Related papers: On Questions of Decay and Existence for the Viscou…

We start with the classic result that the Cauchy problem for ideal compressible gas dynamics is locally well posed in time in the sense of Hadamard; there is a unique solution that depends continuously on initial data in Sobolev space $H^s$…

We first construct the global unique solution by assuming that the initial data is small in the H^3 norm but its higher order derivatives could be large. If further the initial data belongs to \Dot{H}^{-s} (0\le s<3/2) or…

We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are…

We consider the Cauchy problem for the generalized Kadomtsev--Petviashvili--Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable…

It was recently proven by De Lellis, Kappeler, and Topalov that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip (T) endowed with the topology of H^1 (T). We prove here that the Lagrangian…

As a continued work of [18], we are concerned with the Timoshenko system in the case of non-equal wave speeds, which admits the dissipative structure of \textit{regularity-loss}. Firstly, with the modification of a priori estimates in [18],…

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index $s\in \R$ over the whole space $\R^n$ for any spatial dimension $n\geq 1$. Here, the diffusion-type source term…

For a nonlinear system of coupled PDEs, that describes evolution of a viscous thin liquid sheet and takes account of surface tension at the free surface, we show exponential $(H^1,\,L^2)$ asymptotic decay to the flat profile of its…

In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\{{array}{l} (-\Delta)^k u=v^q,u>0 \quad in ~R^n, (-\Delta)^k v=u^p,v>0 \quad in ~R^n,…

We prove space-time decay estimates of suitable weak solutions to the Navier-Stokes Cauchy problem, corresponding to a given asymptotic behavior of the initial data of the same order of decay. We use two main tools. The first is a result…

In the paper, we consider the initial value problem to the Camassa-Holm equation in the real-line case. Based on the local well-posedness result and the lifespan, we proved that the data-to-solution map of this problem is not uniformly…

We study the global well-posedness and asymptotic behavior of solutions for the Cauchy problem of three-dimensional sixth order Cahn-Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a…

We consider a genuinely nonlinear $1$-d system of hyperbolic conservation laws with two unknowns. A famous construction of Glimm & Lax shows that global-in-time "Glimm-Lax" weak entropy solutions exist in this setting for any initial data…

Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in \cite{linares-pazoto} that the damping is active on a set…

In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in the critical Besov space $B_{2,1}^{\frac{1}{2}}(\mathbb{T})$. Our results shows that in…

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish…

We consider an abstract second order evolution equation with damping. The "elastic" term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We…

In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by…

We study the large-time behavior of finite-energy weak solutions for the Vlasov-Navier-Stokes equations in a two-dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the…

The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in $L^{p}$ framework. It is proved that the small global solutions constructed in $L^{2}$-Sobolev spaces in our preceding paper [12]…