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1.
In the present article we consider a special class of equations
when the function (E is a strictly convex Banach space) is V-monotone with respect to (w.r.t.) , i.e. there exists a continuous non-negative function , which equals to zero only on the diagonal, so that the numerical function α(t):= V(x 1(t), x 2(t)) is non-increasing w.r.t. , where x 1(t) and x 2(t) are two arbitrary solutions of (1) defined on . The main result of this article states that every V-monotone Levitan almost periodic (almost automorphic, Bohr almost periodic) Eq. (1) with bounded solutions admits at least one Levitan almost periodic (almost automorphic, Bohr almost periodic) solution. In particulary, we obtain some new criterions of existence of almost recurrent (Levitan almost periodic, almost automophic, recurrent in the sense of Birkgoff) solutions of forced vectorial Liénard equations.   相似文献   

2.
We consider the periodic solutions of
with f being periodic in t and discontinuous in x. Some results of periodic solutions for continuous nonlinearities are generalized via the critical point theorems for locally Lipschitz functionals.  相似文献   

3.
We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension
where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial) viscous parabolic approximation
as . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u t + F(u) x = 0, satisfying the Liu admissibility conditions. These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23]. Dedicated to Prof. Tai Ping Liu on the occasion of his 60 th birthday  相似文献   

4.
We study the large-time behaviour of the solutions u of the evolution equation involving nonlinear diffusion and gradient absorption
We consider the problem posed for and t  >  0 with non-negative and compactly supported initial data. We take the exponent p  >  2 which corresponds to slow p-Laplacian diffusion, and the exponent q in the superlinear range 1  <  q  <  p  −  1. In this range the influence of the Hamilton–Jacobi term is determinant, and gives rise to the phenomenon of localization. The large-time behaviour is described in terms of a suitable self-similar solution that solves a Hamilton–Jacobi equation. The shape of the corresponding spatial pattern is rather conical instead of bell-shaped or parabolic. Dedicated to Pavol Brunovsky.  相似文献   

5.
In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where a c D t α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange
(1)
At last, exact solutions for some Euler–Lagrange equations are presented. In particular, we consider the following equations
(2)
(3)
where g(t) and f(t) are suitable functions. D. Baleanu is on leave of absence from Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania. e-mail: baleanu@venus.nipne.ro.  相似文献   

6.
We investigate the dynamics of the semiflow φ induced on H01(Ω) by the Cauchy problem of the semilinear parabolic equation
on Ω. Here is a bounded smooth domain, and has subcritical growth in u and satisfies . In particular we are interested in the case when f is definite superlinear in u. The set
of attraction of 0 contains a decreasing family of invariant sets
distinguished by the rate of convergence . We prove that the Wk’s are global submanifolds of , and we find equilibria in the boundaries . We also obtain results on nodal and comparison properties of these equilibria. In addition the paper contains a detailed exposition of the semigroup approach for semilinear equations, improving earlier results on stable manifolds and asymptotic behavior near an equilibrium.Supported by DFG Grant BA 1009/15-1.  相似文献   

7.
This paper deals with connected branches of nonstationary periodic trajectories of Hamilton equations
emanating from the degenerate stationary point for H being the generalized Hénon-Heiles (HH) Hamiltonian:
or the generalized Yang-Mills (YM) Hamiltonian:
The existence of such branches has been proved. Minimal periods of searched trajectories near x0 have been described.  相似文献   

8.
  相似文献   

9.
We show that solutions of the 1D Kuramoto-Sivashinsky equation with periodic boundary conditions are asymptotically determined by their values at four points. That is, there existx 1,x 2,x 3, andx 4 in the (periodic) domain such that if
  相似文献   

10.
1IntroductionandPreliminariesLetXbearealBanachspacewithnormIJ'11andadualX'.ThenormalizeddualitymappingJ:X~ZxisdefinedbyJx={x'eX*I(x,x')=11x112=11x if'},where',')denotesthegeneralizeddualitypairing.Itiswell-knownthatifX isstrictlyconvex,Jissingle-valuedandJ(tx)=tjxforallt201xeX.IfX*isuniformlyconvex,thenJisuniformlycontinuousonanyboundedsubsetSofX(of.Browde,fljandBarbuL2]).AnoperatorTwithdomainD(T)andrangeR(T)inXissaidtobeaccretiveifforeveryx,y6D(T),thereexistsajeJ(x--y)suchthat(T…  相似文献   

11.
Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation
in (N ≥ 3) converge to a steady state. Dedicated to Prof. Pavol Brunovsky on the occasion of his 70th birthday.  相似文献   

