Inside dynamics of solutions of integro-differential equations
Olivier Bonnefon Jérôme Coville Jimmy Garnier Lionel Roques
Discrete & Continuous Dynamical Systems - B 2014, 19(10): 3057-3085 doi: 10.3934/dcdsb.2014.19.3057
In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations \begin{equation*} \partial_t u(t,x)= (J\star u)(t,x) -u(t,x) + f(u(t,x)), \ t>0 \hbox{ and } x\in\mathbb{R}, \end{equation*} with both thin-tailed and fat-tailed dispersal kernels $J$ and a monostable reaction term $f.$ The notion of inside dynamics has been introduced to characterize traveling waves of some reaction-diffusion equations [23]. Assuming that the solution is made of several fractions $\upsilon^i\ge 0$ ($i\in I \subset \mathbb{N}$), its inside dynamics is given by the spatio-temporal evolution of $\upsilon^i$. According to this dynamics, the traveling waves can be classified in two categories: pushed and pulled waves. For thin-tailed kernels, we observe no qualitative differences between the traveling waves of the above integro-differential equations and the traveling waves of the classical reaction-diffusion equations. In particular, in the KPP case ($f(u)\leq f'(0)u$ for all $u\in(0,1)$) we prove that all the traveling waves are pulled. On the other hand for fat-tailed kernels, the integro-differential equations do not admit any traveling waves. Therefore, to analyse the inside dynamics of a solution in this case, we introduce new notions of pulled and pushed solutions. Within this new framework, we provide analytical and numerical results showing that the solutions of integro-differential equations involving a fat-tailed dispersal kernel are pushed. Our results have applications in population genetics. They show that the existence of long distance dispersal events during a colonization tend to preserve the genetic diversity.
keywords: long distance dispersal integro-differential equation monostable pulled and pushed solutions thin-tailed/fat-tailed kernel. Traveling waves
Existence of radial stationary solutions for a system in combustion theory
Jérôme Coville Juan Dávila
Discrete & Continuous Dynamical Systems - B 2011, 16(3): 739-766 doi: 10.3934/dcdsb.2011.16.739
In this paper, we construct radially symmetric solutions of a nonlinear non-cooperative 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.
keywords: combustion model flame balls. radial solutions Elliptic system
Nonlocal refuge model with a partial control
Jérôme Coville
Discrete & Continuous Dynamical Systems - A 2015, 35(4): 1421-1446 doi: 10.3934/dcds.2015.35.1421
In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \Omega$$ where $\Omega\subset \mathbb{R}^n$ is a bounded domain, $K\in C(\mathbb{R}^n\times \mathbb{R}^n) $ is non-negative, $a_i,\beta \in C(\Omega)$ and $\lambda\in\mathbb{R}$. Such type of equation appears in some studies of population dynamics where the population evolves in a partially controlled heterogeneous landscape and disperses on long ranges. Under some fairly general assumptions on $K,a_i$ and $\beta$, we first establish a necessary and sufficient condition for the existence of a unique positive solution. Then, we analyse the structure of the set of positive solutions $(\lambda,u_\lambda)$ depending on the presence or absence of a refuge zone (i.e $\omega$ so that $\beta_{|\omega}\equiv 0$).
keywords: asymptotic behaviour non trivial solution principal eigenvalue Nonlocal diffusion operators partially controlled refuge model.
Concentration phenomenon in some non-local equation
Olivier Bonnefon Jérôme Coville Guillaume Legendre
Discrete & Continuous Dynamical Systems - B 2017, 22(3): 763-781 doi: 10.3934/dcdsb.2017037
We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation
$\partial_t u(t,x)\\=\int_{\Omega }m(x,y)\left(u(t,y)-u(t,x)\right)\,dy+\left(a(x)-\int_{\Omega }k(x,y)u(t,y)\,dy\right)u(t,x),$
supplemented by the initial condition
$\Omega $
, where the domain
$\Omega $
is a, the functions
are non-negative kernels satisfying integrability conditions and the function
is continuous. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function
represents the density of individuals characterized by the trait, the domain of trait values
$\Omega $
is a bounded subset of
, the kernels
respectively account for the competition between individuals and the mutations occurring in every generation, and the function
represents a growth rate. When the competition is independent of the trait, that is, the kernel
is independent of
, (
), we construct a positive stationary solution which belongs to
inthe space of Radon measures on
$\Omega $
$\mathbb{M}(\Omega )$
.Moreover, in the case where this measure
is regular and bounded, we prove its uniqueness and show that, for any non-negative initial datum in
$L^1(\Omega )\cap L^{\infty}(\Omega )$
, the solution of the Cauchy problem converges to this limit measure in
$L^2(\Omega )$
. We also exhibit an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. The numerical simulations seem to reveal a dependence of the limit measure with respect to the initial datum.
keywords: Non-local equation demo-genetics concentration phenomenon asymptotic behaviour
Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients
Jérôme Coville Nicolas Dirr Stephan Luckhaus
Networks & Heterogeneous Media 2010, 5(4): 745-763 doi: 10.3934/nhm.2010.5.745
We consider a so-called random obstacle model for the motion of a hypersurface through a field of random obstacles, driven by a constant driving field. The resulting semi-linear parabolic PDE with random coefficients does not admit a global nonnegative stationary solution, which implies that an interface that was flat originally cannot get stationary. The absence of global stationary solutions is shown by proving lower bounds on the growth of stationary solutions on large domains with Dirichlet boundary conditions. Difficulties arise because the random lower order part of the equation cannot be bounded uniformly.
keywords: Random obstacles Qualitative behavior of parabolic PDEs with random coefficients Interface evolution in Random media.
Bistable travelling waves for nonlocal reaction diffusion equations
Matthieu Alfaro Jérôme Coville Gaël Raoul
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 1775-1791 doi: 10.3934/dcds.2014.34.1775
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.
keywords: Travelling waves bistable case nonlocal reaction-diffusion equation ignition case. Leray-Schauder topological degree

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