A note on parabolic equation with nonlinear dynamical boundary condition

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate.     

The existence of strong solution for a class of fully nonlinear equation

This article mainly investigates the existence of global strong solution of a class of fully nonlinear evolution equation and the strong solution of its steady-state equation. By using the T-compulsorily weakly continuous operator theory, the existence of the global strong solution of the fully nonlinear evolution equation is obtained. In addition, based on the acute angle principle, the W^2,p-strong solution for the corresponding stationary equation is also derived.     

A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation

In this work we consider a multi-dimensional higher-order Kirchhoff-type wave equation, with Dirichlet boundary conditions. We establish a blow-up result for certain solutions with positive initial energy.     

A general class of free boundary problems for fully nonlinear parabolic equations

Annali di Matematica Pura ed Applicata (1923 -) - In this paper, we consider the fully nonlinear parabolic free boundary problem $$\begin{aligned} \left\{ \begin{array}{ll} F(D^2u) -\partial _{t}...     

一类四阶非线性奇摄动方程的边值问题

利用匹配渐近展开法,讨论了一类四阶非线性方程的具有两个边界层的奇摄动边值问题．引进伸长变量,根据边界条件与匹配原则,在一定的可解性条件下,给出了外部解和左右边界层附近的内层解,得到了该问题的二阶渐近解,并举例说明了这类非线性问题渐近解的存在性．     

A class of nonlocal singularly perturbed problems for nonlinear hyperbolic differential equation

The author discussed a class of singularly perturbed problems for differential equation fiee {1--7]). Now we consider the non1ocal singu1arly perturbed problem as follows:where E is a positive small parameter anHere x = (xl, x2,' ) x.) E n, fl denotes a bounded region in R", 0fl signilies a boundary offl for class Cl cr (cr 6 (0, 1) is H5lder exponent), T0 is a positive constant, L1 is a uniformlyelliptic operator, L2 is a first order differential operator, T is an integral operator, K(x…     

Solitary wave interaction for a higher-order nonlinear Schrodinger equation

Solitary wave interaction for a higher-order version of thenonlinear Schrödinger (NLS) equation is examined. An asymptotictransformation is used to transform a higher-order NLS equationto a higher-order member of the NLS integrable hierarchy, ifan algebraic relationship between the higher-order coefficientsis satisfied. The transformation is used to derive the higher-orderone- and two-soliton solutions; in general, the N-soliton solutioncan be derived. It is shown that the higher-order collisionis asymptotically elastic and analytical expressions are foundfor the higher-order phase and coordinate shifts. Numericalsimulations of the interaction of two higher-order solitarywaves are also performed. Two examples are considered, one satisfiesthe algebraic relationship derived from asymptotic theory, andthe other does not. For the example which satisfies the algebraicrelationship, the numerical results confirm that the collisionis elastic. The numerical and theoretical predictions for thehigher-order phase and coordinate shifts are also in strongagreement. For the example which does not satisfy the algebraicrelationship, the numerical results show that the collisionis inelastic and radiation is shed by the solitary wave collision.As the bed of radiation shed by the waves decays very slowly(like t^–), it is computationally infeasible to calculatethe final phase and coordinate shifts for the inelastic example.An asymptotic conservation law is derived and used to test thefinite-difference scheme for the numerical solutions.     

A note on a one-sided Jacobi algorithm

Summary We propose a one-sided or implicit variant of the Jacobi diagonalization algorithm for positive definite matrices. The variant is based on a previous Cholesky decomposition and currently uses essentially one square array which, on output, contains the matrix of eigenvectors thus reaching the storage economy of the symmetric QL algorithm. The current array is accessed only columnwise which makes the algorithm attractive for various parallelized and/or vectorized implementations. Even on a serial computer our algorithm shows improved efficiency, in particular if the Cholesky step is made with diagonal pivoting. On matrices of ordern=25–200 our algorithm is about 2–3.5 times slower than QL thus being almost on the halfway between the standard Jacobi and QL algorithms. The previous Cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices.This work was partly done during the first author's visit to the Department of Mathematics, The University of Tennessee-Knoxville, while participating in the Special Year on Numerical Linear Algebra sponsored by the UTK Departments of Computer Science and Mathematics, and the ORNL Mathematical Sciences Section, Engineering Physics and Mathematics Division as well as during a second author visit to the Fernuniversität Hagen. Both authors gratefully acknowledge the support of the respective institutions     

Global attractivity of a higher-order nonlinear difference equation

The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation

     

非线性抛物方程在非线性边界条件下解的爆破时间的下界

研究了非线性抛物方程在非线性边界条件下的解的爆破问题,通过构造一个能量表达式,运用微分不等式的方法,得到该能量表达式所满足的微分不等式,然后通过积分得到当爆破发生时解在非线性边界条件下的爆破时间的下界.     

A note on a nonlinear functional equation and its application

This paper treats the following type of nonlinear functional equations

     

A note on time decay for the nonlinear beam equation

In this note we obtain another form of Morawetz-type identity by Lagrange method and present a simple proof of the time decay estimates for the solution of the nonlinear beam equation, for a nonlinear term with repelling sign, as t→∞. As an immediate consequence, we also get the time-integrability of the local energy of the solution for the beam equation.     

A note on the solution of a class of two-point boundary value problems involving a mixed boundary condition

In this note, we consider a class of two-point boundary value problems involving a pa- rameter in one of the boundary conditions. We shall show that if we know the solution corresponding to a particular value of the parameter, then the solution for any other value of the parameter can be obtained by a simple algebraic method.     

OSCILLATIONS OF SOLUTION FOR A CLASS OF HIGHER-ORDER NEUTRAL DIFFERENTIAL EQUATION

In this paper, the higher order neutral differential equation with continuous distributed delay is concerned and the oscillatory criteria are given.