scholarly journals Remarks on Spreading and Vanishing for Free Boundary Problems of Some Reaction-Diffusion Equations

2014 ◽  
Vol 57 (3) ◽  
pp. 449-465 ◽  
Author(s):  
Yuki Kaneko ◽  
Kazuhiro Oeda ◽  
Yoshio Yamada
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Boundary Problems ◽  
Spreading And Vanishing

On the initial growth of interfaces in reaction–diffusion equations with strong absorption

1993 ◽  
Vol 123 (5) ◽  
pp. 803-817 ◽  
Author(s):  
Luis Alvarez ◽  
Jesus Ildefonso Diaz
Keyword(s):  
Free Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Strong Absorption ◽  
Diffusion Equations ◽  
Initial Growth ◽  
Dirichlet Problems ◽  
Local Solutions

We study the initial growth of the interfaces of non-negative local solutions of the equation ut = (um)xx−λuq when m ≧ 1 and 0<q <1. We show that if with C < C0, for some explicit C0 = C0(λ, m, q), then the free boundary Ϛ(t) = sup {x:u(x, t) > 0} is a ‘heating front’. More precisely Ϛ(t) ≧at(m−q)/2(1−q) for any t small enough and for some a>0. If on the contrary, with C<C0, then Ϛ(t) is a ‘cooling front’ and in fact Ϛ(t) ≧ −atm−q)/2(1−q) for any t small enough and for some a > 0. Applications to solutions of the associated Cauchy and Dirichlet problems are also given.

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