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.

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.

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 Finite-Time Stability Analysis of Discrete-Time Linear Singular Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Songlin Wo ◽  
Xiaoxin Han
Keyword(s):  
Discrete Time ◽  
Finite Time ◽  
Singular Systems ◽  
Linear Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Time Stability ◽  
Finite Time Stability ◽  
Necessary And Sufficient ◽  
Time Linear

The finite-time stability (FTS) problem of discrete-time linear singular systems (DTLSS) is considered in this paper. A necessary and sufficient condition for FTS is obtained, which can be expressed in terms of matrix inequalities. Then, another form of the necessary and sufficient condition for FTS is also given by using matrix-null space technology. In order to solve the stability problem expediently, a sufficient condition for FTS is given via linear matrix inequality (LMI) approach; this condition can be expressed in terms of LMIs. Finally, an illustrating example is also given to show the effectiveness of the proposed method.

scholarly journals Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks

2022 ◽  
Vol 12 (2) ◽  
pp. 883
Author(s):  
Yuxin Cui ◽  
Shu Li ◽  
Yunxiao Shan ◽  
Fengqiu Liu
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
The State ◽  
Reconstruction Technique ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Control Networks ◽  
State Transfer ◽  
Necessary And Sufficient

This study focuses on the finite-time set reachability of probabilistic Boolean multiplex control networks (PBMCNs). Firstly, based on the state transfer graph (STG) reconstruction technique, the PBMCNs are extended to random logic dynamical systems. Then, a necessary and sufficient condition for the finite-time set reachability of PBMCNs is obtained. Finally, the obtained results are effectively illustrated by an example.

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.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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