Bifurcation of positive solutions to scalar reaction–diffusion equations with nonlinear boundary condition

2018 ◽  
Vol 264 (1) ◽  
pp. 425-454
Author(s):  
Ping Liu ◽  
Junping Shi
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations

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 .

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