scholarly journals Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2075
Author(s):  
Ratinan Boonklurb ◽  
Tawikan Treeyaprasert ◽  
Aong-art Wanna
Keyword(s):  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Initial Conditions ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Heat Equations ◽  
Quenching Rate ◽  
Linear Heat ◽  
Quenching Set

This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for (x,t)∈(0,1)×(0,T), where f and g are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given.

Finite Time Blow-up in a Delayed Diffusive Population Model with Competitive Interference

2017 ◽  
Vol 18 (5) ◽  
pp. 435-450 ◽  
Author(s):  
Rana D. Parshad ◽  
Suman Bhowmick ◽  
Emmanuel Quansah ◽  
Rashmi Agrawal ◽  
Ranjit Kumar Upadhyay
Keyword(s):  
Hopf Bifurcation ◽  
Initial Data ◽  
Finite Time ◽  
Blow Up ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Turing Instability ◽  
Functional Responses ◽  
Normal Form Theory ◽  
Delayed Model

AbstractIn the current manuscript, an attempt has been made to understand the dynamics of a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses for large initial data. In Ref. Upadhyay and Agrawal, 83(2016) 821–837, it was shown that the model possesses globally bounded solutions, for small initial conditions, under certain parametric restrictions. Here, we show that actually solutions to this model system can blow-up in finite time, for large initial condition, even under the parametric restrictions derived in Ref. Upadhyay and Agrawal, 83(2016) 821–837. We prove blow-up in the delayed model, as well as the non-delayed model, providing sufficient conditions on the largeness of data, required for finite time blow-up. Numerical simulations show that actually the initial data does not have to be very large, to induce blow-up. The spatially explicit system is seen to possess non-Turing instability. We have also studied Hopf-bifurcation direction in the spatial system, as well as stability of the spatial Hopf-bifurcation using the central manifold theorem and normal form theory.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

scholarly journals Wave Breaking Phenomenon for DGH Equation with Strong Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhengguang Guo ◽  
Min Zhao
Keyword(s):  
Finite Time ◽  
Wave Breaking ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Dissipative Term

The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipative term. We establish some sufficient conditions to guarantee finite time blow-up of strong solutions.

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 Finite-time stabilization of switched linear systems and its application in terminal guidance

2016 ◽  
Vol 13 (6) ◽  
pp. 172988141667980
Author(s):  
Yang Guo ◽  
Xiaoxiang Hu ◽  
Biao Deng ◽  
Shuai Zhang ◽  
Yu Jiang
Keyword(s):  
Linear Systems ◽  
Finite Time ◽  
State Feedback ◽  
State Constraints ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Switched Linear Systems ◽  
Finite Time Stabilization ◽  
Finite Time Boundedness ◽  
Terminal Guidance

This article presents finite-time stabilization methods of switched linear systems with disturbances. After extending finite-time stabilization and finite-time boundedness definitions to switched linear systems, sufficient conditions guaranteeing system finite-time boundedness are proposed, by which the state feedback controller method is obtained. For a class of switched terminal guidance systems, the methods are illustrated by application to guidance design to solve the finite-time stabilization problem considering nonzero initial conditions and state constraints.

Global Existence and Blow-Up in Finite Time of Solutions to One Kind of Reaction-Diffusion Educations with Neumann Boundary Conditions

2014 ◽  
Vol 971-973 ◽  
pp. 1017-1020
Author(s):  
Jun Zhou Shao ◽  
Ji Jun Xu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Blowing Up ◽  
Processing Techniques

This paper deals with the properties of one kind of reaction-diffusion equations with Neumann boundary conditions based on the comparison principles. The relations of parameter and the situation of the coupled about equations are used to construct the global existent super-solutions and the blowing-up sub-solutions, and then we obtain the conditions of the global existence and blow-up in finite time solutions with the processing techniques of inequality.

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