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1.
We establish new properties of solutions of the functional differential equation x′(t) = ax(t) + bx(t − r) + cx′(t − r) + px(qt) + hx′(qt) + f
1(x(t), x(t − r), x′(t − r), x(qt), x′(qt)) in the neighborhood of the singular point t = +∞.
__________
Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 144–160, January–March, 2007. 相似文献
2.
Asymptotic Variational Wave Equations 总被引:1,自引:0,他引:1
Alberto Bressan Ping Zhang Yuxi Zheng 《Archive for Rational Mechanics and Analysis》2007,183(1):163-185
We investigate the equation (u
t
+(f(u))
x
)
x
=f
′ ′(u) (u
x
)2/2 where f(u) is a given smooth function. Typically f(u)=u
2/2 or u
3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u
tt
− c(u) (c(u)u
x
)
x
=0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data. 相似文献
3.
In the present paper the steady boundary-layer flows induced by permeable stretching surfaces with variable temperature distribution
are investigated under the aspect of Reynolds' analogy r = St
x
/C
f(x). It is shown that for certain stretching velocities and wall temperature distributions, “Reynolds' function”r, i.e. the ratio of the local Stanton number St
x
and the skin friction coefficient C
f(x) equals −1/2 for any value of the Prandtl number Pr and of the dimensionless suction/injection velocity f
w. In all of these cases, the dimensionless temperature field ϑ is connected to the dimensionless downstream velocity f
′ by the simple relationship ϑ=(f
′)Pr. It is also shown that in the general case, Reynolds' function r may possess several singularities in f
w. The largest of them represents a critical value, so that for f
w<f
w,crit the solutions of the energy equation (although they still satisfy all the boundary conditions) become nonphysical. 相似文献
4.
A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent
moving-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) 01(t)=0
exp(−λt), (ii) 02(t) =0(t/t
*)exp(−λt), and 03(t)=0[1+a
cos(ωt)], where λ and ω are real parameters and t
* characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized
representation of an incomplete gamma function Γ(α,x;b) and its decomposition C
Γ and S
Γ. The solutions are presented for moving, -point, -line, and -plane heat sources. It is also demonstrated that the present
analysis covers the classical temperature solutions of a constant strength source under quasi-steady state situations.
Received on 13 June 1997 相似文献
5.
Let Ω be a bounded Lipschitz domain in ℝ
n
with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x
n) with x′=(x1, ..., x
n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special
eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with
Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in
3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
6.
D. V. Bel’skii 《Nonlinear Oscillations》2009,12(4):447-455
We establish new properties of C
1(0, +∞)-solutions of systems of linear functional differential equations x′(t) = Ax(t) + Bx(qt) + Cx′(qt) in the neighborhood of the singular point t = 0. 相似文献
7.
Yoshihisa Morita Hirokazu Ninomiya 《Journal of Dynamics and Differential Equations》2006,18(4):841-861
We deal with a reaction–diffusion equation u
t
= u
xx
+ f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c
1
t) (c
1 < 0) and ψ2(x + c
2
t) (c
2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all
. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c
1
t) and ψ2(x + c
2
t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c
1, we show the existence of an entire solution which behaves as ψ1( − x + c
1
t) in
and φ(x + ct) in
for t≈ − ∞. 相似文献
8.
Hermano Frid 《Archive for Rational Mechanics and Analysis》2006,181(1):177-199
We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems
of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems
whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which
are hyperplanes. In particular, we obtain the uniqueness of the self-similar L∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is
in the sense of the convergence as t→∞ in Lloc1 of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in Lloc1(ℝn), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem. 相似文献
9.
Wehavediscussedconceptofequationwithn_turningpointsinmypaper[1],i.e.,asecondorderlinearordinarydifferentialequationd2ydx2+[λ2q1(x)+λq2(x,λ)]y=0,whereq1(x)=(x-μ1)(x-μ2)…(x-μn)f(x),f(x)≠0,andλisalargeparameter.Althroughthefirsttermoftheasymptoticexpan… 相似文献
10.
Alain Haraux 《Journal of Dynamics and Differential Equations》2007,19(4):915-933
Résumé A l’aide d’inégalités différentielles, on établit une estimation proche de l’optimalité pour la norme dans de l’unique solution bornée de u′′ + cu′ + Au = f(t) lorsque A = A
* ≥ λ I est un opérateur borné ou non sur un espace de Hilbert réel H, V = D(A
1/2) et λ, c sont des constantes positives, tandis que .
By using differential inequalities, a close-to-optimal bound of the unique bounded solution of u′′ + cu′ + Au = f(t) is obtained whenever A = A
* ≥ λ I is a bounded or unbounded linear operator on a real Hilbert space H, V = D(A
1/2) and λ, c are positive constants, while .
相似文献
11.
Aloisio Neves 《Journal of Dynamics and Differential Equations》2009,21(3):555-565
This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L2per[0, p]{L^{2}_{\rm per}[0, \pi]} . We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic
problems. 相似文献
12.
The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution
T
w(x) = T
∞+A·x
λ and lateral mass flux v
w(x) = a·x
(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions
are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection
parameter in detail. The results are compared with the numerical findings of previous authors.
Received on 8 March 1999 相似文献
13.
