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 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.

scholarly journals ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS

2014 ◽  
Vol 57 (3) ◽  
pp. 543-554
Author(s):  
JANNE HEITTOKANGAS ◽  
ATTE REIJONEN
Keyword(s):  
Differential Equation ◽  
Bergman Space ◽  
Nontrivial Solution ◽  
Unit Disc ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Zeros ◽  
Zero Sequence ◽  
Zeros Of Solutions ◽  
Necessary And Sufficient

AbstractIf A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

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 Lyapunov-Krasovskii stability analysis of nonlinear integro-differential equation

2018 ◽  
Vol 7 (2) ◽  
pp. 53
Author(s):  
Prebo Jackreece
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Integro Differential Equation ◽  
Necessary And Sufficient

The purpose of this paper is to develop a qualitative stability analysis of a class of nonlinear integro-differential equation within the framework of Lyapunov-Krasovskii. We show that the existence of a Lyapunov-Krasovskii functional is a necessary and sufficient condition for the uniform asymptotic stability of the nonlinear Volterra integro-differential equations.

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