Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space
Henri Berestycki Luca Rossi
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 41-67 doi: 10.3934/dcds.2008.21.41
This paper is concerned with time-dependent reaction-diffusion equations of the following type:

$\partial_t u=$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 $c'\in\R$.

keywords: principal eigenvalues travelling waves Reaction-diffusion equations extinction. time periodic parabolic equations persistence forced speed
Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains
Henri Berestycki Luca Rossi
Discrete & Continuous Dynamical Systems - A 2009, 25(1): 19-61 doi: 10.3934/dcds.2009.25.19
This work is the continuation of our previous paper [6]. There, we dealt with the reaction-diffusion equation

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

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.

keywords: Reaction-diffusion equations bifurcation. travelling waves forced speed principal eigenvalues asymptotically cylindrical domains
Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains
Luca Rossi
Communications on Pure & Applied Analysis 2008, 7(1): 125-141 doi: 10.3934/cpaa.2008.7.125
In this paper we consider fully nonlinear elliptic operators of the form $F(x,u,Du,D^2u)$. Our aim is to prove that, under suitable assumptions on $F$, the only nonnegative viscosity super-solution $u$ of $F(x,u,Du,D^2u)=0$ in an unbounded domain $\Omega$ is $u\equiv 0$. We show that this uniqueness result holds for the class of nonnegative super-solutions $u$ satisfying

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.

keywords: unbounded domains viscosity solutions. positive solutions Fully nonlinear elliptic equations
The periodic patch model for population dynamics with fractional diffusion
Henri Berestycki Jean-Michel Roquejoffre Luca Rossi
Discrete & Continuous Dynamical Systems - S 2011, 4(1): 1-13 doi: 10.3934/dcdss.2011.4.1
Fractional diffusions arise in the study of models from population dynamics. In this paper, we derive a class of integro-differential reaction-diffusion equations from simple principles. We then prove an approximation result for the first eigenvalue of linear integro-differential operators of the fractional diffusion type, and we study from that the dynamics of a population in a fragmented environment with fractional diffusion.
keywords: Fractional diffusion reaction-diffusion equation KPP nonlinearity persistence.
On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary
Daniele Castorina Annalisa Cesaroni Luca Rossi
Communications on Pure & Applied Analysis 2016, 15(4): 1251-1263 doi: 10.3934/cpaa.2016.15.1251
We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in [1]. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive problem converge uniformly, in the reference frame moving with constant velocity $c$, to a unique steady state solving a suitable ergodic problem.
keywords: ergodic problem Hamilton-Jacobi-Bellman operators Large time behavior characteristic boundary points initial boundary value problems.

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