Related papers: On the decay estimate for small solutions to nonli…
We consider Klein-Gordon equations with an external potential $V$ and a quadratic nonlinearity in $3+1$ space dimensions. We assume that $V$ is regular and decaying and that the (massive) Schr\"odinger operator $H=-\Delta+V+m^2$ has a…
We study the Cauchy problem for the semi-linear damped wave equation in any space dimension. We assume that the time-dependent damping term is effective. We prove the global existence of small energy data solutions in the supercritical…
In this paper, we are concerned with the Cauchy problem for isentropic gas dynamics. Through the contribution of many researchers such as Lax, P. D., Glimm, J., DiPerna, R. J. and Liu, T. P., the decay of solutions was established. They…
This paper studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial…
This paper deals with stability and the large-time decay to any given global smooth solutions of the 3D density-dependent incompressible Boussinesq system. The decay rate for solutions of the corresponding Cauchy problem is obtained in this…
Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More preciously, we investigate some $L^2$-estimates for the solution to the homogeneous…
Building on the hyperboloidal foliation approach of Lefloch and Ma, we extend Klainerman's physical-space approach to dispersive estimates to recover the frequency-restricted $L^1$--$L^\infty$ dispersive estimates for Klein-Gordon…
We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak solutions in the physical energy space. Our method relies on a…
The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the…
We study the decay/growth rates in all $L^p$ norms of solutions to an inhomogeneous nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $\alpha$-time derivative and a power $\beta$ of the Laplacian when the dimension is large, $N>…
For the linear damped wave equation (DW), the $L^p$-$L^q$ type estimates have been well studied. Recently, Watanabe showed the Strichartz estimates for DW when $d=2,3$. In the present paper, we give Strichartz estimates for DW in higher…
The global existence issue in critical regularity spaces for the full Navier-Stokes equationssatisfied by compressible viscous and heat-conductive gases has been first addressed in \cite{D2}, then recently extended to the general $L^p$…
We consider damped wave equations with a potential and rotational inertia terms. We study the Cauchy problem for this model in the one dimensional Euclidean space and we obtain fast energy decay and L^2-decay of the solution itself as time…
The aim of this paper is to develop the general $L_p$ theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in $\mathbb{R}^N$ ($N\geq 3$). By the spectral analysis, we obtain…
In this paper, we study the compressible viscoelastic equations in an exterior domain. We prove the $L_2$ estimates for the solution to the linearized problem and show the decay estimates for the solution to the nonlinear problem. In…
In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{it\phi(\sqrt{-\Delta})}$, where $\phi: \mathbb{R}^+\to \mathbb{R}$ is smooth away from the origin. Especially, the decay…
In this paper, developing a new approach based on Fourier analysis methods for dispersive PDEs, we establish a low regularity NLS approximation for the one-dimensional cubic Klein-Gordon equation. Our main result includes energy class…
We study the 2D coupled wave-Klein-Gordon systems with semi-linear null nonlinearities $Q_0$ and $Q_{\alpha\beta}$. The main result states that the solution to the 2D coupled systems exists globally provided that the initial data are small…
In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leq p\leq q\leq \infty$, for the solutions to the $\sigma$-evolution equation, $\sigma>1$, with structural damping and power nonlinearity $|u|^{1+\alpha}$ or…
We study long-time dynamics of the damped focusing cubic Klein-Gordon equation on a compact three-dimensional Riemannian manifold, together with its space-independent reduction, the damped focusing Duffing equation. Under the geometric…