• Lotka-Volterra
• Reaction-Diffusion equations

In this work we will examine the complexity of a Lotka-Volterra diffusive system for two species. This system of PDE belongs to a large class of system named Reaction-Diffusion system.\\
The reaction-diffusion systems involve a variety of systems and its applications vary widely in the spectrum of applied sciences. They move from Chemistry, for example the Gierer-Meinhordt model that describes the densities of chemical substances like activators and inhibitors, and go through several disciplines as the Neurosciences (Hogde-Huxley model for the axons) or Nuclear engineering (nuclear reactor dynamics). Further applications associated to the Navier-Stokes equations are for instance in the branch of Biology that studies the chemotaxic process or involved in Biomedicine for the tumour-growth model. The historical reason of the Lotka-Volterra model comes from biology and specifically from the study of two species population dynamics. To all of these applications correspond a specific nonlinearity that models the physical system. The richness of the mathematical background needed, that includes dynamical system theory, operator theory, stability theory and many others reflects the presence of several phenomena like for example multiple steady-states, the formation of spatial patterns, the existence of solitary waves and oscillatory phenomena.

In this work we will study the diffusive system in the case of isolated system that is modelled by the Laplace operator with Neumann boundary condition. Intuitively, while it is fairly simple to understand that the property of isolated system will be related to the Neumann condition of null flux on the boundary, from a mathematical point of view it makes the study of the asymptotic behaviour interesting. This aspect is closely bound to the form of the spectrum of that operator and to the fact that in presence of Neumann condition, the zero belongs to the spectrum. Unlike the case of Dirichlet condition, where all the orbits of the partial differential system are forced to converge to the unique equilibrium point of the associated ordinary system, in the case of Neumann condition the asymptotic study tell us that it is possible also to converge to a periodic state. In this sense it is interesting to study the stability of the unique equilibrium point of the ordinary system.

The last chapter of this work is devoted to investigate the existence of global solutions for the linearized system around the equilibrium point mentioned above with low regular initial data. The only tools exploited in the chapter are a version of the Gagliardo-Niremberg inequality with some interpolation results and Sobolev theorems. In fact we will follow a direct method for the proof. The main ideas will be suggested by the previous consideration about the spectrum of the Laplace operator and do not use massively the theory of dynamical system. Thanks to the explicit form of the solution of the linearized system we will deduce the stability property of the equilibrium point and we will characterize the evolution of the orbits for large time.

In the first part of the second chapter are presented the existence theorems known in literature. The main theorem of the section involves as well low regular initial data but does not give any information about the asymptotic behaviour of the solutions. The scheme of the section is the following: the first result will be a classical theorem of existence for parabolic type equation with regular initial data; thanks to an argument of density it will be extended also for less regular initial data. The crucial point of this argument will be to gain some a-priori estimates. These estimates will be obtained in the same chapter by following two different ways: the starting point for both of them will be to improve a primary apriori estimate from a Lyapunov function that will be the lowest estimate. The procedure then will follow two different ways: the first case will take care of low regular data and will use a feed-back and bootstrap argument that will provide in a finite number of steps the needed estimate and it will be based only on interpolation theory. The latter will study a more regular case and will use a limiting procedure: manipulating the original equation and multiplying it by a suitable quantity we will obtain information on the
$L^{\infty}$ norm of the solution. The tools used in the second case belong to sectorial operators and fractional powers operator theory too. Thanks to the higher regularity for the estimate, it will be possible to prove a compactness property for the orbit. This will be fundamental in order to apply the Invariance Principle for the study of the asymptotic behaviour in the last part of the second chapter . It will be proven that in case of Neumann boundary condition the solution will tend asymptotically to a periodic state in the sense of the spaces associated to the fractional operators. Moreover, thanks to the high initial regularity, it will imply a quite strong convergence in an Holderian sense.