Convergence to equilibrium for positive solutions of some mutation-selection model

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.