We consider an initial boundary value problem for the equation $u_{tt}-\Delta u-\nabla \varphi ·\nabla u+f\left(u\right)+g\left(u_t\right)=0$. We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.

In this work we study the existence, uniqueness and decay of solutions to a class of viscoelastic equations in a separable Hilbert space $H$given by $$\begin{array}{c}{u}_{tt}+M\left(\left[u\right]\right)Au-{\int }_0^{t}g(t-\tau )N\left(\left[u\right]\right)Aud\tau =0,\text{in}{L}^2(0,T;H)\\ u\left(0\right)=u_0,u_t\left(0\right)=u_1\end{array}$$ where by$\left[u\left(t\right)\right]$we are denoting $$\left[u\left(t\right)\right]=\left((u\left(t\right),u_t\left(t\right),(Au\left(t\right),u_t\left(t\right)),\parallel A^{\frac{1}{2}}u\left(t\right){\parallel }^2,\parallel A^{\frac{1}{2}}{u}_t\left(t\right){\parallel }^2,\parallel Au\left(t\right){\parallel }^2\right)\in {ℝ}^5$$ $A:D\left(A\right)\subset H\to H$ is a nonnegative, self-adjoint operator, $M$, $N:{\mathbb{R}}^5\to \mathbb{R}$ are $C^2$- functions and $g:\mathbb{R}\to \mathbb{R}$ is a $C^3$-function with appropriates conditions. We show that there exists global solution in time for small initial data. When $\left[u\left(t\right)\right]={∥A^{\frac{1}{2}}u∥}^2$ and $N=1$, we show the global existence for large initial data $\left(u_0,u_1\right)$ taken in the space $D\left(A\right)D\left(A^{1/2}\right)$ provided they are...

We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in $O\left({\mathrm{e}}^{-\mathrm{c}\sqrt{\mathrm{n}}}\right)$.

In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $$u_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta u(t,x)+\delta |u_t{(t,x)|}^{p-1}{u}_t(t,x)={\mu \left|u(t,x)\right|}^{q-1}u(t,x),x\in \Omega ,t\ge 0,v_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta v(t,x)+\delta |v_t{(t,x)|}^{p-1}{v}_t(t,x)={\mu \left|v(t,x)\right|}^{q-1}v(t,x),x\in \Omega ,t\ge 0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in \Omega ,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),x\in \Omega ,u{{|}_{{}_{\partial \Omega }}=v|}_{{}_{\partial \Omega }}=0$$ where $q>1$, $p\ge 1$, $\delta >0$, $\alpha >0$, $\beta \ge 0$, $\mu \in ℝ$ and $\Delta$ is the Laplacian in ${ℝ}^N$.

This paper deals with the asymptotic behavior as $t\to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+𝒫\left[u\right]$, where $𝒫$ is a Preisach hysteresis operator, $\psi \in L^{\infty }(0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi$ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples...

Let $\Omega$ be a bounded domain in ${ℝ}^n$ with a smooth boundary $\Gamma$. In this work we study the existence of solutions for the following boundary value problem: $$\frac{{\partial }^{2}y}{\partial t^2}-M\left({\int }_{\Omega }{\left|\nabla y\right|}^2\mathrm{d}x\right)\Delta y-\frac{\partial }{\partial t}\Delta y=f\left(y\right)\text{in}Q=\Omega \times (0,\infty ),.1y=0\text{in}{\Sigma }_1={\Gamma }_1\times (0,\infty ),M\left({\int }_{\Omega }{\left|\nabla y\right|}^2\mathrm{d}x\right)\frac{\partial y}{\partial \nu }+\frac{\partial }{\partial t}\left(\frac{\partial y}{\partial \nu }\right)=g\text{in}{\Sigma }_0={\Gamma }_0\times (0,\infty ),y\left(0\right)=y_0,\frac{\partial y}{\partial t}\left(0\right)=y_1\text{in}\Omega ,\left(1\right)$$ where $M$ is a $C^1$-function such that $M\left(\lambda \right)\ge {\lambda }_0>0$ for every $\lambda \ge 0$ and $f\left(y\right)={\left|y\right|}^{\alpha }y$ for $\alpha \ge 0$.

CONTENTS1. Introduction..................................................................................................................................... 5 1.1. Main Theorem 1.1................................................................................................................. 8 1.2. Main Theorem 1.2................................................................................................................. 92. Radon transform.......................................................................................................................................