Related papers: Sharp decay estimates and smoothness for solutions…
Process of the nonlinear deformation of the shallow water wave in a basin of constant depth is studied. The characteristics of the first breaking are analyzed in details. The Fourier spectrum and steepness of the nonlinear wave is…
We consider a family of spherically symmetric, asymptotically Euclidean manifolds with two trapped sets, one which is unstable and one which is semi-stable. The phase space structure is that of an inflection transmission set. We prove a…
We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded…
We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian…
We obtain sharp estimates for certain trilinear oscillatory integrals. In particular, we extend Phong and Stein's seminal result to a trilinear setting. This result partially answers a question raised by Christ, Li, Tao and Thiele…
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity…
This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where…
We develop an optimal regularity theory for $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly…
We consider the long-time behavior of small amplitude solutions of the semilinear wave equation $\Box \phi =\phi^p$ in odd $d\geq 5$ spatial dimensions. We show that for the quadratic nonlinearity ($p=2$) the tail has an anomalously small…
A numerical explicit method to evaluates transient solutions of linear partial differential non-homogeneous equation with constant coefficients is proposed.
We prove spatiotemporal algebraically decaying estimates for the density of the solutions of the linearly damped nonlinear Schr\"odinger equation with localized driving, when supplemented with vanishing boundary conditions. Their derivation…
Process of the nonlinear deformation of the surface wave in shallow water is studied. Main attention is paid to the relation between the Fourier-spectrum and wave steepness. It is shown that the spectral harmonics of the initially sine wave…
We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of…
In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of…
We consider on Riemannian manifolds the nonlinear evolution equation \begin{equation*} \partial _{t}u=\Delta _{p}(u^{1/(p-1)}), \end{equation*}% where $p>1$. This equation is also known as a doubly non-linear parabolic equation or…
We determine the central simple algebras D over a functionfield K of trancendence degree two which admit a model of smooth Cayley-Hamilton algebras. This happens if and only if there is a smooth model S of K such that the ramification…
Penalized spline estimation with discrete difference penalties (P-splines) is a popular estimation method for semiparametric models, but the classical least-squares estimator is highly sensitive to deviations from its ideal model…
Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in a previous work. It is demonstrated…
We show the global existence of smooth solutions of a nonlinear partial differential equation modeling the dynamics of spinodal decomposition in diffusive materials
In this paper, we establish refined regularity estimates for nonnegative solutions to the fractional Poisson equation $$ (-\Delta)^s u(x) =f(x),\,\, x\in B_1(0). $$ Specifically, we have derived H\"{o}lder, Schauder, and Ln-Lipschitz…