Reactiondiffusion
equations for population dynamics with forced speed II 
cylindricaltype domains
Henri Berestycki  EHESS, CAMS, 54 Boulevard Raspail, F75006, Paris, France (email) Abstract: This work is the continuation of our previous paper [6]. There, we dealt with the reactiondiffusion equation $\partial_t u=\Delta u+f(xcte,u),\qquad t>0,\quad x\in\R^N,$
where $e\in S^{N1}$ and $c>0$ are given and $f(x,s)$ satisfies
some usual assumptions in population dynamics, together with
$f_s(x,0)<0$ for $x$ large. The interest for such equation comes
from an ecological model introduced in [1]
describing the effects of global
warming on biological species. In [6],we proved that
existence and uniqueness of travelling wave solutions of the type
$u(x,t)=U(xcte)$ and the large time behaviour of solutions with
arbitrary nonnegative bounded initial datum depend on the sign of
the generalized principal in $\R^N$ of an associated linear operator.
Here, we establish analogous results for the Neumann problem in
domains which are asymptotically cylindrical, as well as for the problem in
the whole space with $f$ periodic in some space variables,
orthogonal to the direction of the shift $e$. $\partial_t u=\Delta u+f(t,xcte,u),$ when $f(t,x,s)$ is periodic in $t$. This for instance represents the seasonal dependence of $f$. In both cases, we obtain a necessary and sufficient condition for the existence, uniqueness and stability of pulsating travelling waves, which are solutions with a profile which is periodic in time.
Keywords: Reactiondiffusion equations, travelling waves, forced speed,
asymptotically cylindrical domains, principal eigenvalues, bifurcation.
Received: August 2008; Revised: February 2009; Available Online: June 2009. 
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