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On the retention of the interfaces in some elliptic and parabolic nonlinear problems
Reactiondiffusion equations for population dynamics with forced speed II  cylindricaltype domains
1.  EHESS, CAMS, 54 Boulevard Raspail, F75006, Paris 
$\partial_t u=\Delta u+f(xcte,u),\qquad t>0,\quad x\in\R^N,$
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 $e$.
The $L^1$ convergence of solution $u(t,x)$ as $t\to\infty$ is established
next. In this paper, we also show
that a bifurcation from the zero solution takes place as the principal crosses $0$. We are
able to describe the shape of solutions close to extinction
thus answering a question raised by M.~Mimura.
These two results are new even in the framework
considered in [6].
Another type of problem is obtained by adding to the previous one a term
$g(xc'te,u)$ periodic in $x$ in the direction $e$.
Such a model arises when considering
environmental change on two different scales.
Lastly, we also solve the case of an equation
$\partial_t u=\Delta u+f(t,xcte,u),$
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.
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2018 Impact Factor: 1.143
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