Related papers: $L^p$--$L^q$ decay estimates for wave equations wi…

We prove local energy decay for the damped wave equation on R^d. The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption…

We study the Cauchy problem for the improved Boussinesq equation \[ u_{tt}-u_{xx}-u_{xxtt}-(u^2)_{xx}=0 \] on the real line with spatially quasi-periodic initial data. For a non-resonant frequency vector $\omega\in\mathbb R^\nu$, we prove…

We study the large time behavior of solutions to the dissipative Korteweg-de Vrie equations $u_t+u_{xxx}+|D|^{\alpha}u+uu_x=0$ with $0<\alpha<2$. We find $v$ such that $u-v$ decays like $t^{-r(\alpha)}$ as $t\to\infty$ in various Sobolev…

We establish the existence of weak solutions $u$ of the semilinear wave equation $\partial_t^2 u-\textrm{div}_x(a(t,x)\nabla_xu)=f_k(u)$ where $a(t,x)$ is equal to $1$ outside a compact set with respect to $x$ and a non-linear term $f_k$…

We consider the Cauchy problem in the whole space for strongly damped Klein-Gordon equations. We derive asymptotic profles of solutions with weighted initial data by a simple method introduced by R. Ikehata. The obtained results show that…

We consider solutions to the 2d Navier-Stokes equations on $\mathbb{T}\times\mathbb{R}$ close to the Poiseuille flow, with small viscosity $\nu>0$. Our first result concerns a semigroup estimate for the linearized problem. Here we show that…

In this paper, by using Fourier splitting method and the expanded properties of decay character $r^*$, we establish the algebraic decay rate of higher order derivative of solutions to 2D dissipative quasi-geostrophic flows.

This paper focuses the study on the $L^p-L^q$ estimates on the solutions to an asymmetric wave equation with dissipation which arises in the study for the magneto-hydrodynamics by using the method of Green function.

We consider the 2D Boussinesq equations with a velocity damping term in a strip $\mathbb{T}\times[-1,1]$, with impermeable walls. In this physical scenario, where the \textit{Boussinesq approximation} is accurate when density/temperature…

We consider the following Cauchy problem for weakly coupled systems of semi-linear damped elastic waves with a power source non-linearity in three-dimensions: \begin{equation*} U_{tt}-a^2\Delta U-\big(b^2-a^2\big)\nabla\text{div }…

We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the…

In this work we study the asymptotic behavior of solutions for a general linear second-order evolution differential equation in time with fractional Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with less demand on…

This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-\mathcal{L}_{\mathbb{H}})^{\alpha}$, $\alpha>0$ on the Heisenberg group $\mathbb{H}^{n}$ with power type…

We consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - \Delta u + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) =…

In this paper, we discuss the asymptotic behaviour of the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear Laplacian viscosity. Firstly, we obtain the existence, uniqueness and regularity of…

For the one-dimensional case, we establish the long-time asymptotics of solution to Cauchy problem and prove existence of modified wave operators. In particular, we show that the part of the wave travels ballistically if the potential is…

Global time estimates of Lp-Lq norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are…

We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low…

We continue the analysis on the model equation arising in the theory of viscoelasticity $$ \partial_{tt} u(t)-\big[1+k_t(0)\big]\Delta u(t) -\int_0^\infty k'_t(s)\Delta u(t-s) d s + f(u(t)) = g $$ in the presence of a (convex, nonnegative…

We prove a full asymptotic stability result for solitary wave solutions of the mKdV equation. We consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the Cauchy problem evolving from…