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$\partial_t u=$Δ$u+f(x-cte,u),t>0,x\in\R^N.$

These kind of equations have been introduced in [1] in
the case $N=1$ for studying the impact of a climate shift on the
dynamics of a biological species.

In the present paper, we first extend the results of
[1] to arbitrary dimension $N$ and to a greater
generality in the assumptions on $f$. We establish a necessary
and sufficient condition for the existence of travelling wave
solutions, that is, solutions of the type $u(t,x)=U(x-cte)$. This
is expressed in terms of the sign of the generalized principal eigenvalue $\l$ of
an associated linear elliptic operator in $\R^N$. With this
criterion, we then completely describe the large time dynamics for
this equation. In particular, we characterize situations in which
there is either extinction or persistence.

Moreover, we consider the problem obtained by adding a term
$g(x,u)$ periodic in $x$ in the direction $e$:

$\partial_t u=$Δ$u+f(x-cte,u)+g(x,u),t>0,x\in\R^N.$

Here, $g$ can be viewed as representing geographical characteristics of the territory which are not subject to shift. We derive analogous results as before, with $\l$ replaced by the generalized principal eigenvalue of the parabolic operator obtained by linearization about $u\equiv0$ in the whole space. In this framework, travelling waves are replaced by pulsating travelling waves, which are solutions of the form $U(t,x-cte)$, with $U(t,x)$ periodic in $t$. These results still hold if the term $g$ is also subject to the shift, but on a different time scale, that is, if $g(x,u)$ is replaced by $g(x-c'te,u)$, with $c'\in\R$.

$\partial_t u=\Delta u+f(x-cte,u),\qquad t>0,\quad x\in\R^N,$

where $e\in S^{N-1}$ 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
$u(x,t)=U(x-cte)$ 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
$g(x-c'te,u)$ 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

$\partial_t u=\Delta u+f(t,x-cte,u),$

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.

limin$f_{x\in\Omega, |x|\to\infty}\frac{u(x)+1}{\dist(x,\partial\Omega)}=0,$

and then, in particular, for strictly sublinear super-solutions in
a domain $\Omega$ containing an open cone. In the special case that
$\Omega=\mathbb R^N$, or that $F$ is the Bellman operator, we show that the
same result holds for the whole class of nonnegative
super-solutions.

Our principal assumption on the operator $F$ involves its zero
and first order dependence when
$|x|\to\infty$. The same kind of assumption was introduced in a recent
paper
in collaboration with H. Berestycki and F. Hamel [4] to establish a Liouville type result for semilinear equations. The
strategy we follow to prove our main results is the same as in
[4], even if here we consider fully nonlinear
operators, possibly unbounded solutions and more general domains.

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