In this paper we consider Bao-Lackey's extension of the Laplace operator on a Finsler space. We prove that this operator is of Laplace type on scalars and on top degree forms, and compute the first heat coefficients. In exchange, the BL Laplacian on 1-forms is nonminimal and a study of its heat kernel asymptotics is more difficult. The results obtained in this paper for the 1-formed Laplacian concern Finsler surfaces and direct products of Finsler surfaces. We apply our computation of the heat coefficients to prove that, on Randers spaces, the scalar BL Laplacian and the scalar Laplacian of the metric $a_{ij}$ have the same eigenvalues if and only if the Randers space is Riemann.

In this paper, we obtain a sufficient condition for the univalence of analytic functions in the open unit disk $\mathbb{U}$. This condition involves two arbitrary functions $ g(z) $ and $ h(z) $ analytic in $\mathbb{U} $. Replacing $g(z)$ and $h(z)$ by some particular functions, we find the well-known conditions for univalency established by Z.Nehari (Bull. Amer. Math. Soc. 55(1949)) and S.Ozaki and M.Nunokawa (Proc. Amer. Math. Sco., 33(1972))}. Likewise we find other new sufficient conditions.

The goal of the present paper is to derive $L^p-L^q$ estimates for wave equations with a bounded time dependent coefficient. A classification of the oscillating behaviour of the coefficient is given. This classification determines in an essential way the possibility of deriving $L^p-L^q$ decay estimates.

A weighted weak type estimate with power weights is established for sublinear operators which satisfy certain size condition.

We study the global existence and large time asymptotic behavior of solutions to the initial-boundary value problems for the nolinear nonlocal equation on a segment $\left( 0,a\right) $ \begin{equation} \left\{ \begin{array}{c} u_{t}+u\text{ }u_{x}+\frac{C_{1}}{\pi }\partial _{x}\int_{0}^{x}\frac{% u_{s}(s,t)}{\sqrt{x-s}}ds=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=0,t>0% \end{array}% \right. \label{1} \end{equation} and \begin{equation} \left\{ \begin{array}{c} u_{t}+u\text{ }u_{x}+\frac{C_{1}}{\pi }\int_{0}^{x}\frac{u_{ss}(s,t)}{\sqrt{% x-s}}ds=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=u_{x}(0,t)=0,t>0% \end{array} \right. \label{2} \end{equation} where the constant $C_{1}$ is choosen by a dissipative condition, such that ${\mathrm Re} C_{1}p^{\frac{3}{2}}>0$ for ${\mathrm Re}p=0.$ We prove that if the initial data $u_{0}\in \mathbf{L}^{\infty }\left( 0,a\right) $ is small enough$,$ then there exists a unique solution of problems (0.1) and (0.2) $u\in \mathbf{C}([0,+\infty );\mathbf{L}^{2}\left( 0,a\right) ) $ $\cap \mathbf{C}(\mathbf{R}^{+};\mathbf{H}^{1}\left( 0,a\right) ).$ Moreover there exists a constant $A$ such that the solution has the following large time asymptotics uniformly with respect to $x\in \left( 0,a\right) $ \begin{equation*} u(x,t)=At^{-\frac{2}{3}}\Lambda \left( xt^{-\frac{2}{3}}\right) +O\left( t^{-% \frac{2(1+\delta )}{3}}\right) , \end{equation*}% where $\delta \in \left( 0,\frac{2}{3}\right) $ and \begin{equation*} \Lambda (s)=\frac{e^{-i\frac{\pi }{4}}\sqrt{2}}{2\pi i}\int_{0}^{+i\infty }e^{sz-C_{1}z^{\frac{3}{2}}}dz,\text{ }s>0. \end{equation*}

We establish local and global existence of periodic solutions for KdV type equations, employing Fourier series and a fixed point argument. We also investigate the polynomial growth of the solutions.

Given a collection of complex numbers $\gamma \equiv \{\gamma _{ij}\}\,\,\,(0\leq i+j\leq 2n,\,\,\,|i-j|\leq n)$ with $\gamma _{00}>0$ and $\gamma _{ji}=\bar{\gamma}_{ij},$ we consider the moment problem for $\gamma$ in the case of $n=2$, which is refered to Embry quartic moment problem. In this note we give a solution for the singular case.

In this article we introduce the sequence space $m\left( f,\phi ,\Delta ^{n},p,q\right) ,$ $1\leq p<\infty ,$ using modulus functions. We study its different properties like completeness, solidity etc. Also we obtain some inclusion results involving the space $m\left( f,\phi ,\Delta ^{n},p,q\right) .

We study the codimension-one totally geodesic foliation perpendicular to a non-singular Killing field of a Lorentzian manifold. We determine the structure of the totally geodesic foliation perpendicular to a non-singular Killing field of a two-dimensional Lorentzian torus containing at least two kinds of leaves.

We present new criteria for a self-adjoint operator to have a ground state. Although the proof which we give is easy, it has many applications. And in various models, it is easy to check the criteria. As an application, we consider models of "quantum particles" coupled to a massive Bose field and prove the existence of a ground state of them, where the particle Hamiltonian does not necessarily have compact resolvent.