共查询到20条相似文献,搜索用时 31 毫秒
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Bilender P. Allahverdiev 《Mathematical Methods in the Applied Sciences》2019,42(1):229-236
In this study, maximal dissipative second‐order dynamic operators on semi‐infinite time scale are studied in the Hilbert space , that the extensions of a minimal symmetric operator in limit‐point case. We construct a self‐adjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations so that we can determine the scattering function of the dilation as stated in the scheme of Lax‐Phillips. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl‐Titchmarsh function of a self‐adjoint second‐order dynamic operator. Finally, we prove the theorems on completeness of the system of root functions of the dissipative and accumulative dynamic operators. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(12):4691-4713
The Euler‐Bernoulli beam model with fully nonconservative boundary conditions of feedback control type is investigated. The output vector (the shear and the moment at the right end) is connected to the observation vector (the velocity and its spatial derivative on the right end) by a 2 × 2 matrix (the boundary control matrix), all entries of which are nonzero real numbers. For any combination of the boundary parameters, the dynamics generator, , of the model is a non–self‐adjoint matrix differential operator in the state Hilbert space. A set of 4 self‐adjoint operators, defined by the same differential expression as on different domains, is introduced. It is proven that each of these operators, as well as , is a finite‐rank perturbation of the same self‐adjoint dynamics generator of a cantilever beam model. It is also shown that the non–self‐adjoint operator, , shares a number of spectral properties specific to its self‐adjoint counterparts, such as (1) boundary inequalities for the eigenfunctions, (2) the geometric multiplicities of the eigenvalues, and (3) the existence of real eigenvalues. These results are important for our next paper on the spectral asymptotics and stability for the multiparameter beam model. 相似文献
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Flank David M. Bezerra Marcelo José D. Nascimento 《Mathematical Methods in the Applied Sciences》2014,37(5):663-675
A damped hyperbolic equation with a dissipative nonlinearity posed in the energy space is considered. The differential operator involved is not the Laplace operator but rather the operator ? div(a?(x) ? ( · )) that has its coefficient depending on a parameter ?. We analyze the behavior of the global attractors as the parameter ? tends to 0. It is assumed that the coefficient a?(x) has a well determined behavior with the parameter, and the idea is to relate the distance of the global attractors with the magnitude a? ? a0, measured in an appropriate norm. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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Víctor Barrera‐Figueroa Vladimir S. Rabinovich 《Mathematical Methods in the Applied Sciences》2019,42(15):5072-5093
In this paper, we consider one‐dimensional Schrödinger operators Sq on with a bounded potential q supported on the segment and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in defined by the Schrödinger operator and matrix point conditions at the ends h0, h1. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator . Moreover, we provide closed‐form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self‐adjoint and nonself‐adjoint problems involving general point interactions described in terms of δ‐ and δ′‐distributions. 相似文献
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In this paper, we focus on the Schrödinger–Kirchhoff‐type equation (SK) where a,b > 0 are constants, may not be radially symmetric, and f(x,u) is asymptotically linear with respect to u at infinity. Under some technical assumptions on V and f, we prove that the problem (SK) has a positive solution. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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Complete systems of recursive integrals and Taylor series for solutions of Sturm–Liouville equations
Vladislav V. Kravchenko Samy Morelos Sébastien Tremblay 《Mathematical Methods in the Applied Sciences》2012,35(6):704-715
Given a regular nonvanishing complex valued solution y0 of the equation , x ∈ (a,b), assume that it is n times differentiable at a point x0 ∈ [a,b]. We present explicit formulas for calculating the first n derivatives at x0 for any solution of the equation . That is, a map transforming the Taylor expansion of y0 into the Taylor expansion of u is constructed. The result is obtained with the aid of the representation for solutions of the Sturm‐Liouville equation in terms of spectral parameter power series. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
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A new approach toward boundedness in a two‐dimensional parabolic chemotaxis system with singular sensitivity
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Johannes Lankeit 《Mathematical Methods in the Applied Sciences》2016,39(3):394-404
We consider the parabolic chemotaxis model in a smooth, bounded, convex two‐dimensional domain and show global existence and boundedness of solutions for χ∈(0,χ0) for some χ0>1, thereby proving that the value χ = 1 is not critical in this regard. Our main tool is consideration of the energy functional for a > 0, b≥0, where using nonzero values of b appears to be new in this context. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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Jia Baoguo 《Mathematical Methods in the Applied Sciences》2016,39(18):5355-5364
