Large Time Behavior of Solutions of a Semilinear Parabolic Equation with a Nonlinear Dynamical Boundary Condition

1999 ◽  
pp. 251-272 ◽  
Author(s):  
Marek Fila ◽  
Pavol Quittner
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Large Time ◽  
Large Time Behavior ◽  
Semilinear Parabolic Equation ◽  
Time Behavior ◽  
Nonlinear Dynamical ◽  
Dynamical Boundary Condition ◽  
Semilinear Parabolic

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 Large time behavior of solutions to the compressible Navier–Stokes equations in an infinite layer under slip boundary condition

2016 ◽  
Vol 26 (14) ◽  
pp. 2617-2649 ◽  
Author(s):  
Abulizi Aihaiti ◽  
Shota Enomoto ◽  
Yoshiyuki Kagei
Keyword(s):  
Boundary Condition ◽  
Large Time ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Large Time Behavior ◽  
Navier Stokes ◽  
Time Behavior ◽  
Navier Stokes Equations ◽  
Infinite Layer ◽  
Slip Boundary

This paper is concerned with large time behavior of solutions to the compressible Navier–Stokes equations in an infinite layer of [Formula: see text] under slip boundary condition. It is shown that if the initial data is sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one-dimensional diffusion waves.

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