Blow up and global solutions for a quasilinear riser problem

In this paper, we consider an initial–boundary value problem to a riser vibrating with dissipative term in the equation. It is proved that under suitable conditions that the solution with a negative initial energy blows up in finite time. And we show that the solution with a nonnegative initial energy is global.     

Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

For the Cauchy problem for the nonlinear wave equation with nonlinear damping and source terms we define stable and unstable sets for the initial data. We prove that, if during the evolution the solution enters into the stable set, the solution is global and we are able to estimate the decay rate of the energy. If during the evolution the solution enters into the unstable set, the solution blows up in finite time.     

具阻尼项的非线性波动方程的稳定集与不稳定集

本文利用势井理论讨论一类非线性波动方程的初边值问题 .我们构造其稳定集 W和不稳定集 V,证明了当初值属于 W时 ,对 β∈ R整体弱解存在并且利用乘子法得到当 β>0解的指数衰减估计 ;当初值属于 V时 ,而 β<0时 ,解将爆破     

Global existence and nonexistence for a nonlinear wave equation with damping and source terms

In this paper we consider a nonlinear wave equation with damping and source term on the whole space. For linear damping case, we show that the solution blows up in finite time even for vanishing initial energy. The criteria to guarantee blowup of solutions with positive initial energy are established both for linear and nonlinear damping cases. Global existence and large time behavior also are discussed in this work. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence and blowup of a localized problem with free boundary

This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large.     

Porous elastic system with nonlinear damping and sources terms

We study the long-time behavior of porous-elastic system, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. We also prove the existence of a global attractor.     

带线性坍塌项和竞争势的非线性波动方程柯西问题的稳定集和不稳定集(英文)

本文考虑带线性坍塌项和竞争势的非线性波动方程柯西问题,定义了新的稳定集和不稳定集,证明了如果初值进入不稳定集,则解在有限时间爆破;如果初值进入稳定集,则整体解存在.运用势井讨论,回答了当初值在多么小的时候,该柯西问题的整体解存在.     

Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation

We study a 1D transport equation with nonlocal velocity and supercritical dissipation. We show that for a certain class of initial data the solution blows up in finite time.     

Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.     

一类非线性高阶Kirchhoff型方程的初边值问题

本文研究了一类具有非线性耗散项的高阶Kirchhoff型方程的初边值问题.通过构造稳定集讨论了此问题整体解的存在性,应用Nakao的差分不等式建立了解能量的衰减估计.在初始能量为正的条件下,证明了解在有限时间内发生blow-up,并且给出了解的生命区间估计.     

Blow-up and global existence for heat flows of harmonic maps

Summary In this paper it is proved that the solution to the evolution problem for harmonic maps blows up in finite time, if the initial map belongs to some nontrivial homotopy class and the initial energy is sufficiently small.     

Blow-up result and energy decay rates for binary mixtures of solids with nonlinear damping and source terms

In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin.     

A Blow-Up Result in a Nonlinear Wave Equation with Delay

In this paper, we consider a nonlinear wave equation with delay. We show that under suitable conditions on the initial data, the weights of the damping, the delay term and the nonlinear source, the energy of solutions blows up in a finite time.     

Blow-up of solutions for a viscoelastic equation with nonlinear damping

The initial boundary value problem for a viscoelastic equation with nonlinear damping in a bounded domain is considered. By modifying the method, which is put forward by Li, Tasi and Vitillaro, we sententiously proved that, under certain conditions, any solution blows up in finite time. The estimates of the life-span of solutions are also given. We generalize some earlier results concerning this equation.      

Blow up of Solutions of a Nonlinear Viscoelastic Wave Equation

We consider a nonlinear viscoelastic wave equation with nonlinear source term. Under suitable conditions on g, it is proved that any weak solution with negative initial energy blows up in finite time if p>2.     

Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.     

Global Existence and Blowup for a Parabolic Equation with a Non-Local Source and Absorption

In this paper we consider a double fronts free boundary problem for a parabolic equation with a non-local source and absorption. The long time behaviors of the solutions are given and the properties of the free boundaries are discussed. Our results show that if the initial value is sufficiently large, then the solution blows up in finite time, while the global fast solution exists for sufficiently small initial data, and the intermediate case with suitably large initial data gives the existence of the global slow solution.     

一类具有非线性阻尼和源项的Petrovsky方程初边值问题解的爆破

研究一类具有非线性阻尼和源项的Petrovsky方程u_(tt)+△~2u+au_t|u_t|~(m-2)=bu|u|~(p-2)的初边值问题解的爆破,利用不稳定集证明了当m相似文献   

Blow-up of Solutions to Quasilinear Parabolic Equations with Singular Absorption and a Positive Initial Energy

Mediterranean Journal of Mathematics - In this paper, the authors prove that the solution blows up in a finite time for a larger class of initial data, namely positive initial energy. The results...     

带势的非线性Klein-Gordon方程柯西问题的稳定和不稳定集

对带势的非线性Klein-Gordon方程柯西问题,我们定义了新的对于初值的稳定和不稳定集.我们证明了如果发展进入了不稳定集,解在有限时间内爆破;如果发展进入了稳定集,解整体存在.运用势并讨论,我们回答了当初值为多少时,柯西问题的整体解存在.