Related papers: Singularity Formation and Global Well-Posedness fo…
We consider the defocusing periodic fractional nonlinear Schr\"odinger equation $$ i \partial_t u +\left(-\Delta\right)^{\alpha}u=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< \alpha < 1$ and the operator $(-\Delta)^\alpha$ is the…
We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $\kappa(-\Delta)^{\alpha/2}\theta\ (0<\alpha\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a…
We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schr\"odinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all…
In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H-system in a bounded smooth domain of $R^2$. Global existence and finite time blowup of solutions are proved when…
We study solutions of a Euclidean weighted porous medium equation when the weight behaves at spacial infinity like $|x|^{-\gamma}$, for $\gamma\in [0,2)$, and is allowed to be singular at the origin. In particular we show local-in-time…
We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schr\"odinger equation $iu_t + \Delta u = |u|^{4/n} u$ for large spherically symmetric $L^2_x(\R^n)$ initial data in…
We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global…
Consideration here is a generalized $\mu$-type integrable equation, which can be regarded as a generalization to both the $\mu$-Camassa-Holm and modified $\mu$-Camassa-Holm equations. It is shown that the proposed equation is formally…
In this paper, we consider the long time behavior for the solution of a class of variable coefficient wave equation with nonlinear damping and logarithmic source. The existence and uniqueness of local weak solution can be obtained by using…
We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…
In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schr\"odinger equation \begin{align*} i\partial_t u + \frac12\Delta u = |u| u, \end{align*} in its sharp critical weighted space $\mathcal F \dot…
In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of…
We study singularity formation in two one-dimensional nonlinear wave models with quadratic time-derivative nonlinearities. The non-null model violates the null condition and typically develops finite-time blow-up; the null-form model is…
The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent…
This paper is devoted to global well-posedness, self-similarity and symmetries of solutions for a superdiffusive heat equation with superlinear and gradient nonlinear terms with initial data in new homogeneous Besov-Morrey type spaces.…
The well-posedness of the growth-coagulation equation is established for coagulation kernels having singularity near the origin and growing atmost linearly at infinity. The existence of weak solutions is shown by means of the method of the…
We study the Kolomogorov two-equation model of turbulence in one space dimension. Two are the main results of the paper. First of all, we establish a local well-posedness theory in Sobolev spaces even in the case of vanishing mean turbulent…
This work investigates the semilinear wave equation featuring the displacement dependent term $\sigma(u)\partial_t u $ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity…
We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation$$ u\_t-D\_x^\alpha u\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le \alpha\le 2,$$is locally well-posed in the…
In this paper, we study a new fifth-order Camassa-Holm type equation derived by Li \cite{Li.Z}. We firstly establish the local well-posedness in the sense of Hadamard for the Cauchy problem of the new fifth-order Camassa-Holm type equation…