Related papers: Asymptotic parabolicity for strongly damped wave e…

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$ \left\{ \begin{array}{lll} \ds u_{tt} + Au - A^\alpha w = 0, \\ w_t + A^\alpha u_t + A^\beta w = 0,\\ u(0) =…

The contraction semigroup $S(t)={\rm e}^{t\mathbb{A}}$ generated by the abstract linear dissipative evolution equation $$ \ddot u + A u + f(A) \dot u=0 $$ is analyzed, where $A$ is a strictly positive selfadjoint operator and $f$ is an…

In this paper we obtain higher order asymptotic profilles of solutions to the Cauchy problem of the linear damped wave equation in $\textbf{R}^n$ \begin{equation*} u_{tt}-\Delta u+u_t=0, \qquad u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x),…

Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $\mu $ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite…

We construct unique martingale solutions to the damped stochastic wave equation $$ \mu \frac{\partial^2u}{\partial t^2}(t,x)=\Delta u(t,x)-\frac{\partial u}{\partial t}(t,x)+b(t,x,u(t,x))+\sigma(t,x,u(t,x))\frac{dW_t}{dt},$$ where $\Delta$…

An evolution problem for abstract differential equations is studied. The typical problem is: $$\dot{u}=A(t)u+F(t,u), \quad t\geq 0; \,\, u(0)=u_0;\quad \dot{u}=\frac {du}{dt}\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert…

Under fairly general assumptions, we prove that every compact invariant subset $\mathcal I$ of the semiflow generated by the semilinear damped wave equation \epsilon u_{tt}+u_t+\beta(x)u-\sum_{ij}(a_{ij} (x)u_{x_j})_{x_i}&=f(x,u),&&…

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field…

We investigate the long time behavior of solutions to the differential equation $\ddot{x}(t)+\frac{c}{\left( t+1\right) ^{\alpha}}\dot{x}(t)+\nabla \Phi\left( x(t)\right) =g(t),~t\geq0, $ where $c$ is nonnegative constant,…

Let $U$ be an operator in a Hilbert space $\mathcal{H}_{0}$, and let $\mathcal{K}\subset\mathcal{H}_{0}$ be a closed and invariant subspace. Suppose there is a period-2 unitary operator $J$ in $\mathcal{H}_{0}$ such that $JUJ=U^*$, and $PJP…

In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but,…

We consider gradient descent equations for energy functionals of the type S(u) = 1/2 < u(x), A(x)u(x) >_{L^2} + \int_{\Omega} V(x,u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent…

In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations $\partial_{t}^{2} u - \Delta u + \nu (-\Delta)^{\sigma} \partial_{t} u=0$, where $\nu >0$ and $0< \sigma \le1$.…

In this paper, we investigate the existence of positive solution for the following class of elliptic equation $$ - \epsilon^{2}\Delta u +V(x)u= f(u) \,\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon >0$ is a positive parameter,…

We obtain estimate of the exponential decay rate of semigroup associated with second order linear differential equation $u"+Du'+Au=0$ in Hilbert space. We assume that $A$ is a selfadjoint positive definite operator, $D$ is an accretive…

We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+…

We consider the generalized Korteweg-de Vries equation $\partial_t u = -\partial_x(\partial_x^2 u + f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is…

In this article, we study the blow-up of the damped wave equation in the \textit{scale-invariant case} and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}-\Delta…

According to the wave power rule, the second derivative of a function with respect to the variable t is equal to negative n times the function raised to the power of 2n-1. Solving the ordinary differential equations numerically results in…

This paper is concerned with damped hyperbolic gradient systems of the form \[ \alpha u_{tt} + u_t = -\nabla V(u) + u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, $\alpha$ is a…