Our goal is to study the global existence and large time asymptotic behavior of solutions to the Neumann initial-boundary value problem for the nonlinear nonlocal equation on a half-line

$$\{\begin{array}{l}{u}_t+\mathcal{N}(u,u_x)+ℒu=f,(t,x)\in {\mathbf{R}}^{+}\times {\mathbf{R}}^{+},\hfill \\ u(0,x)=u_0(x),x\in {\mathbf{R}}^{+},\hfill \\ {\partial }_{x}u(t,0)=h(t),t\in {\mathbf{R}}^{+},\hfill \end{array}$$

where the nonlinear term is $\mathcal{N}(u,u_x)=u^{\rho }{u}_x^{\sigma }$, with , and $ℒ$ is a pseudodifferential operator defined by the inverse Laplace transform

$$ℒu=\frac{1}{2\pi i}{\displaystyle {\int }_{-i\infty }^{i\infty }{e}^{px}}E{p}^{\frac{5}{2}}\left(\widehat{u}(t,p)-\left(\frac{u(t,0)}{p}+\frac{{\partial }_{x}u(t,0)}{p^2}\right)\right)dp$$

where $\widehat{u}(p)={\displaystyle {\int }_0^{\infty }{e}^{-px}u(x)dx}$. We prove that if the initial data $u_0\in {\mathbf{L}}^{1,a}\cap {\mathbf{L}}^{\infty }$ for $a\in [0,1)$, then there exists a unique solution

$$u\in \mathbf{C}([0,\infty );{\mathbf{L}}^{1,a})\cap \mathbf{C}((0,\infty );{\mathbf{L}}^{\infty }\cap {\mathbf{W}}_{\infty }^1),$$

for the inital-boundary value problem. We also obtain the large time asymptotic formulas for the solutions...

The notion of generalized $\eta$-Einstein trans-Sasakian manifold is introduced. Conformally flat trans-Sasakian manifolds are studied and introduced the idea of a manifold of hyper generalized quasi-constant curvature with various non-trivial examples.

In the present paper, the authors derive differential sandwich theorems involving convolution product for certain subclasses of multivalent normalized analytic functions in the open unit disk. The results in this paper generalize many earlier results in the literature.

The object of the present paper is to study locally $\varphi$-symmetric three-dimensional quasi-Sasakian manifolds and such manifolds with $\eta$-parallel Ricci tensor and cyclic parallel Ricci tensor. An example of a locally $\varphi$-symmetric three-dimensional quasi-Sasakian manifold is also given.

In this report, we consider pseudo-umbilical $CR$-submanifolds in a locally conformal Kaehler space form and we mainly get a relation of the scalar curvature and the coefficient functions of the shape operator of a pseudo-umbilical $CR$-submanifold in a locally conformal Kaehler space form.