no. 1
Long time behavior of solutions of a reaction–diffusion equation on unbounded intervals with Robin boundary conditions
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 67-92.

We study the long time behavior, as t→∞, of solutions of

{ut=uxx+f(u),x>0,t>0,u(0,t)=bux(0,t),t>0,u(x,0)=u0(x)⩾0,x⩾0,
where b⩾0 and f is an unbalanced bistable nonlinearity. By investigating families of initial data of the type {σϕ}σ>0, where ϕ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value σ⁎ such that the solution converges uniformly to 0 for any 0<σ<σ⁎, and locally uniformly to a positive stationary state for any σ>σ⁎. In the threshold case σ=σ⁎, the profile of the solution approaches the symmetrically decreasing ground state with some shift, which may be either finite or infinite. In the latter case, the shift evolves as Cln⁡t where C is a positive constant we compute explicitly, so that the solution is traveling with a pulse-like shape albeit with an asymptotically zero speed. Depending on b, but also in some cases on the choice of the initial datum, we prove that one or both of the situations may happen.

DOI : 10.1016/j.anihpc.2014.08.004
Classification : 35K57, 35K15, 35B40
Mots clés : Reaction–diffusion equation, Long time behavior, Robin boundary condition, Sharp threshold 
@article{AIHPC_2016__33_1_67_0,
     author = {Chen, Xinfu and Lou, Bendong and Zhou, Maolin and Giletti, Thomas},
     title = {Long time behavior of solutions of a reaction{\textendash}diffusion equation on unbounded intervals with {Robin} boundary conditions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {67--92},
     publisher = {Elsevier},
     volume = {33},
     number = {1},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.08.004},
     mrnumber = {3436427},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.08.004/}
}
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Chen, Xinfu; Lou, Bendong; Zhou, Maolin; Giletti, Thomas. Long time behavior of solutions of a reaction–diffusion equation on unbounded intervals with Robin boundary conditions. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 67-92. doi : 10.1016/j.anihpc.2014.08.004. http://www.numdam.org/articles/10.1016/j.anihpc.2014.08.004/

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