0$ with Dirichlet boundary condition, is studied on an n-dimensional bounded domain Ω. If it is star-shaped, then we have an upper bound of λ for the existence of the solution. We also have infinitely many bendings in λ of the connected component of the solution set in λ,v if Ω is a ball and 3 ≤ n ≤ 9.

Global Attractor for the Convective Cahn-Hilliard Equation in $H^{k}$

Xiaopeng Zhao, Ning Duan (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider the convective Cahn-Hilliard equation with periodic boundary conditions. Based on the iteration technique for regularity estimates and the classical theorem on existence of a global attractor, we prove that the convective Cahn-Hilliard equation has a global attractor in $H^{k}$ .

On the radius of spatial analyticity for the higher order nonlinear dispersive equation

Aissa Boukarou, Kaddour Guerbati, Khaled Zennir (2022)

Mathematica Bohemica

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In this work, using bilinear estimates in Bourgain type spaces, we prove the local existence of a solution to a higher order nonlinear dispersive equation on the line for analytic initial data $u_{0}$ . The analytic initial data can be extended as holomorphic functions in a strip around the $x$ -axis. By Gevrey approximate conservation law, we prove the existence of the global solutions, which improve earlier results of Z. Zhang, Z. Liu, M. Sun, S. Li, (2019).

Solvability for semilinear PDE with multiple characteristics

Alessandro Oliaro, Luigi Rodino (2003)

Banach Center Publications

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We prove local solvability in Gevrey spaces for a class of semilinear partial differential equations. The linear part admits characteristics of multiplicity k ≥ 2 and data are fixed in $G^{σ}$ , 1 < σ < k/(k-1). The nonlinearity, containing derivatives of lower order, is assumed of class $G^{σ}$ with respect to all variables.

Second order quasilinear functional evolution equations

László Simon (2015)

Mathematica Bohemica

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We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in $(0, T)$ is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in $(0, \infty)$ (boundedness and stabilization as $t \to \infty$ ) are shown.

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki, P. Kacprzyk (2008)

Applicationes Mathematicae

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{j k}$ . Moreover, we assume that the stored energy function is $C^{\infty}$ with respect to x and $e_{j k}$ . In our description we assume that Piola-Kirchhoff’s stress tensor $p_{j k}$ depends on the tensor...

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta, Anibal Rodriguez-Bernal, Philippe Souplet (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$ : either the space derivative $u_{x}$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^{1}$ norm to the unique steady state. The main difficulty is to prove $C^{1}$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out...