Related papers: Asymptotic behavior for dissipative Korteweg-de Vr…

We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to…

We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$,…

Let $n\geq 3$, $0< m<\frac{n-2}{n}$ and $T>0$. We construct positive solutions to the fast diffusion equation $u_t=\Delta u^m$ in $\mathbb{R}^n\times(0,T)$, which vanish at time $T$. By introducing a scaling parameter $\beta$ inspired by…

In this paper we report the asymptotic behaviors of viscosity solutions of the following degenerate elliptic equations \begin{equation*}\label{main-Eq} Lu=x_n^{2\alpha}\sum_{i,j=1}^{n-1}a_{ij}(x)D_{ij}u(x)…

We consider the compressive wave for the modified Korteweg--de Vries equation with background constants $c>0$ for $x\to-\infty$ and $0$ for $x\to+\infty.$ We study the asymptotics of solutions in the transition zone $4c^2t-\varepsilon…

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation $$(-\Delta)^s u + \mu u = (I_{\alpha}*F(u))f(u) \quad \hbox{on $\mathbb{R}^N$}$$ where $s \in (0,1)$, $N\geq 2$, $\alpha \in…

We study asymptotics for solutions of Maxwell's equations, in fact of the Hodge-de Rham equation $(d+\delta)u=0$ without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure…

We consider coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two. We show temporal decay estimates of solutions with small initial data and obtain their asymptotic profiles as…

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system…

We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is the study…

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,…

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - \Delta u + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a…

We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schr\"odinger equation $ i\partial_tu+\Delta u+\mu|x|^{-b}|u|^{\alpha}u+iau=0$, $\mu \in\mathbb{C} $, $b>0$, $a \in \mathbb{C}$ such that $\Re…

We consider semilinear wave equations with small initial data in two space dimensions. For a class of wave equations with cubic nonlinearity, we show the global existence of small amplitude solutions, and give an asymptotic description of…

We study finite total curvature solutions of the Liouville equation $\Delta u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases:…

We study the asymptotics for the Ablowitz-Ladik equation. By taking appropriate continuum limits, it can be shown that the behavior of the equation near degenerate frequencies is well approximated by a complex modified Korteweg-de Vries…

In this paper, we mainly consider large time behavior for the classical free wave equation $u_{tt}-\Delta u=0$ in $\mathbb{R}^n$. We derive some large time optimal estimates for the quantity of solution $\|u(t,\cdot)\|_{L^2}$ with initial…

We study the variable bottom generalized Korteweg-de Vries (bKdV) equation dt u=-dx(dx^2 u+f(u)-b(t,x)u), where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water.…

We consider the late-time asymptotic behavior for solutions of Einstein's equations with the wave map matter. Solutions starting from small compactly supported $\ell$-equivariant initial data with $\ell\geq 1$ are shown to decay as…

In this paper, we investigate the quantitative exponential stability of the Korteweg-de Vries equation on a finite interval with its length close to the critical set. Sharp decay estimates are obtained via a constructive PDE control…