Related papers: Global Well-Posedness of a Class of Strictly Hyper…
We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at…
We establish the global existence of a class of strongly coupled parabolic systems. The necessary apriori estimates will be obtained via our new approach to the regularity theory of parabolic scalar equations with integrable data and new…
This work focuses on the mathematical analysis of the Cauchy problem associated with a two-dimensional equation describing the dynamics of a thin fluid film flowing down an inclined flat plate under the influence of gravity and an electric…
Systems of PDEs comprised of a combination of constraints and evolution equations are ubiquitous in physics. For both theoretical and practical reasons, such as numerical integration, it is desirable to have a systematic understanding of…
In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial…
We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth…
We study the well-posedness, steady states and long time behavior of solutions to Vlasov-Fokker-Planck equation with external confinement potential and nonlinear self-consistent interactions. Our analysis introduces newly characterized…
We address the Cauchy problem for the $k$-generalized Zakharov-Kuznetsov equation ($k$-gZK) posed on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$. By applying established and recently developed linear and bilinear Strichartz-type…
In this work we prove local and global well-posedness results for the Cauchy problem of a family of regularized nonlinear Benjamin-type equations in both periodic and nonperiodic Sobolev spaces.
In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schr\"{o}dinger equation posed on $\T^3$ with intial data lying in its critical space $H^\frac{1}{2}(\T^3)$. By establishing the linear profile…
We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times {\mathbb R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial {\mathbb R}^d_+$, where…
We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type set in the whole space. We assume that the solutions may have arbitrary growth. A complete study of the structure of solutions of…
In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension $N\geq1$. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz…
We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First,…
This work is devoted to study the relation between the regularity index and the decay for solutions of the modified Korteweg-de Vries equation in weighted Sobolev spaces. We show if a solution belongs to a Sobolev space and has a large…
This paper investigates the asymptotic behaviors of global solutions to fourth-order parabolic and hyperbolic equations with Dirichlet boundary conditions. The equations model Micro-Electro-Mechanical Systems (MEMS) and are depending on a…
This paper mainly discuss the regularity behavior of the hyperbolic magnetic Schroedinger equation with singular coefficients near the origin. We apply the techniques from the microlocal analysis to explore the upper bound of loss of…
This is a survey of the author's recent work rather than a broad survey of the literature. The survey is concerned with the global in time solutions of the Cauchy problem for matter waves propagating in the curved spacetimes, which can be,…
Existence of global regular solution branches of the Boltzmann Cauchy problem with continuously differentiable data in phase space dimension $2d\geq 6$ with polynomial decay at infinity of order greater than $2d$ is proved. There are data…
Answering a question left open in \cite{MZ2}, we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski…