We study the energy decay rate for the isotropic elasticity systems in a bounded domain under weak growth assumptions on the feedback function. This work improves some earlier results of Lagnese [9] and Komornik [5].

We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations

$$\{\begin{array}{c}i{u}_t=-\frac{1}{2}\mathrm{\Delta }u+f(|u{|}^2)u,(t,x)\in \mathbf{R}\times {\mathbf{R}}^n,\\ u(0,x)=u_0(x),x\in {\mathbf{R}}^n,n\ge 2,\end{array}$$

where the nonlinear interaction term is $f(|u{|}^2)=V\ast |u{|}^2,V(x)=\lambda |x{|}^{-\delta }$, . We suppose that the initial data $u_0\in H^{0,l}$ and the value $ϵ={\Vert u_0\Vert }_{H^{0,l}}$ is sufficiently small, where $l$ is an integer satisfying $l\ge \left[\frac{n}{2}\right]+3$, and $\left[s\right]$ denotes the largest integer less than $s$. Then we prove that there exists a unique final state $u+\in H^{0,l-2}$ such that for all

$$u(t,x)=\frac{1}{{(it)}^{\frac{n}{2}}}\widehat{u}+(\frac{x}{t})\exp (\frac{i{x}^2}{2t}-\frac{i{t}^{1-\delta }}{1-\delta }f(|{\widehat{u}}_{+}{|}^2)(\frac{x}{t})+O(1+t^{1-2\delta }))+O(t^{-n/2-\delta })$$

uniformly with respect to $x\in {\mathbf{R}}^n$ with the following decay estimate ${\Vert u(t)\Vert }_{L^p}\le Cϵt^{\frac{n}{p}-\frac{n}{2}}$, for all $t\ge 1$ and for every $2\le p\le \infty$. Furthermore we show that for there exists a unique final state $u_{+}\in H^{0,l-2}$ such that for all $t\ge 1$

$$\Vert u(t)-\exp (-\frac{i{t}^{1-\delta }}{1-\delta }f(|{\widehat{u}}_{+}{|}^2)(\frac{x}{t}))U(t)u_{+}{\Vert }_{L^2}=o(t^{1-2\delta })$$

and uniformly with respect to $x\in {\mathbf{R}}^n$

$$u(t,x)=\frac{1}{{(it)}^{\frac{n}{2}}}\widehat{u}+(\frac{x}{t})\exp (\frac{i{x}^2}{2t}-\frac{i{t}^{1-\delta }}{1-\delta }f(|{\widehat{u}}_{+}{|}^2)(\frac{x}{t}))+O(t^{-n/2+1-2\delta }),$$

where $\widehat{\varphi }$ denotes the Fourier transform of the function . In [5] we assumed that $u_0\in H^{m,0}\cap H^{0,m},(m=n+2)$, and showed the same results as in this paper. Here we show that we do not need regularity conditions on the initial data by showing the local existence theorem in lower order Sobolev spaces.

In this paper the family of the Schrödinger operator $H_k=-d^2/d{x}^2+q(x)+\delta (x-1)/k$ in $L^2(0,\infty )$ is investigated. Roughly speaking, if $q(x)=o(x^{-2})$, then the spectral concentration occurs as $k\to 0$.

Necessary conditions and sufficient conditions are given in order that the Laplace Transform is bounded between two Lebesgue spaces with weights. Such a boundedness is characterized for a large class of weights.

We consider a family of orders of complex cubic fields which is similar to one introduced by Levesque and Rhim. We find the Voronoi-algorithm expansions and the fundamental units.

The 2-generators of $A_7$, $A_8$, $A_9$, $M_{11}$, $M_{12}$ and $M_{22}$ are computed up to equivalence of their automorphism groups and the automorphism group of a free group of rank 2. The actions of the automorphism group of the free group on their defining subgroups are also computed.

This paper defines a concept of a semi-$k$-set-contraction operator, and establishes a degree theory for it. As its application, we discuss the existence for the solution of two-point boundary value problems for nonlinear second order integro-differential equations in Banach spaces.