Existence and uniqueness of global weak solutions to a generalized Camassa-Holm equation

This paper is concerned with the existence and uniqueness of global weak solutions to a generalized Camassa-Holm equation on real line. By introducing some new variables, the equation is transformed into two different semi-linear systems. Then the existence and uniqueness of global weak solutions to the original equation are obtained from that of the two semi-linear systems, respectively.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1601.03889, arXiv:1509.08569, arXiv:1401.0312 by other author

Pohozaev identity for the anisotropic pp-Laplacian and estimates of torsion function

In this paper we prove the Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of our identity, we deduce the nonexistence of nontrivial solutions of the Dirichlet problem for the weighted anisotropic $p$-Laplacian in star-shaped domains of $\mathbb{R}^n$. We also provide an upper bound estimate for the first Dirichet eigenvalue of the anisotropic $p$-Laplacian on bounded domains of $\mathbb{R}^n$, some sharp estimates for the torsion function of compact manifolds with boundary and a nonexistence result for the solutions of the Laplace equation on closed Riemannian manifolds.Comment: 21 page

On Payne-Schaefer's Conjecture about an Overdetermined Boundary Problem of Sixth Order

This paper considers overdetermined boundary problems. Firstly, we give a proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two dimensional case and under an additional condition for the case of dimension no less than three. Secondly, we prove an integral identity for an overdetermined problem of fourth order which can be used to deduce Bennett's symmetry theorem. Finally, we prove a symmetry result for an overdetermined problem of second order by integral identities.Comment: 17 page

On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by Payne-P\'olya-Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh-Benguria.Comment: 12 pages, comments are welcom

On a Conjecture of Ashbaugh and Benguria about Lower Eigenvalues of the Neumann Laplacian

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szeg\"o-Weinberger inequality and supports strongly an important conjecture of Ashbaugh-Benguria.Comment: 16 page

Inequalities for the Steklov Eigenvalues

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq \lambda_2\leq ...$ denote the eigenvalues of the Steklov problem: $\Delta u=0$ in $\Omega$ and $(\partial u)/(\partial \nu)=\lambda u$ on $\partial \Omega$. Then $\sum_{i=1}^{n} \lambda^{-1}_i \geq (n^2|\Omega|)/(|\partial\Omega|)$ with equality holding if and only if $\Omega$ is isometric to an $n$-dimensional Euclidean ball. Let $M$ be an $n(\geq 2)$-dimensional compact connected Riemannian manifold with boundary and non-negative Ricci curvature. Assume that the mean curvature of \pa M is bounded below by a positive constant $c$ and let $q_1$ be the first eigenvalue of the Steklov problem: $\Delta^2 u= 0$ in $M$ and $u= (\partial^2 u)/(\partial \nu^2) -q(\partial u)/(\partial \nu) =0$ on $\partial M$. Then $q_1\geq c$ with equality holding if and only if $M$ is isometric to a ball of radius $1/c$ in ${\bf R}^n$.Comment: 17 page

Eigenvalues of the Wentzell-Laplace Operator and of the Fourth Order Steklov Problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Stekolv problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Stekolv problems on a Euclidean ball.Comment: 20 page

Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator

We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and the corresponding eigenfunctions of a Wentzell-type bi-Laplace problem on Euclidean balls.Comment: In this version, we corrected some typo

A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment

In this paper, we study the dynamics of a two-species competition model with two different free boundaries in heterogeneous time-periodic environment, where the two species adopt a combination of random movement and advection upward or downward along the resource gradient. We show that the dynamics of this model can be classified into four cases, which forms a spreading-vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Rough estimates of the asymptotic spreading speed of free boundaries and the long time behavior of solutions are also established when spreading occurs. Furthermore, some sufficient conditions for spreading and vanishing are provided.Comment: 26 page

A diffusive logistic problem with a free boundary in time-periodic environment: favorable habitat or unfavorable habitat

We study the diffusive logistic equation with a free boundary in timeperiodic environment. To understand the effect of the dispersal rate $d$, the original habitat radius $h_0$, the spreading capability $\mu$, and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d$, $h_0$, $\mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha, \gamma, h(t), T)$ to determine the spreading and vanishing of the invasive species. We prove that if $\lambda_1(d, \alpha, \gamma, h_0, T)\leq 0$, the spreading must happen; while if $\lambda_1(d, \alpha, \gamma, h_0, T)> 0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the dispersal rate is slow or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small, while survive successfully if the initial value is big. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1311.7254 by other author