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1.
We consider the Cauchy problem in R n for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L1,1( R n) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L2‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference. 相似文献
2.
《Mathematical Methods in the Applied Sciences》2018,41(14):5423-5458
We investigate lp boundedness, the topological structure of solutions set and the asymptotic periodicity of Volterra functional difference equations. The theoretical results are complemented with a set of applications. 相似文献
3.
We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C 1, α on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C 2, α on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions. 相似文献
4.
V. B. Levenshtam 《Journal of Mathematical Sciences》2009,163(1):89-110
A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential
equations with rapidly oscillating coefficients, some of which may be proportional to ω
n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential
equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with
rapidly oscillating terms proportional to powers ω
d
. For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term
and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem. 相似文献
5.
We study the numerical solution procedure for two-dimensional Laplace’s equation subjecting to non-linear boundary conditions.
Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature
methods are presented for solving the equations, which possess high accuracy order O(h
3) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration
computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with
odd powers is shown. Based on the asymptotic expansion, the h
3 −Richardson extrapolation algorithms are used and the accuracy order is improved to O(h
5). The efficiency of the algorithms is illustrated by numerical examples. 相似文献
6.
Ping Zhang 《Applications of Mathematics》2006,51(4):427-466
In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and
some related problems. We first introduce the main tools, the L
p
Young measure theory and related compactness results, in the first section. Then we use the L
p
Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear
wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove
the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed.
In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic
equation, which is also the so-called vortex density equation arising from sup-conductivity. 相似文献
7.
《Mathematical Methods in the Applied Sciences》2018,41(13):5074-5090
We consider the Cauchy problem in R n for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L1,1( R n) initial data by using a simple method introduced in by the first author. The obtained results will include regularity loss type estimates, which are essentially new in this kind of equation. 相似文献
8.
Dongho Chae 《偏微分方程通讯》2013,38(3):535-557
In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ? N , N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ? N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case. 相似文献
9.
Joelma Azevedo Claudio Cuevas Herme Soto 《Mathematical Methods in the Applied Sciences》2017,40(18):6944-6975
We investigate the asymptotic periodicity, Lp‐boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications. 相似文献
10.
P. Poláčik 《偏微分方程通讯》2013,38(11):1567-1593
We consider quasilinear parabolic equations on ? N satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry. 相似文献
11.
We consider a Ginzburg-Landau type functional on S
2 with a 6
th
order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when
a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly
detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg
problem for the prescribed Gauss curvature equation on S
2.
Received: December 3, 1997 相似文献
12.
M. D. Surnachev 《Journal of Mathematical Sciences》2011,177(1):148-207
We study the asymptotic behavior of positive solutions to nonlinear elliptic equations of Emden–Fowler type with absorption
term. For operators with variable coefficients we obtain conditions on coefficients under which the solutions have the same
asymptotics as solutions to the model equation Δu = −x|
p
|u|
σ−1
u. For positive solutions we obtain lower order terms of the asymptotic expansion at infinity. Bibliography: 10 titles. 相似文献
13.
We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? n . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function. 相似文献
14.
For a large class of partial differential equations on exterior domains or on ?N we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi equations. 相似文献
15.
We consider the asymptotic behavior of the strong solution to the incompressible magnetohydrodynamics (MHD) equations in a half space. The Lr‐decay rates of the strong solution and its derivatives with respect to space variables and time variable, including the L1 and L ∞ decay rates of its first order derivatives with respect to space variables, are derived by using Lq ? Lr estimates of the Stokes semigroup and employing a decomposition for the nonlinear terms in MHD equations. In addition, if the given initial data lie in a suitable weighted space, we obtain more rapid decay rates than observed in general. Similar results are known for incompressible Navier–Stokes equations in a half space under same assumption. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
16.
Shell theory equations are constructed by the method in [1] to the accuracy of quantities of the order of h*2+k, where
and k = 2−4t for
(h* is the relative semithickness of the shell and t is the index of the state of stress variation). Without being within the framework of the Lovetype theory, the equations obtained are compared with the Reissner-Naghdi equations. [2, 3] in which the transverse shear is taken into account, and it is shown that from the asymptotic viewpoint these latter are inconsistent. It is also shown that if the shell resists shear weakly, then from the asymptotic viewpoint the Reissner-Naghdi theory is completely well founded.The three-dimensional equations of elasticity theory are reduced to two-dimensional equations in [1] by using an asymptotic method, i.e. all members of the same order relative to the small parameter h* are taken into account at each stage of the calculations. It has been shown that without going outside the framework of the ordinary concepts of the Love-type theory of shells (in particular, without taking account of transverse shear), the shell theory equations can be constructed to the accuracy of quantities of the order of h2−2t*, but it is impossible to exceed this limit without a qualitative complication in the theory. 相似文献
17.
We establish sufficient algebraic coefficient conditions for the asymptotic stability of solutions of systems of linear difference
equations with continuous time and delay in the case of a rational correlation between delays. We use (n
2 + m)-parameter Lyapunov functions (n is the dimension of the system of equations and m is the number of delays).
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 516–522, April, 1998.
This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science
Foundation (grant No. K42199). 相似文献
18.
Rodica Luca-Tudorache 《PAMM》2007,7(1):2030023-2030024
We study the existence, uniqueness and asymptotic behavior of the strong and weak solutions of a nonlinear differential system with 2N equations in a real Hilbert space H, subject to a boundary condition and initial data. This problem is a discrete version with respect to spatial variable x of some partial differential problems (with H = ℝn ), which have applications in integrated circuits modelling 相似文献
19.
In this paper, we study a system of elliptic equations in R2 which arises from the self-dual equations for the Abelian Chern–Simons system with two Higgs fields and two gauge fields. We provide a new proof for the existence of topological solutions by constructing explicit supersolutions and subsolutions. We also study the asymptotic behavior of condensate solutions on the torus. It is shown that the maximal solutions converge uniformly to zero away from the vortex points, and the convergence rate is computed. 相似文献
20.
Thomas M. Fischer George C. Hsiao Wolfgang L. Wendland 《Journal of Mathematical Analysis and Applications》1985,110(2):583-603
In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A0 + A1Re + O(Re2 ln Re−1) as the Reynolds number Re → 0+, where the vectors A0 and A1 can be obtained from the method of matched inner and outer expansions. 相似文献