Related papers: Long time asymptotics for the nonlocal mKdV equati…
We employ the $\bar{\partial}$-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}):…
We formulate a $3\times3$ Riemann-Hilbert problem to solve the Cauchy problem for the Sasa-Satsuma equation on the line, which allows us to give a representation for the solution of Sasa-Satsuma equation. We then apply the method of…
We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+\lambda u=f(x,u)+g(x,u)\cdot\mu$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form…
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we…
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…
The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case…
In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly…
This manuscript studies some qualitative properties of solutions to the Cauchy problem for the viscous Moore-Gibson-Thompson (MGT) equation. For one thing, by applying the WKB analysis and diagonalization procedure, we derive some $L^p-L^q$…
The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schr\"odinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a step-like…
In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions.…
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…
We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\Delta u+(|\cdot|^{-3}\ast|u|^{2})u=0$ in $\mathbb{R}^{5}$ with the initial data in $H^1$, and study the divergent property of infinite-variance and nonradial…
We study the long-time asymptotic behavior of the Cauchy problem for the modified Korteweg - de Vries equation with an initial function of the step type. This function rapidly tends to zero as $x\to+\infty$ and to some positive constant c…
The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with time-dependent coefficients whose derivatives are integrable. For this purpose, the…
In this work, we consider the long-time asymptotics of the modified Landau-Lifshitz equation with nonzero boundary conditions (NZBCs) at infinity. The critical technique is the deformations of the corresponding matrix Riemann-Hilbert…
We derive full asymptotics of the modified KdV equation (mKdV) with a higher-order perturbative term. We make use of the perturbative theory of infinite-dimensional integrable systems developed by P. Deift and X. Zhou \cite{DZ-2}, and some…
In this paper, we investigate the Cauchy problem of the nonhomogeneous incompressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field density and prove that the 2D Cauchy problem has a unique local strong solution provided that…
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations. First, we obtain the local existence and uniqueness of the smooth non-negative solution or the…
We study the large-space and large-time asymptotic behavior of the soliton gas of genus $2n-1$ for the mKdV equation with $n\in \mathbb{N}_+$. As $x \to +\infty$, we show that the large-space asymptotics of the mKdV soliton gas can be…
We explore the $N_{\infty}$-soliton asymptotics for the modified Camassa-Holm (mCH) equation with linear dispersion and boundaries vanishing at infinity: $m_t+(m(u^2-u_x^2)^2)_x+\kappa u_x=0,\quad m=u-u_{xx}$ with $\lim_{x\rightarrow \pm…