Consider the following ν‐th order Caputo delta fractional equation (0.1) The following asymptotic results are obtained. Theorem A. Assume 0 < ν < 1 and there exists a constant b2 such that c(t)≥b2>0. Then the solutions of the equation (0.1) with x(a) > 0 satisfy 相似文献
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Characterizations of Sobolev spaces associated to operators satisfying off‐diagonal estimates on balls
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Junqiang Zhang Der‐Chen Chang Dachun Yang 《Mathematical Methods in the Applied Sciences》2017,40(8):2907-2929
Let be a metric measure space of homogeneous type and L be a one‐to‐one operator of type ω on for ω ∈[0, π /2). In this article, under the assumptions that L has a bounded H ∞ ‐functional calculus on and satisfies (p L , q L ) off‐diagonal estimates on balls, where p L ∈[1, 2) and q L ∈(2, ∞ ], the authors establish a characterization of the Sobolev space , defined via L α /2, of order α ∈(0, 2] for p ∈(p L , q L ) by means of a quadratic function S α , L . As an application, the authors show that for the degenerate elliptic operator L w : =? w ? 1div(A ?) and the Schrödinger type operator with a ∈(0, ∞ ) on the weighted Euclidean space with A being real symmetric, if n ?3, with q ∈[1, 2], , p ∈(1, ∞ ) and with , then, for all , , where the implicit equivalent positive constants are independent of f , denotes the class of Muckenhoupt weights, the reverse Hölder class, and D (L w ) and the domains of L w and , respectively. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Congming Peng Ying Zhang Caochuan Ma 《Mathematical Methods in the Applied Sciences》2019,42(18):6896-6905
The purpose of this work is to investigate the blow‐up dynamics of L2?critical focusing inhomogeneous fractional nonlinear Schrödinger equation: with 0<b<1. For this, we establish a new compactness lemma related to the equation. By applying this lemma, we study the dynamical behavior for blow‐up solutions for initial data satisfying , where Q is the ground state solution of our problem. 相似文献
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Li Wang 《Mathematical Methods in the Applied Sciences》2013,36(9):1123-1132
In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: where N ≥ 3,λ1,λ2 ∈ [0,ΛN), is the best Hardy constant. is the critical Sobolev exponent. a1,a2, b are positive parameters, α,β > 0 and 1 < α + β : = q < 2 < 2*. with . By means of the concentration‐compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a1,a2,b and λ1,λ2. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
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In this paper, we are concerned with the existence of positive solutions for the following nonlocal BVP of fractional DEs with p‐Laplacian operator By using the fixed point theorem in a cone, multiplicity solutions of the BVP are obtained. An example is also given to show the effectiveness of the obtained result. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition
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In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a1,b0,b1,c1,d0, and d1 are real constants that satisfy the conditions Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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On the mixed problem for the semilinear Darcy‐Forchheimer‐Brinkman PDE system in Besov spaces on creased Lipschitz domains
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Robert Gutt Mirela Kohr Sergey E. Mikhailov Wolfgang L. Wendland 《Mathematical Methods in the Applied Sciences》2017,40(18):7780-7829
The purpose of this paper is to study the mixed Dirichlet‐Neumann boundary value problem for the semilinear Darcy‐Forchheimer‐Brinkman system in L p ‐based Besov spaces on a bounded Lipschitz domain in , with p in a neighborhood of 2. This system is obtained by adding the semilinear term | u | u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well‐posedness results for the Dirichlet and Neumann problems in L p ‐based Besov spaces on bounded Lipschitz domains in (n ≥3) are also presented. Then, using integral potential operators, we show the well‐posedness in L 2‐based Sobolev spaces for the mixed problem of Dirichlet‐Neumann type for the linear Brinkman system on a bounded Lipschitz domain in (n ≥3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann‐to‐Dirichlet operator, we extend the well‐posedness property to some L p ‐based Sobolev spaces. Next, we use the well‐posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy‐Forchheimer‐Brinkman system in L p ‐based Besov spaces, with p ∈(2?ε ,2+ε ) and some parameter ε >0. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(10):3672-3683
In this paper, we consider the endpoint case regularity for the 3D liquid crystals system. We prove that if , then weak solution (v,d) is smooth, and our main observation is that the condition is not necessary in this situation. The proof is based on the blow‐up analysis and backward uniqueness for the parabolic operator developed by Escauriaza‐Seregin‐S̆verák. 相似文献
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Fabrizio Colombo Marco M. Peloso Stefano Pinton 《Mathematical Methods in the Applied Sciences》2019,42(18):6259-6276
In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S‐spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let e?, e?=1,2,3 be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation is obtained replacing the Fourier law given by into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, such that the fractional powers of T exist in the sense of the S‐spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure. 相似文献
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Bilender P. Allahverdiev Hüseyin Tuna 《Mathematical Methods in the Applied Sciences》2017,40(18):7287-7306
In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples. 相似文献