12.
For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:
For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and that of the Hamilton–Jacobi equation . Supported by NSF DMS 0601162. Supported by NSF DMS 0601403.  相似文献   

13.
We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation
where is a ring-shaped domain, a and μ are given positive constants, is the Heaviside maximal monotone graph: if s > 0, if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as combustion. We show that under certain conditions on the initial data the level sets are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ μ is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential equation expresses the velocity of advancement of the level surface Γ μ through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory of each of the fluid particles.  相似文献   

14.
We prove the existence of positive radial solutions of the following equation:
and give sufficient conditions on the positive functions K1(r) and K2 (r) for the existence and nonexistence of ground states (G.S.) and Singular ground states (S.G.S.), when or . We also give sufficient conditions for the existence of radial S.G.S. and G.S. of equation
when and , respectively. We are also able to classify all the S.G.S. of this equation. The proofs use a new Emden–Fowler transform which allow us to use techniques taken from dynamical system theory, in particular the ones developed in Johnson et al. (Nonlinear Anal, T.M.A. 20, 1279–1302 (1993)) for the problems obtained by substituting the ordinary Laplacian Δ for the m-Laplacian Δm in the preceding equations.MSC: 37B55, 35H30, 35J70  相似文献   

15.
For periodic solutions to the autonomous delay differential equation
with rational periods we derive a characteristic equation for the Floquet multipliers. This generalizes a result from an earlier paper where only periods larger than 2 were considered. As an application we obtain a criterion for hyperbolicity of certain periodic solutions, which are rapidly oscillating in the sense that the delay 1 is larger than the distance between consecutive zeros. The criterion is used to find periodic orbits which are unstable and hyperbolic. An example of a non-autonomous periodic linear delay differential equation with a monodromy operator which is not hyperbolic shows how subtle the conditions of the hyperbolicity criteria in the present paper and in its predecessor are. We also derive first results on Floquet multipliers in case of irrational periods. These are based on approximations by periodic solutions with rational periods.  相似文献   

16.
We are concerned with the existence of a weak solution to the degenerate quasi-linear Dirichlet boundary value problem
It is assumed that 1  <  p  <  ∞, p  ≠  2, Ω is a bounded domain in is a given function, and λ stands for the (real) spectral parameter near the first (smallest) eigenvalue λ1 of the positive p-Laplacian  − Δ p , where . Eigenvalue λ1 being simple, let φ1 denote the eigenfunction associated with it. We show the existence of a solution for problem (P) when f “nearly” satisfies the orthogonality condition ∫Ω f φ1  dx  =  0 and λ  ≤  λ1  +  δ (with δ >  0 small enough). Moreover, we obtain at least three distinct solutions if either p < 2 and λ1  −  δ ≤  λ  <  λ1, or else p > 2 and λ1  <  λ  ≤  λ1  +  δ. The proofs use a minimax principle for the corresponding energy functional performed in the orthogonal decomposition induced by the inner product in L 2(Ω). First, the global minimum is taken over , and then either a local minimum or a local maximum over lin {φ1}. If the latter is a local minimum, the local minimizer in thus obtained provides a solution to problem (P). On the other hand, if it is a local maximum, one gets only a pair of sub- and supersolutions to problem (P), which is then used to obtain a solution by a topological degree argument.  相似文献   

17.
Existence of a Solution “in the Large” for Ocean Dynamics Equations   总被引：1，自引：0，他引：1  
For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u 3 under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions
a weak solution exists and is unique and and the norms are continuous in t. The work was carried out under partial support of Russian Foundation for Basic Research (project 05-01-00864).  相似文献   

18.
The divergence identity for punctured domain B1(0)\ {0}
suggest a viewpoint on describing the behavior of a function uC2(B1(0)\{0}) near the origin. This is useful especially on describing the singular behavior of solutions of polyharmonic equations. In this paper we mainly show that the solution u of the equation
satisfies the identity that, letting vi=(–)iu
provided there exist s0>0 and t0 0 such that f(x,t) c|x|–tq for 0<|x|<s0 and t t0 with n–q(n–2p) and q>1.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.  相似文献   

19.
The article deals with positive solutions of the Dirichlet problem for
where f(s)>0 for s>0 and f(0)=0. The asymptotic behavior of solutions is discussed for a rather large class of g. For g regular near zero, stability properties of equilibria are investigated.  相似文献   

20.
The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x3 and find the solution (u, p) having the following form
where x′  =  (x1, x2). Such solution generalize the nonstationary Poiseuille solutions.  相似文献   

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