Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems

TL;DR

This paper investigates the local regularity and long-term behavior of solutions to a degenerate parabolic system modeling multi-species population densities, establishing continuity, convergence to Barenblatt profiles, and eventual concavity.

Contribution

It introduces new regularity results for degenerate systems and proves convergence of species densities to Barenblatt profiles under certain conditions.

Findings

Solutions are locally H"older continuous.

Population densities converge to Barenblatt profiles as time approaches infinity.

Each density becomes concave after finite time.

Abstract

We study the local H\"older continuity and the asymptotic behaviour of solution, $\mathbf{u}=(u^1,\cdots, u^k)$, of the degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(m\,U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$ } \end{equation*} which describes the populations density of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. For the local H\"older continuity, we adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto. Under some regularity conditions, we also prove that the population density function of $i$-th species with the population $M_i$ converges in $C_s^{\infty}$ to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ as $t\to \infty$ where $\mathcal{B}_M$ is the Barenblatt profile of the standard porous medium equation with $L^1$ mass $M=M_1+\cdots+M_k$. As a consequence of asymptotic behaviour, it is shown that each density function becomes a concave function after a finite time.