Localization for a Fast Diffusion Equation with Absorption

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m-拉普拉斯型抛物方程解的局部化新证明

采用De Giorgi迭代技巧,给出m-拉普拉斯型抛物方程解的局部化新证明,并得到一些先验估计。     

具位势项的快扩散方程的第二临界指标

This paper considers a fast diffusion equation with potential ut= um V (x)um+upin Rn×(0,T), where 1 2αm+n< m ≤ 1, p > 1, n ≥ 2, V (x) ～ω|x|2with ω≥ 0 as |x| →∞,and α is the positive root of αm(αm + n 2) ω = 0. The critical Fujita exponent was determined as pc= m +2αm+nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p > pcvia the critical decay rates of the initial data.With u0(x) ～ |x| aas |x| →∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.     

Multiscale decompositions on bounded domains

A construction of multiscale decompositions relative to domains is given. Multiscale spaces are constructed on which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.

     

Properties of the Fast Diffusion Equation and Its Multidimensional Exact Solutions

We prove invariance of the fast diffusion equation in the two-dimensional coordinate space and give its reduction to a one-dimensional analog in the space variable. Using these results, we construct new exact multidimensional solutions which depend on arbitrary harmonic functions. As a consequence, we obtain new exact solutions to the well-known Liouville equation, the stationary analog of the fast diffusion equation with a linear source. We consider some generalizations to the case of systems of quasilinear parabolic equations.     

Convergence of the Finite Difference Scheme for a Fast Diffusion Equation in Porous Media

We are concerned with an implicit scheme for the finite difference solution to a nonlinear parabolic equation with a multivalued coefficient that describes the fast diffusion in a porous medium. The boundary conditions contain the multivalued function as well. We prove the stability and the convergence of the scheme, emphasizing the precise nature of convergence in this specific case, and compute the error level of the approximating solution. The method is aimed to simplify the numerical computations for the solutions to equations of this type, without performing an approximation of the multivalued function. The theory is illustrated by numerical results.     

Blowup analysis for integral equations on bounded domains

Consider the integral equation

$f^{q?1}(x)=\underset{\mathrm{\Omega }}{\int }\frac{f(y)}{|x?y{|}^{n?\alpha }}dy,f(x)>0,x\in \stackrel{￣}{\mathrm{\Omega }},$

where $\mathrm{\Omega }?{\mathbb{R}}^n$ is a smooth bounded domain. For $1<\alpha <n$, the existence of energy maximizing positive solution in the subcritical case $2<q<\frac{2n}{n+\alpha }$, and nonexistence of energy maximizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are proved in [6]. For $\alpha >n$, the existence of energy minimizing positive solution in the subcritical case $0<q<\frac{2n}{n+\alpha }$, and nonexistence of energy minimizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are also proved in [4]. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{+}$ (in the case of $1<\alpha <n$), and the blowup behaviour of energy minimizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{?}$ (in the case of $\alpha >n$) are analyzed. We see that for $1<\alpha <n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical Sobolev exponent. But for $\alpha >n$, different phenomena appear.     

Infinite-Time Quenching in a Fast Diffusion Equation with Strong Absorption

This work is concerned with the fast diffusion equation , where 0 < m < 1 and κ < 1. A global positive solution is said to quench regularly in infinite time if for some bounded sequence and some , and if for all compact . It is shown that such regular quenching in infinite time occurs for a large class of initial data if κ > m , whereas it is impossible in one space dimension when κ < −m and the solution is radially symmetric and nondecreasing for x > 0.      

Fast and Robust Sixth Order Multigrid Computation for 3D Convection Diffusion Equation

We present a sixth-order explicit compact finite difference scheme to solve the three-dimensional (3D) convection-diffusion equation. We first use a multiscale multigrid method to solve the linear systems arising from a 19-point fourth-order discretization scheme to compute the fourth-order solutions on both a coarse grid and a fine grid. Then an operator-based interpolation scheme combined with an extrapolation technique is used to approximate the sixth-order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid-independent convergence rate for solving convection-diffusion equations with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth-order compact (SOC) scheme, compared with the previously published fourth-order compact (FOC) scheme.     

New phenomena on bounded nonhomogeneous domains

In this paper we get the invariant functions and the invariant harmonic functions under Aut (D) for certain Reinhardt domainsD. By using the invariant functions, we get much more invariant Kahler metrics. And the Ricci curvatures, scalar curvatures and holomorpic sectional curvatures are also obtained, which are very different for the bounded homogeneous domains.     

Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

Angular derivatives on bounded symmetric domains

In this paper we generalise the classical Julia-Wolff-Carathéodory theorem to holomorphic functions defined on bounded symmetric domains. This work comprises part of the Ph.D. thesis [13] of the first author who gratefully acknowledges the support of Forbairt Basic Research grant SC/97/614.     

Distributed-order fractional diffusions on bounded domains

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. This paper provides explicit strong solutions and stochastic analogues for distributed-order time-fractional diffusion equations on bounded domains, with Dirichlet boundary conditions.     

A condition for the extinction of a bounded branching process

A subcritical Galton-Watson process is investigated under the condition that the number of particles is bounded above (superfluous particles are annihilated). A necessary and sufficient condition is obtained for the probability of extinction of such a process to be equal to unity.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 9–18, July, 1970.The author wishes to thank B. A. Sevast'yanov for posing this problem and for his help in its solution.     

On the Chern-Simons limit for a Maxwell-Chern-Simons model on bounded domains

This paper concerns the Chern-Simons limit for the Abelian Maxwell-Chern-Simons model on bounded domains with vanishing gauge fields. We prove that every sequence of solutions of the Maxwell-Chern-Simons equations has a subsequence converging to a solution of the Chern-Simons equation in any Ck norms. We also show that the Maxwell-Chern-Simons equations with the nontopological type boundary condition do not admit any nontrivial solutions on star-shaped domains.     

带梯度吸收项的快扩散方程的自相似奇性解

研究一类带有非线性梯度吸收项的快速扩散方程的自相似奇性解．通过自相似变换,该自相似奇性解满足一个非线性常微分方程的边值问题,再利用打靶法技巧研究该常微分方程初值问题解的存在唯一性并根据初值的取值范围对其解进行了分类．通过对这些解类的性质的分析研究,得出了自相似强奇性解存在唯一性的充分必要条件,此时自相似奇性解就是强奇性解．     

Some Gradient Estimates and Liouville Properties of the Fast Diffusion Equation on Riemannian Manifolds*

In the paper, the authors provide a new proof and derive some new elliptic type (Hamilton type) gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. These results generalize earlier results in the literature. And some parabolic type Liouville theorems for ancient solutions are obtained.     

Biholomorphic mappings on bounded starlike circular domains

Let Ω⊂Cn be a bounded starlike circular domain with 0∈Ω. In this paper, we introduce a class of holomorphic mappings Mg on Ω. Let f(z) be a normalized locally biholomorphic mapping on Ω such that and z=0 is the zero of order k+1 of f(z)−z. We obtain a sharp growth theorem and sharp coefficient bounds for f(z). As applications, sharp distortion theorems for a subclass of starlike mappings are obtained. These results unify and generalize many known results.