Factorization of linear and nonlinear differential operators: necessary and sufficient conditions

Inhomogeneous Mixed-boundary value problem for one dimensional nonlinear Schrödinger equations via factorization techniques

We consider the inhomogeneous Mixed-boundary value problem for the cubic nonlinear Schrödinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques

Necessary and sufficient conditions on existence of radial solutions for exterior Dirichlet problem of fully nonlinear elliptic equations

In this paper, we study the exterior Dirichlet problem for the fully nonlinear elliptic equation $f(λ(D^{2}u))=1$. We obtain the necessary and sufficient conditions of existence of radial solutions with prescribed asymptotic behavior at infinity.

Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation

In this paper, we consider the nonlinear Schrödinger equation with the super critical power of nonlinearity in the attractive case. We give a sufficient condition and a necessary condition to obtain global or blowing up solutions. These conditions coincide in the critical case, thereby extending the results of Weinstein. Furthermore, we improve a blow-up condition.

Criteria for commutative factorization of a class of algebraic matrices

The problem of matrix factorization motivated by diffraction or elasticity is studied. A powerful tool for analyzing its solutions is introduced, namely analytical continuation formulae are derived. Necessary condition for commutative factorization is found for a class of "balanced" matrices. Together with Moiseyev's method and Hurd's idea, this gives a description of the class of commutatively solvable matrices. As a result, a simple analytical procedure is described, providing an answer, whether a given matrix is commutatively factorizable or not.

A Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth

For proper minimizers of parabolic variational integrals with linear growth with respect to $|Du|$, we establish a necessary and sufficient condition for $u$ to be continuous at a point $(x_o,t_o)$, in terms of a sufficient fast decay of the total variation of $u$ about $(x_o,t_o)$ (see (1.4) below). These minimizers arise also as {proper} solutions to the parabolic $1$-laplacian equation. Hence, the continuity condition continues to hold for such solutions (§ 3).

近似Beals-Kartashova分解的量词消去

Quantifier elimination for approximate Beals-Kartashova factorization

线性偏微分算子（LPDO）的唯一已知的构造性因子分解算法是Beals-Kartashova（BK）因子分解\cite{bk2005}。BK-因子分解最有趣的特征之一：在开始时，所有一阶因子都被构造，然后应该检查因子分解条件。这导致了重要的应用领域-即，数值模拟，可以大大简化，如果不是计算与一个LPDE的顺序$n$，我们将能够继续计算与$n$ LPDE的顺序1。在数值模拟中，不需要完全满足因式分解条件，但需要满足一定的精度，我们称之为近似因式分解。本文的思想是探讨用圆柱代数分解的量词消去法解决这类问题的可行性。