Nonuniform Exponential Dichotomies and Lyapunov
Regularity 总被引:2,自引:0,他引:2
The notion of exponential dichotomy plays a central role in the Hadamard–Perron theory of invariant manifolds for dynamical systems. The more general notion of nonuniform exponential dichotomy plays a similar role under much weaker assumptions. On the other hand, for nonautonomous linear equations v′ = A(t)v with global solutions, we show here that this more general notion is in fact as weak as possible: namely, any such equation possesses a nonuniform exponential dichotomy. It turns out that the construction of invariant manifolds under the existence of a nonuniform exponential dichotomy requires the nonuniformity to be sufficiently small when compared to the Lyapunov exponents. Thus, it is crucial to estimate the deviation from the uniform exponential behavior. This deviation can be measured by the so-called regularity coefficient, in the context of the classical Lyapunov–Perron regularity theory. We obtain here lower and upper sharp estimates for the regularity coefficient, expressed solely in terms of the matrices A(t). 相似文献
14.
I. Kiguradze 《Nonlinear Oscillations》2008,11(4):521-526
For the differential equation u″ = f(t, u, u′), where the function f: R × R
2 → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found.
Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008. 相似文献
15.
Aloisio Neves 《Journal of Dynamics and Differential Equations》2010,22(3):617-627
This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L2per[0, p]{L^2_{{\rm per}}[0, \pi]}. We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic
problems. 相似文献
16.
Hans-Otto Walther 《Journal of Dynamics and Differential Equations》2009,21(1):195-232
Systems of the form
generalize differential equations with delays r(t) < 0 which are given implicitly by the history x
t
of the state. We show that the associated initial value problem generates a semiflow with differentiable solution operators
on a Banach manifold. The theory covers reaction delays, signal transmission delays, threshold delays, and delays depending
on the present state x(t) only. As an application we consider a model for the regulation of the density of white blood cells and study monotonicity
properties of the delayed argument function . There are solutions (r, x) with τ′(t) > 0 and others with τ′(t) < 0. These other solutions correspond to feedback which reverses temporal order; they are short-lived and less abundant.
Transient behaviour with a sign change of τ′ is impossible.
相似文献
17.
Kenneth R. Meyer Patrick McSwiggen Xiaojie Hou 《Journal of Dynamics and Differential Equations》2010,22(3):367-380
The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search
for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ
1 as t → −∞ and u(t) → γ
2 as t → ∞ where γ
1, γ
2 are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that
f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions
we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied. 相似文献
18.
Interest in nonlinear wave equations has been stimulated bynumerous physical applications, such as telecommunication (e.g.nonlinear telegrapher equation), gasdynamics, anisotropic plasticity andnonlinear elasticity, etc. Mathematical models of these phenomena canoften be reduced to particular types of the equation u
tt
= f(x, u
x
) u
xx
+ g(x, u
x
). In this paper, the problem ofclassification of the latter equation with respect to admitted contacttransformation groups is reduced to the investigation of pointtransformation groups of the equivalent system of first-orderquasi-linear equations v
t
=a(x, v)w
x
, w
t
= b(x,v)v
x
. 相似文献
19.
We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u
tt
− c(u)(c(u)u
x
)
x
= 0. We allow for initial data u|
t = 0 and u
t
|
t=0 that contain measures. We assume that
0 < k-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (ut2+c(u)2ux2)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby
singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times
of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples. 相似文献
20.
General nonlocal diffusive and dispersive transport theories are derived from molecular hydrodynamics and associated theories of statistical mechanical correlation functions, using the memory function formalism and the projection operator method. Expansion approximations of a spatially and temporally nonlocal convective-dispersive equation are introduced to derive linearized inverse solutions for transport coefficients. The development is focused on deriving relations between the frequency-and wave-vector-dependent dispersion tensor and measurable quantities. The resulting theory is applicable to porous media of fractal character.Nomenclature
C
v
(t)
particle velocity correlation function
-
C
v
,(t)
particle fluctuation velocity correlation function
-
C
j
(x,t)
current correlation function
-
D(x,t)
dispersion tensor
-
D(x,t)
fluctuation dispersion tensor
-
f
0(x,p)
equilibrium phase probability distribution function
-
f(x, p;t)
nonequilibrium phase probability distribution function
-
G(x,t)
conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially
-
(k,t)
Fourier transform ofG(x,t)
-
G(x,t)
fluctuation conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially
-
k
wave vector
-
K(t)
memory function
-
L
Liouville operator
-
m
mass
-
p(t)
particle momentum coordinate
-
P
= (0)( , (0))
projection operator
-
Q
=I-P
projection operator
-
s
real Laplace space variable
-
S(k, )
time-Fourier transform of(k,t)
-
t
time
-
v(t)
particle velocity vector
-
v(t)
particle fluctuation velocity vector
-
V
phase space velocity
-
time-Fourier variable
-
(itn)(k)
frequency moment of(k,t)
-
x(t)
particle displacement coordinate
-
x(t)
particle displacement fluctuation coordinate
-
friction coefficient
- (t)
normalized correlation function
General Functions
()
Dirac delta function
- ()
Gamma function
Averages 0
Equilibrium phase-space average
-
Nonequilibrium phase-space average
- (,)
L
2 inner product with respect tof
0 相似文献