scholarly journals Global Existence, Asymptotic Behavior and Blow-up of Solutions for a Suspension Bridge Equation with Nonlinear Damping and Source Terms

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Hefeng Zhuang
Keyword(s):  
Global Existence ◽  
Source Term ◽  
Blow Up ◽  
Suspension Bridge ◽  
Nonlinear Damping ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Damping Term ◽  
Necessary And Sufficient ◽  
Damping And Source Terms

In this paper, we consider a fourth-order suspension bridge equation with nonlinear damping term |ut|m-2ut and source term |u|p-2u.  We give necessary and sufficient condition for global existence and energy decay results without considering the relation between m and p. Moreover, when p>m, we give sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time Tmax is also established. It worth to mention that our obtained results extend the recent results of Wang (J. Math. Anal. Appl., 2014) to the nonlinear damping case.

scholarly journals Global Existence, Asymptotic Behavior and Blow-up of Solutions for a Suspension Bridge Equation with Nonlinear Damping and Source Terms

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Hefeng Zhuang
Keyword(s):  
Global Existence ◽  
Source Term ◽  
Blow Up ◽  
Suspension Bridge ◽  
Nonlinear Damping ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Damping Term ◽  
Necessary And Sufficient ◽  
Damping And Source Terms

In this paper, we consider a fourth-order suspension bridge equation with nonlinear damping term |ut|m-2ut and source term |u|p-2u.  We give necessary and sufficient condition for global existence and energy decay results without considering the relation between m and p. Moreover, when p>m, we give sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time Tmax is also established. It worth to mention that our obtained results extend the recent results of Wang (J. Math. Anal. Appl., 2014) to the nonlinear damping case.

scholarly journals Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Quang-Minh Tran ◽  
Hong-Danh Pham
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Decay Rate ◽  
Wave Equations ◽  
Source Term ◽  
Blow Up ◽  
Suspension Bridge ◽  
Nonlinear Damping ◽  
Damping Term ◽  
General Decay Rate

<p style='text-indent:20px;'>The paper deals with global existence and blow-up results for a class of fourth-order wave equations with nonlinear damping term and superlinear source term with the coefficient depends on space and time variable. In the case the weak solution is global, we give information on the decay rate of the solution. In the case the weak solution blows up in finite time, estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate. Finally, if our problem contains an external vertical load term, a sufficient condition is also established to obtain the global existence and general decay rate of weak solutions.</p>

scholarly journals A blow-up dichotomy for semilinear fractional heat equations

2020 ◽  
Author(s):  
Robert Laister ◽  
Mikołaj Sierżęga
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Heat Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Whole Space ◽  
Necessary And Sufficient

Abstract We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.

Global Existence and Blow-Up of Weak Solutions for Nonlinear Wave Equations

2014 ◽  
Vol 635-637 ◽  
pp. 1565-1568
Author(s):  
Yun Zhu Gao ◽  
Wei Guo ◽  
Tian Luan
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Weak Solutions ◽  
Wave Equations ◽  
Blow Up ◽  
Nonlinear Damping ◽  
Nonlinear Wave Equations ◽  
Potential Well ◽  
Source Terms ◽  
Damping And Source Terms

In this paper, we discuss the nonlinear wave equations with nonlinear damping and source terms. By using the potential well methods, we get a result for the global existence and blow-up of the solutions.

scholarly journals A necessary and sufficient condition for exponentially bounded existence and uniqueness

1973 ◽  
Vol 8 (1) ◽  
pp. 133-135 ◽  
Author(s):  
David Lowell Lovelady
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
Necessary And Sufficient

A condition which was previously found to be sufficient for global existence and uniqueness of solutions of an ordinary differential equation is shown herein to be necessary, if it is also required that solutions are exponentially bounded.

Non-simultaneous Blow-up in a Nonlinear Parabolic System

2003 ◽  
Vol 3 (3) ◽  
Author(s):  
Fernando Quirós ◽  
Julio D. Rossi
Keyword(s):  
Porous Medium ◽  
Parabolic System ◽  
Finite Time ◽  
Blow Up ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Nonlinear Parabolic ◽  
Blowing Up ◽  
Flux Boundary Conditions ◽  
Necessary And Sufficient

AbstractWe prove the existence of a nontrivially coupled parabolic system such that one of its components becomes unbounded at a finite time while the other remains bounded, a situation that we denote as non-simultaneous blow-up. Our system consists of two porous medium equations with coupled nonlinear flux boundary conditions. As a preliminary step, we will obtain a necessary and sufficient condition for blow-up. Next we characterize completely, in the case of increasing in time solutions, the set of parameters appearing in the system for which nonsimultaneous blow-up indeed occurs. In the course of our proofs we will obtain a necessary and sufficient condition for the blow-up of solutions to general porous medium type equations on the half-line with a prescribed flux at the boundary blowing up at a finite time, a result of independent interest.

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