scholarly journals On a class of nonlocal parabolic equations of Kirchhoff type: Nonexistence of global solutions and blow‐up

2021 ◽  
Author(s):  
Uğur Sert ◽  
Sergey Shmarev
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Global Solutions ◽  
Kirchhoff Type ◽  
Nonlocal Parabolic Equations

scholarly journals General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping

2014 ◽  
Vol 2014 ◽  
pp. 1-21 ◽  
Author(s):  
Wenjun Liu ◽  
Gang Li ◽  
Linghui Hong
Keyword(s):  
Blow Up ◽  
Global Solutions ◽  
Initial Energy ◽  
General Decay ◽  
Strong Damping ◽  
Kirchhoff Type ◽  
Positive Initial Energy ◽  
Relaxation Functions ◽  
Viscoelastic Equations ◽  
Arbitrarily Positive Initial Energy

The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping is considered. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai. Then, we give a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.

scholarly journals Uniqueness of limit flow for a class of quasi-linear parabolic equations

2017 ◽  
Vol 6 (2) ◽  
pp. 243-276 ◽  
Author(s):  
Marco Squassina ◽  
Tatsuya Watanabe
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Global Solutions ◽  
Content Type ◽  
Linear Parabolic Equations ◽  
Whole Space ◽  
Characterization Of Solutions ◽  
Entire Trajectory ◽  
Relevant Class

AbstractWe investigate the issue of uniqueness of the limit flow for a relevant class of quasi-linear parabolic equations defined on the whole space. More precisely, we shall investigate conditions which guarantee that the global solutions decay at infinity uniformly in time and their entire trajectory approaches a single steady state as time goes to infinity. Finally, we obtain a characterization of solutions which blow up, vanish or converge to a stationary state for initial data of the form ${\lambda\varphi_{0}}$ while ${\lambda>0}$ crosses a bifurcation value ${\lambda_{0}}$.

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