On the blow-up rate for the heat equation with a nonlinear boundary condition

2000 ◽  
Vol 23 (15) ◽  
pp. 1323-1330 ◽  
Author(s):  
Miroslav Chleb�k ◽  
Marek Fila
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Blow Up Rate

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.

NUMERICAL BLOW-UP FOR A NONLINEAR PROBLEM WITH A NONLINEAR BOUNDARY CONDITION

2002 ◽  
Vol 12 (04) ◽  
pp. 461-483 ◽  
Author(s):  
RAÚL FERREIRA ◽  
PABLO GROISMAN ◽  
JULIO D. ROSSI
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Nonlinear Problem ◽  
Finite Time ◽  
Numerical Scheme ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Mesh Parameter

In this paper we study numerical approximations for positive solutions of a nonlinear heat equation with a nonlinear boundary condition. We describe in terms of the nonlinearities when solutions of a semidiscretization in space exist globally in time and when they blow up in finite time. We also find the blow-up rates and the blow-up sets. In particular we prove that regional blow-up is not reproduced by the numerical scheme. However, in the appropriate variables we can reproduce the correct blow-up set when the mesh parameter goes to zero.

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