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1.
For elliptic equations ε2Δu − V(x) u + f(u) = 0, x ∈ R
N
, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component
of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal.
An erratum to this article can be found at 相似文献
2.
We find conditions for the unique solvability of the problem u
xy
(x, y) = f(x, y, u(x, y), (D
0
r
u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D
0
r
u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r
1, r
2), 0 < r
1, r
2 < 1, in the class of functions that have the continuous derivatives u
xy
(x, y) and (D
0
r
u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method.
__________
Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005. 相似文献
3.
Stanislaus Maier-Paape Thomas Wanner 《Archive for Rational Mechanics and Analysis》2000,151(3):187-219
This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation
where Ω⊂ℝ
n
, n∈{1,2,3 }, is a bounded domain with sufficiently smooth boundary, and f is cubic-like, for example f(u) =u−u
3. Based on the results of [26] the nonlinear Cahn-Hilliard equation will be discussed. This equation generates a nonlinear
semiflow in certain affine subspaces of H
2(Ω). In a neighborhood U
ε with size proportional to ε
n
around the constant solution , where μ lies in the spinodal region, we observe the following behavior. Within a local inertial manifold containing there exists a finite-dimensional invariant manifold
which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1. The dimension of is proportional to ε−
n
and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to ε.
(Accepted May 25, 1999) 相似文献
4.
Sergio Conti 《Continuum Mechanics and Thermodynamics》2006,17(6):469-476
The Kohn-Müller model for the formation of domain patterns in martensitic shape-memory alloys consists in minimizing the sum of elastic, surface and boundary energy in a simplified scalar setting, with a nonconvex constraint representing the presence of different variants. Precisely, one minimizes
among all u:(0,l)×(0,h)→ ℝ such that ∂
y
u = ± 1 almost everywhere. We prove that for small ε the minimum of J
ε, β scales as the smaller of ε1/2β1/2 l
1/2 h and ε2/3 l
1/3 h, as was conjectured by Kohn and Müller. Together with their upper bound, this shows rigorously that a transition is present between a laminar regime at ε/l≫ β3 and a branching regime at ε/l≪ β3.
PACS 64.70.Kb, 62.20.-x, 02.30.Xx 相似文献
5.
In this paper, we consider v(t) = u(t) − e
tΔ
u
0, where u(t) is the mild solution of the Navier–Stokes equations with the initial data
u0 ? L2(\mathbb Rn)?Ln(\mathbb Rn){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L
2 norm of D
β
v(t) decays like
t-\frac |b|-1 2-\frac n4{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u
1(t) such that the L
2 norm of D
β
(v(t) − u
1(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions. 相似文献
6.
Bifurcations of one kind of reaction-diffusion equations, u″+μ(u-uk)=0(μ is a parameter,4≤k∈Z+), with boundary value condition u(0)=u(π)=0 are discussed. By means of singularity theory based on the method of Liapunov-Schmidt reduction, satisfactory results can be acquired. 相似文献
7.
We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u
1(x),...,u
N
(x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be
, a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14]. 相似文献
8.
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
9.
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
10.
We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ωε that is the union of a domain Ω0 and a large number 3N of thin rods with thickness of order
. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides
of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions
and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding
estimates in the Sobolev space H
1(Ωε) as ε → 0 (N → +∞).
Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006. 相似文献
11.
Nicolae Tarfulea 《应用数学和力学(英文版)》2000,21(3):283-290
IntroductionInthispaper,weconsidertheellipticsystem(1λ) -Δu=f(λ,x,u)-v (inΩ),-Δv=δu-γv(inΩ),u=v=0(onΩ),whereΩisasmoothboundeddomaininRN(N≥2)andλisarealparameter.Thesolutions(u,v)ofthissystemrepresentsteadystatesolutionsofreactiondiffusionsystemsderivedfromseveralap… 相似文献
12.
Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in
modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation
of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution
equation for CDa-e0+u(t)=A CDd0+u(t)+f(t),^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t),
u(0)=u
o
, α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded C
o
-semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified. 相似文献
13.
