共查询到20条相似文献,搜索用时 31 毫秒
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Michael Winkler 《Journal of Differential Equations》2018,264(3):2310-2350
The chemotaxis system is considered under homogeneous Neumann boundary conditions in the ball , where and .Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data satisfying and , the present paper establishes the existence of a globally defined pair of radially symmetric functions which are continuous in and smooth in , and which solve the corresponding initial-boundary value problem for (?) with in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (?) involving arbitrarily large initial data. 相似文献
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Liangchen Wang Chunlai Mu Xuegang Hu Pan Zheng 《Journal of Differential Equations》2018,264(5):3369-3401
This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractantunder homogeneous Neumann boundary conditions in a bounded domain () with smooth boundary, where the initial data and are non-negative and the parameters , , and . The chemotactic function () is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for ,(i) and(ii) .Moreover, we prove asymptotic stabilization of solutions in the sense that:? If and , then any global bounded solution exponentially converge to as ;? If and , then any global bounded solution exponentially converge to as ;? If and , then any global bounded solution algebraically converge to as . 相似文献
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Consider the Hénon equation with the homogeneous Neumann boundary condition where and . We are concerned on the asymptotic behavior of ground state solutions as the parameter . As , the non-autonomous term is getting singular near . The singular behavior of for large forces the solution to blow up. Depending subtly on the dimensional measure and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and . In particular, the critical exponent for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any and a smooth domain Ω. 相似文献
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Stefan Steinerberger 《Journal of Functional Analysis》2018,274(6):1611-1630
Let be a bounded convex domain in the plane and consider If u assumes its maximum in , then the eccentricity of level sets close to the maximum is determined by the Hessian . We prove that is negative definite and give a quantitative bound on the spectral gap This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if is continuous and has n sign changes, then This statement immediately implies estimates on higher derivatives of harmonic functions u in the unit ball: if u is very flat in the origin, then the boundary function has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with cannot decay faster than . 相似文献
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In this paper we study the global boundedness of solutions to the fully parabolic attraction–repulsion chemotaxis system with logistic source: , , , subject to homogeneous Neumann boundary conditions in a bounded and smooth domain (), where χ, α, ξ, γ, β and δ are positive constants, and is a smooth function generalizing the logistic source for all with , and . It is shown that when the repulsion cancels the attraction (i.e. ), the solution is globally bounded if , or with . Therefore, due to the inhibition of repulsion to the attraction, in any spatial dimension, the exponent θ is allowed to take values less than 2 such that the solution is uniformly bounded in time. 相似文献
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In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is , where is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if is a simple compact Lie superalgebra with , then each (projective) unitary representation of factors through a (projective) unitary representation of itself, and these are known by Jakobsen's classification. If , then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford–Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan. 相似文献
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Tej-Eddine Ghoul Van Tien Nguyen Hatem Zaag 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(6):1577-1630
We consider the following parabolic system whose nonlinearity has no gradient structure: in the whole space , where and . We show the existence of initial data such that the corresponding solution to this system blows up in finite time simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics: with and . The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case ; and the fact that the case breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data. 相似文献
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Gerd Grubb 《Journal of Functional Analysis》2018,274(9):2634-2660
This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:
(*)
1) For strongly elliptic pseudodifferential operators (ψdo's) P on of order , a symbol calculus on is introduced that allows showing optimal regularity results, globally over and locally over : for , . The are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal (), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition , the initial condition , and , (*) has a unique solution with . Here if , and is contained in if , but contains nontrivial elements from if (where ). The interior regularity of u is lifted when f is more smooth. 相似文献
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Michael Winkler 《Journal of Differential Equations》2018,264(10):6109-6151
A class of chemotaxis-Stokes systems generalizing the prototype is considered in bounded convex three-dimensional domains, where is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that
(0.1)
Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state in the large time limit.This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1). 相似文献
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A.P. Bergamasco P.L. Dattori da Silva R.B. Gonzalez 《Journal of Differential Equations》2018,264(5):3500-3526
Let be a vector field defined on the torus , where , are real-valued functions and belonging to the Gevrey class , , for . We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets. 相似文献
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In a previous work, it was shown how the linearized strain tensor field can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain , instead of the displacement vector field in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition on a portion of the boundary of Ω can be recast, again as boundary conditions on , but this time expressed only in terms of the new unknown . 相似文献
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Let be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as an “obstacle”. We are interested in the behavior of the first Dirichlet eigenvalue .First, we prove an upper bound on in terms of the distance of the set to the set of maximum points of the first Dirichlet ground state of Ω. In short, a direct corollary is that if
(1)
is large enough in terms of , then all maximizer sets of are close to each maximum point of .Second, we discuss the distribution of and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if is sufficiently large with respect to , then all maximizers of contain all maximum points of . 相似文献
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In this work, we prove the existence of convex solutions to the following k-Hessian equation in the neighborhood of a point , where , is nonnegative near , and . 相似文献
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Yohei Fujishima 《Journal of Differential Equations》2018,264(11):6809-6842
We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity
(P)
where is a continuous initial function. In this paper, we consider the case where decays to ?∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for is related to the decay rate of forward self-similar solutions of . 相似文献
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In this paper we define odd dimensional unitary groups . These groups contain as special cases the odd dimensional general linear groups where R is any ring, the odd dimensional orthogonal and symplectic groups and where R is any commutative ring and further the first author's even dimensional unitary groups where is any form ring. We classify the E-normal subgroups of the groups (i.e. the subgroups which are normalized by the elementary subgroup ), under the condition that R is either a semilocal or quasifinite ring with involution and . Further we investigate the action of by conjugation on the set of all E-normal subgroups. 相似文献