Blow-up phenomena for a class of nonlinear reaction-diffusion equations under nonlinear boundary conditions

2018 ◽  
Vol 98 (16) ◽  
pp. 2868-2883 ◽  
Author(s):  
Ling-Ling Zhang ◽  
Hui-Min Tian
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Reaction

scholarly journals Blow-Up Phenomena for Nonlinear Reaction-Diffusion Equations under Nonlinear Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Juntang Ding
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Nonlinear Reaction

This paper deals with blow-up and global solutions of the following nonlinear reaction-diffusion equations under nonlinear boundary conditions:g(u)t=∇·au∇u+fu  in  Ω×0,T,  ∂u/∂n=bx,u,t  on  ∂Ω×(0,T),  u(x,0)=u0(x)>0,  in  Ω¯,whereΩ⊂RN  (N≥2)is a bounded domain with smooth boundary∂Ω. We obtain the conditions under which the solutions either exist globally or blow up in a finite time by constructing auxiliary functions and using maximum principles. Moreover, the upper estimates of the “blow-up time,” the “blow-up rate,” and the global solutions are also given.

scholarly journals Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

2017 ◽  
Vol 147 (3) ◽  
pp. 649-671
Author(s):  
Nsoki Mavinga ◽  
Rosa Pardo
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotically Linear ◽  
Bifurcation From Infinity

We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

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