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Z. Abdul-Hadi, D. Bschouty, W. Hengartner (1985)
Matematički Vesnik
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Z. Abdul-Hadi, D. Bschouty, W. Hengartner (1985)
Matematički Vesnik
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Daomin Cao, Qing Guo (2011)
Colloquium Mathematicae
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We consider the Cauchy problem for the focusing Hartree equation in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of 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...
Marek Jarnicki, Peter Pflug (1997)
Annales Polonici Mathematici
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We present various characterizations of n-circled domains of holomorphy with respect to some subspaces of .
A. Zięba (1972)
Colloquium Mathematicae
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M. Sango (2003)
Colloquium Mathematicae
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We study the initial boundary value problem for the system of thermoelasticity in a sequence of perforated cylindrical domains , s = 1,2,... We prove that as s → ∞, the solution of the problem converges in appropriate topologies to the solution of a limit initial boundary value problem of the same type but containing some additional terms which are expressed in terms of quantities related to the geometry of . We give an explicit construction of that limit problem.
M. Prizzi, K. P. Rybakowski (2003)
Studia Mathematica
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We study a family of semilinear reaction-diffusion equations on spatial domains , ε > 0, in lying close to a k-dimensional submanifold ℳ of . As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by . The definition of , given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable...
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 , 1 < σ < k/(k-1). The nonlinearity, containing derivatives of lower order, is assumed of class with respect to all variables.
J. Siciak (1985)
Matematički Vesnik
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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 for the initial data u₀(x) in the Besov space with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given -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.
Kyong T. Hahn, Josephine Mitchell (1973)
Annales Polonici Mathematici
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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 . Here we prove a similar result for large initial data in all dimensions .
Zujin Zhang, Xian Yang (2016)
Colloquium Mathematicae
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We study the Cauchy problem for the 3D MHD system with damping terms and (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.
Edgar Lee Stout (2006)
Annales Polonici Mathematici
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We construct a domain of holomorphy in , N≥ 2, whose envelope of holomorphy is not diffeomorphic to a domain in .
Grzegorz Karch (2000)
Studia Mathematica
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Large time behavior of solutions to the generalized damped wave equation for is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where , , and the nonlinear term is either or . In this case, the asymptotic profile of solutions...
M. Sango (2001)
Colloquium Mathematicae
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We investigate the behaviour of a sequence , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains , s = 1,2,..., obtained by removing from a given domain Ω a set whose diameter vanishes when s → ∞. We estimate the deviation of from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.
J. Kisyński, A. Pelczar
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CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES AND ......................... 71. Basic function classes ......................................................................................................................................................................................
A. Aytuna, A. Rashkovskii, V. Zahariuta (2002)
Annales Polonici Mathematici
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For complete Reinhardt pairs “compact set - domain” K ⊂ D in ℂⁿ, we prove Zahariuta’s conjecture about the exact asymptotics , s → ∞, for the Kolmogorov widths of the compact set in C(K) consisting of all analytic functions in D with moduli not exceeding 1 in D, τ(K,D) being the condenser pluricapacity of K with respect to D.
Alessandro Perotti (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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Viene studiata l'equazione per le forme regolari sulla chiusura dell'intersezione di domini pseudoconvessi. Si costruisce un operatore soluzione in forma integrale e sotto ipotesi opportune si ottengono stime della soluzione nelle norme .
Wojciech M. Zajączkowski
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CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in ...............................................346. Existence of solutions of (3.14) in ............417. Green function................................................................528....
Sameer Chavan (2013)
Colloquium Mathematicae
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Let T be a spherical 2-expansive m-tuple and let denote its spherical Cauchy dual. If is commuting then the inequality 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.
Jing Lu (2016)
Colloquium Mathematicae
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We study the Cauchy problem in ℝ³ for the modified Davey-Stewartson system , . 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].
Li Chen, Yujuan Chen, Dang Luo (2013)
Annales Polonici Mathematici
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We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system in a smooth bounded domain of , where is the p-Laplacian operator defined by with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.
Sławomir Michalik (2004)
Studia Mathematica
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A new representation of the Cauchy kernel for an arbitrary acute convex cone Γ in ℝⁿ is found. The domain of holomorphy of is described. An estimation of the growth of near the singularities is given.
Victor Galaktionov (2004)
Journal of the European Mathematical Society
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We consider th-order semilinear parabolic equations in , with Dirac’s mass as the initial function. We show that for , the Cauchy problem admits a solution which is bounded and smooth for small , while for such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.
J. W. Cholewa, T. Dłotko (2003)
Banach Center Publications
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An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on . This semigroup possesses an -global attractor that is closed, bounded, invariant in , and attracts bounded subsets of in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system. ...
Yongxiang Li, Huanhuan Zhang (2016)
Annales Polonici Mathematici
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We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩, where , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the...
Jill Pipher (1992-1993)
Publications mathématiques et informatique de Rennes
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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 in , where is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.
Grzegorz Łysik (2003)
Colloquium Mathematicae
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It is proved that the solution to the initial value problem , u(0,x) = 1/(1+x²), does not belong to the Gevrey class in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.
Noureddine Igbida, Mokhtar Kirane (2002)
Colloquium Mathematicae
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This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form , , , for , t > 0, p > 0, q > 0, r > 0, , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for , where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...