scholarly journals On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Zhilei Liang
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Newtonian Equation ◽  
Comparison Methods ◽  
Self Similar

We identify the blow-up set of solutions to the problem , , , , , and , , where . We obtain that the blow up set satisfies . The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods.

Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

2012 ◽  
Vol 67 (8-9) ◽  
pp. 479-482 ◽  
Author(s):  
Junping Zhao
Keyword(s):  
Boundary Condition ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Existence Theorems ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Gradient Term

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)▽u)+ f (x;u; |▽u|2; t) under nonlinear boundary condition ¶u=¶n+g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T* may depend on a(u), while being independent of g(u) and f .

Bounds for blow-up time for the heat equation under nonlinear boundary conditions

2009 ◽  
Vol 139 (6) ◽  
pp. 1289-1296 ◽  
Author(s):  
L. E. Payne ◽  
P. W. Schaefer
Keyword(s):  
Boundary Condition ◽  
Lower Bound ◽  
Heat Equation ◽  
Differential Inequality ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Nonlinear Boundary Conditions ◽  
Sufficient Condition ◽  
Inequality Technique

A differential inequality technique is used to determine a lower bound on the blow-up time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur. In addition, a sufficient condition which implies that blow-up does occur is determined.

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