scholarly journals Boundedness of Global Solutions for a Heat Equation with Exponential Gradient Source

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Zhengce Zhang ◽  
Yanyan Li
Keyword(s):  
Large Time ◽  
Global Solutions ◽  
Complete Classification ◽  
Large Time Behavior ◽  
Classical Solutions ◽  
Semilinear Parabolic Equation ◽  
Time Behavior ◽  
Semilinear Parabolic ◽  
Do So

We consider a one-dimensional semilinear parabolic equation with exponential gradient source and provide a complete classification of large time behavior of the classical solutions: either the space derivative of the solution blows up in finite time with the solution itself remaining bounded or the solution is global and converges inC1norm to the unique steady state. The main difficulty is to proveC1boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov's functional by carrying out the method of Zelenyak.

scholarly journals The delayed Cucker-Smale model with short range communication weights

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zili Chen ◽  
Xiuxia Yin
Keyword(s):  
Initial Data ◽  
Short Range ◽  
Large Time ◽  
Large Time Behavior ◽  
Classical Solutions ◽  
Time Behavior ◽  
Potential Applications ◽  
Communication Ability ◽  
General Communication ◽  
Flocking Behavior

<p style='text-indent:20px;'>Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cucker<inline-formula><tex-math id="M2">\begin{document}$ - $\end{document}</tex-math></inline-formula>Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.</p>

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