Abstract:
This paper is devoted to the study of maximum principles holding for some nonlocal diffusion operators defined in (half-) bounded domains and its applications to obtain qualitative behaviors of solutions of some nonlinear problems. It is shown that, as in the classical case, the nonlocal diffusion considered satisfies a weak and a strong maximum principle. Uniqueness and monotonicity of solutions of nonlinear equations are therefore expected as in the classical case. It is first presented a simple proof of this qualitative behavior and the weak/strong maximum principle. An optimal condition to have a strong maximum for operator $mathcal{M}[u] :=Jstar u -u$ is also obtained. The proofs of the uniqueness and monotonicity essentially rely on the sliding method and the strong maximum principle.

Abstract:
In this note, we study the existence of a strong maximum principle for the nonlocal operator $$ mathcal{M}[u](x) :=int_{G}J(g)u(x*g^{-1})dmu(g) - u(x), $$ where $G$ is a topological group acting continuously on a Hausdorff space $X$ and $u in C(X)$. First we investigate the general situation and derive a pre-maximum principle. Then we restrict our analysis to the case of homogeneous spaces (i.e., $ X=G /H$). For such Hausdorff spaces, depending on the topology, we give a condition on $J$ such that a strong maximum principle holds for $mathcal{M}$. We also revisit the classical case of the convolution operator (i.e. $G=(mathbb{R}^n,+), X=mathbb{R}^n, dmu =dy$).

Abstract:
In this paper, we establish some Harnack type inequalities satisfied by positive solutions of nonlocal inhomogeneous equations arising in the description of various phenomena ranging from population dynamics to micro-magnetism. For regular domains, we also derive an inequality up to the boundary. The main difficulty in such context lies in a precise control of the solutions outside a compact set and the existence of local uniform estimates. We overcome this problem by proving a contraction result which makes the $L^1$ norms of the solutions on two compact sets $\o_1\subset\subset\o_2$ equivalent. We also construct the principal positive eigenfunctions associated to particular nonlocal operators by using the corresponding Harnack type inequalities.

Abstract:
In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\lambda,\phi)$ of a nonlocal operator. $$\int_{\O}K(x,y)\phi(y)\, dy +a(x)\phi(x) =-\lambda \phi(x),$$ where $\O\subset\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\lambda_p:=\sup \{\lambda \in \R \, |\, \exists \, \phi \in C(\O), \phi > 0 \;\text{so that}\; \oplb{\phi}{\O}+ a(x)\phi + \lambda\phi\le 0\}$ there exists always a solution $(\mu, \lambda_p)$ of the problem in the space of signed measure. Moreover $\mu$ a positive measure. When $\mu$ is absolutely continuous with respect to the Lebesgue measure, $\mu =\phi_p(x)$ is called the principal eigenfunction associated to $\lambda_p$. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.

Abstract:
In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \times \O$$ where $\Omega\subset \R^n$ is a bounded open set, $K\in C(\R^n\times \R^n) $ is nonnegative, $a_i,\beta \in C(\Omega)$ and $\lambda\in\R$. Such type of equation appears in some studies of population dynamics where the above solutions are the stationary states of the dynamic of a spatially structured population evolving in a heterogeneous partially controlled landscape and submitted to a long range dispersal. Under some fairly general assumptions on $K,a_i$ and $\beta$ we first establish a necessary and sufficient criterium for the existence of a unique positive solution. Then we analyse the structure of the set of positive solution $(\lambda,u_\lambda)$ with respect to the presence or absence of a refuge zone (i.e $\omega$ so that $\beta_{|\omega}\equiv 0$).

Abstract:
In this paper we are interested in the long time behaviour of the positive solutions of the mutation selection model with Neumann Boundary condition: $$ \frac{\partial u(x,t)}{dt}=u\left[r(x)-\int_{\O}K(x,y)|u|^{p}(y)\,dy\right]+\nabla\cdot\left(A(x)\nabla u(x)\right),\qquad \text{in}\quad \R^+\times\O$$ where $\O\subset \R^N$ is a bounded smooth domain, $k(.,.) \in C(\bar \O \times C(\bar\O), \R), p\ge 1$ and $A(x)$ is a smooth elliptic matrix. In a blind competition situation, i.e $K(x,y)=k(y)$, we show the existence of a unique positive steady state which is positively globally stable. That is, the positive steady state attracts all the possible trajectories initiated from any non negative initial datum. When $K$ is a general positive kernel, we also present a necessary and sufficient condition for the existence of a positive steady states. We prove also some stability result on the dynamic of the equation when the competition kernel $K$ is of the form $K(x,y)=k_0(y)+\eps k_1(x,y)$. That is, we prove that for sufficiently small $\eps$ there exists a unique steady state, which in addition is positively asymptotically stable. The proofs of the global stability of the steady state essentially rely on non-linear relative entropy identities and an orthogonal decomposition. These identities combined with the decomposition provide us some a priori estimates and differential inequalities essential to characterise the asymptotic behaviour of the solutions.

Abstract:
In this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro-magnetism. More precisely, we study the following eigenvalue problem: $$\int_{\O}J(\frac{x-y}{g(y)})\frac{\phi(y)}{g^n(y)}\, dy +a(x)\phi =\rho \phi,$$ where $\O\subset\R^n$ is an open connected set, $J$ a nonnegative kernel and $g$ a positive function. First, we establish a criterion for the existence of a principal eigenpair $(\lambda_p,\phi_p)$. We also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator. As an application of these results we construct and characterize the solutions of some nonlinear nonlocal reaction diffusion equations.

Abstract:
In this paper, we construct radially symmetric solutions of a nonlinear noncooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.

Abstract:
In this paper we are investigating the long time behaviour of the solution of a mutation competition model of Lotka-Volterra's type. Our main motivation comes from the analysis of the Lotka-Volterra's competition system with mutation which simulates the demo-genetic dynamics of diverse virus in their host : $$ \frac{dv_{i}(t)}{dt}=v_i\[r_i-\frac{1}{K}\Psi_i(v)\]+\sum_{j=1}^{N} \mu_{ij}(v_j-v_i). $$ In a first part we analyse the case where the competition terms $\Psi_i$ are independent of the virus type $i$. In this situation and under some rather general assumptions on the functions $\Psi_i$, the coefficients $r_i$ and the mutation matrix $\mu_{ij}$ we prove the existence of a unique positive globally stable stationary solution i.e. the solution attracts the trajectory initiated from any nonnegative initial datum. Moreover the unique steady state $\bar v$ is strictly positive in the sense that $\bar v_i>0$ for all $i$. These results are in sharp contrast with the behaviour of Lotka-Volterra without mutation term where it is known that multiple non negative stationary solutions exist and an exclusion principle occurs (i.e For all $i\neq i_0, \bar v_{i}=0$ and $\bar v_{i_0}>0$). Then we explore a typical example that has been proposed to explain some experimental data. For such particular models we characterise the speed of convergence to the equilibrium. In a second part, under some additional assumption, we prove the existence of a positive steady state for the full system and we analyse the long term dynamics. The proofs mainly rely on the construction of a relative entropy which plays the role of a Lyapunov functional.

Abstract:
We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is "above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.