The global solution and nonlinear stability for the coupled CGL–Burgers equations for sequential flames in ℝN

2016 ◽  
Vol 15 (04) ◽  
pp. 477-504 ◽  
Author(s):  
Xinglong Wu ◽  
Boling Guo
Keyword(s):  
Lower Bound ◽  
Global Solution ◽  
Chemical Reaction ◽  
Nonlinear Stability ◽  
Finite Time ◽  
Blow Up ◽  
Ginzburg Landau ◽  
Burgers Equations ◽  
Blow Up Rate

The present paper is devoted to the study of the global solution and nonlinear stability to the coupled complex Ginzburg–Landau and Burgers (CGL–Burgers) equations for sequential flames which describe the interaction of the excited oscillatory and the damped monotonic mode governing a sequential chemical reaction. If the solution blows up in finite time, we derive the lower bound of blow-up rate of blow-up solution.

The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

scholarly journals A system of Schrödinger equations with general quadratic-type nonlinearities

2020 ◽  
pp. 2050023
Author(s):  
Norman Noguera ◽  
Ademir Pastor
Keyword(s):  
Global Existence ◽  
Variational Methods ◽  
Nonlinear Stability ◽  
Ground State ◽  
Finite Time ◽  
Blow Up ◽  
Quadratic Growth ◽  
Schrodinger Equations ◽  
Concentration Compactness ◽  
Schrödinger Equations

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

Critical exponent in a Stefan problem with kinetic condition

1997 ◽  
Vol 8 (5) ◽  
pp. 525-532 ◽  
Author(s):  
ZHICHENG GUAN ◽  
XU-JIA WANG
Keyword(s):  
Free Boundary ◽  
Critical Exponent ◽  
Global Solution ◽  
Stefan Problem ◽  
Finite Time ◽  
Blow Up ◽  
One Dimensional ◽  
Kinetic Condition ◽  
Dirac Function ◽  
The One

In this paper we deal with the one-dimensional Stefan problemut−uxx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ+, u(x, 0) =u0(x)with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u)[ges ] Ceγ1[mid ]u[mid ] for some γ1 large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u)[ges ] Ce√2πe[mid ]u[mid ].

scholarly journals Blow-up for discrete reaction-diffusion equations on networks

2015 ◽  
Vol 9 (1) ◽  
pp. 103-119 ◽  
Author(s):  
Soon-Yeong Chung ◽  
Jae-Hwang Lee
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Blow Up Rate

In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results.

scholarly journals Lower Bound on the Blow-up Rate of the Axisymmetric Navier–Stokes Equations

2008 ◽  
Vol 2008 ◽  
Author(s):  
Chiun-Chuan Chen ◽  
Robert M. Strain ◽  
Horng-Tzer Yau ◽  
Tai-Peng Tsai
Keyword(s):  
Lower Bound ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Blow Up Rate

scholarly journals Blow up of nonlinear multi-time fractional differential equations

2021 ◽  
Author(s):  
Weike Tang
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Time ◽  
Blow Up ◽  
Well Posedness ◽  
Burgers Equations ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we study the well-posedness of nonlinear multi-time fractional differential equations and show that the solutions of the system will blow up in finite time under certain assumptions. In particular, we apply the results to the nonlinear time fractional Burgers equations.

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