H. Watanabe Tomohiro Sato Motoyuki Hirose Kunihiro Osaki Ming-Long Yao 《Rheologica Acta》1998,37(6):519-527
Dielectric relaxation behavior was examined for 4-4′-n-pentyl-cyanobiphenyl (5CB) and 4-4′-n-heptyl-cyanobiphenyl (7CB) under flow. In quiescent states at all temperatures examined, both 5CB and 7CB exhibited dispersions
in their complex dielectric constant ε*(ω) at characteristic frequencies ω
c
above 106 rad s–1. This dispersion reflected orientational fluctuation of individual 5CB and 7CB molecules having large dipoles parallel to
their principal axis (in the direction of C≡N bond). In the isotropic state at high temperatures, these molecules exhibited no detectable changes of ε*(ω) under flow at
shear rates . In contrast, in the nematic state at lower temperatures the terminal relaxation intensity of ε*(ω) as well as the static
dielectric constant ε′(0) decreased under flow at . This rheo-dielectric change was discussed in relation to the flow effects on the nematic texture (director distribution)
and anisotropy in motion of individual molecules with respect to the director.
Received: 14 April 1998 Accepted: 29 July 1998 相似文献
14.
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
15.
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
16.
c ). To begin with, we assume that the flux-function f(u) is piecewise genuinely nonlinear, in the sense that it exhibits finitely many (at most p, say) points of lack of genuine nonlinearity along each wave curve. Importantly, our analysis applies to arbitrary large p, in the sense that the constant c restricting the total variation is independent of p. Second, by an approximation argument, we prove that the existence theory above extends to general flux-functions f(u) that can be approached by a sequence of piecewise genuinely nonlinear flux-functions f
ε(u).
The main contribution in this paper is the derivation of uniform estimates for the wave curves and wave interactions (which are entirely independent of the properties of the flux-function) together
with a new wave interaction potential which is decreasing in time and is a fully local functional depending upon the angle
made by any two propagating discontinuities. Our existence theory applies, for instance, to the p-system of gas dynamics for general pressure-laws p=p(v) satisfying solely the hyperbolicity condition p′(v)<0 but no convexity assumption.
(Accepted December 30, 2002)
Published online April 23, 2003
Communicated by C. M. Dafermos 相似文献
17.
Xiao-Biao Lin 《Journal of Dynamics and Differential Equations》2007,19(4):1037-1074
We study the spectral and linear stability of Riemann solutions with multiple Lax shocks for systems of conservation laws.
Using a self-similar change of variables, Riemann solutions become stationary solutions for the system u
t
+ (Df(u) − x
I)u
x
= 0. In the space of O((1 + |x|)−η) functions, we show that if , then λ is either an eigenvalue or a resolvent point. Eigenvalues of the linearized system are zeros of the determinant of
a transcendental matrix. On some vertical lines in the complex plane, called resonance lines, the determinant can be arbitrarily small but nonzero. A C
0 semigroup is constructed. Using the Gearhart–Prüss Theorem, we show that the solutions are O(e
γ t
) if γ is greater than the real parts of the eigenvalues and the coordinates of resonance lines. We study examples where Riemann
solutions have two or three Lax-shocks.
Dedicated to Professor Pavol Brunovsky on his 70th birthday. 相似文献
18.
Anne-Laure Dalibard 《Archive for Rational Mechanics and Analysis》2009,192(1):117-164
We study the limit as ε → 0 of the entropy solutions of the equation . We prove that the sequence u
ε
two-scale converges toward a function u(t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation
law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence
result in . 相似文献
19.
M. J. Giles 《Applied Scientific Research》1996,57(3-4):223-234
The exact expression for the probability distribution function (pdf),P(Δur), of a velocity difference Δur, over a distancer, in incompressible fluid turbulence, obtained from the Navier-Stokes equations, is used as a basis for deriving approximate
profiles forP(Δur). These approximate forms are deduced from an approximate factorisation of the underlying functional probability distribution
of the flow field, in which the individual factors capture different physical effects.P(Δur) is represented as the integral, with respect to the spatially averaged dissipation rateε
r, of the product of the conditionalpdf of Δur givenε
r, and thepdf ofε
r. The approximation yields the latter as a log-Poissonpdf, while the conditionalpdf is found to be a Gaussian for a transverse increment, and the product of a Gaussian and a cubic polynomial for a longitudinal
increment. This approximation is equivalent to the refined similarity hypothesis coupled with the log-Poisson distribution,
and it possesses the characteristic features ofP(Δur), including symmetric profiles for transverse increments, asymmetric profiles for longitudinal increments, and the development
of pronounced non-Gaussian features at small separations. The associated scaling exponents for longitudinal and transverse
structure functions are shown to be identical, in this approximation, and to assume the log-Poisson form. 相似文献
20.
Matteo Bonforte Gabriele Grillo Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》2010,196(2):631-680
We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ut=D(um/m) = div (um-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for
x ? \mathbb Rd{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular
to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted
here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold
(\mathbb Rd,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard
\mathbb Rd{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated
with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics,
which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear
Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known
exponential decay of such representation for all other values of m. 相似文献