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By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term hh affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)u=b(x)g(u)+λh(x), u>0u>0 in ΩΩ, u|Ω=∞u|Ω=, where ΩΩ is a bounded domain with smooth boundary in RNRN, λ>0λ>0, g∈C1[0,∞)gC1[0,) is increasing on [0,∞)[0,), g(0)=0g(0)=0, gg is regularly varying at infinity with positive index ρρ, the weight bb, which is non-trivial and non-negative in ΩΩ, may be vanishing on the boundary, and the inhomogeneous term hh is non-negative in ΩΩ and may be singular on the boundary.  相似文献   

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The paper deals with the radially symmetric solutions of ut=Δu+um(x,t)vn(0,t)ut=Δu+um(x,t)vn(0,t), vt=Δv+up(0,t)vq(x,t)vt=Δv+up(0,t)vq(x,t), subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that uu (vv) blows up alone if and only if m>p+1m>p+1 (q>n+1q>n+1), which means that any blow-up is simultaneous if and only if m≤p+1mp+1, q≤n+1qn+1. (ii) Any blow-up is uu (vv) blowing up with vv (uu) remaining bounded if and only if m>p+1m>p+1, q≤n+1qn+1 (m≤p+1mp+1, q>n+1q>n+1). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1m>p+1, q>n+1q>n+1. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.  相似文献   

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We consider the semilinear parabolic equation ut=Δu+uput=Δu+up on RNRN, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈RNxRN and t∈RtR. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t?0t?0, then it necessarily converges to 0, as t→∞t, uniformly with respect to x∈RNxRN.  相似文献   

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In this paper, we consider a continuous map f:X→Xf:XX, where XX is a compact metric space, and prove that for any positive integer NN, ff is Schweizer–Smital chaotic if and only if fNfN is too.  相似文献   

6.
We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W1,p(1≤p≤∞)W1,p(1p) norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new LpLp-energy method and L1L1-estimates.  相似文献   

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We study boundary value problems of the form -Δu=f-Δu=f on ΩΩ and Bu=gBu=g on the boundary ∂ΩΩ, with either Dirichlet or Neumann boundary conditions, where ΩΩ is a smooth bounded domain in RnRn and the data f,gf,g are distributions  . This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution uu in terms of f and g  . For f=0f=0, taking advantage of the fact that uu is harmonic on ΩΩ, we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for f≠0f0 and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case.  相似文献   

9.
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α<1/2α<1/2) dissipation α(−Δ)(Δ)α. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α=1/2α=1/2) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L2L2 to LL, from LL to Hölder (CδCδ, δ>0δ>0), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be LL, but it does not appear that their approach can be easily extended to establish the Hölder continuity of LL solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2αC12α. Higher regularity starting from CδCδ with δ>1−2αδ>12α can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press].  相似文献   

10.
We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|p|u|p. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power pp of the nonlinear term satisfies p>6p>6.  相似文献   

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In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion XX in a bounded κκ-fat open set; if uu is a positive harmonic function with respect to XX in a bounded κκ-fat open set DD and hh is a positive harmonic function in DD vanishing on DcDc, then the non-tangential limit of u/hu/h exists almost everywhere with respect to the Martin-representing measure of hh.  相似文献   

14.
We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C2C2. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C2C2-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RNRN of some substance whose density is initially a characteristic function of the complement of a domain with bounded C2C2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SNSN and the hyperbolic space HNHN.  相似文献   

15.
For certain Gaussian processes X(t)X(t) with trend −ctβctβ and variance V2(t)V2(t), the ruin time is analyzed where the ruin time is defined as the first time point tt such that X(t)−ctβ≥uX(t)ctβu. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞u showing that the limiting distribution depends on the parameters ββ, V(t)V(t) and the correlation function of X(t)X(t).  相似文献   

16.
We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the LBLB norm of the gradient of vector-valued functions and the LALA norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in LpLp, with 1<p<∞1<p<, and an Orlicz version involving a Young function A   satisfying both the Δ2Δ2 and the 22 condition.  相似文献   

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We consider families of Ehrenfest chains and provide a simple criterion on the LpLp-cutoff and the LpLp-precutoff with specified initial states for 1≤p<∞1p<. For the family with an LpLp-cutoff, a cutoff time is described and a possible window is given. For the family without an LpLp-precutoff, the exact order of the LpLp-mixing time is determined. The result is consistent with the well-known conjecture on cutoffs of Markov chains proposed by Peres in 2004, which says that a cutoff exists if and only if the multiplication of the spectral gap and the mixing time tends to infinity.  相似文献   

19.
A class of second-order abstract dissipative evolution differential operators DD with 0∈kerD0kerD is shown for which the fact that a non-zero t?u(t)t?u(t) belongs to a cone and −DuDu to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0Du=0 are given.  相似文献   

20.
In the Hammersley harness processes the RR-valued height at each site i∈ZdiZd is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in ZdZd. With this representation we compute covariances and show L2L2 and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L2L2 at speed t1−d/2t1d/2 (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.  相似文献   

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