System of Porous Medium Equations

Abstract: We investigate the evolution of population density vector, $\bold{u}=\left(u^1,\cdots,u^k\right)$, of $k$-species whose diffusion is controlled by its absolute value $\left|\bold{u}\right|$. More precisely we study the properties and asymptotic large time behaviour of solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ of degenerate parabolic system \begin{equation*} \left(u^i\right)_t=\nabla\cdot\left(\left|\bold{u}\right|^{m-1}\nabla u^i\right) \qquad \mbox{for $m>1$ and $i=1,\cdots,k$}. \end{equation*} Under some regularity assumption, we prove that the function $u^i$ which describes the population density of $i$-th species with population $M_i$ converges to $\frac{M_i}{\left|\bold{M}\right|}\mathcal{B}{\left|\bold{M}\right|}$ in space with two different approaches where $\mathcal{B}{\left|\bold{M}\right|}$ is the Barenblatt solution of the porous medium equation with $L^1$-mass $\left|\bold{M}\right|=\sqrt{M_1^2+\cdots+M_k^2}$. \indent As an application of the asymptotic behaviour, we establish a suitable harnack type inequality which makes the spatial average of $u^i$ under control by the value of $u^i$ at one point. We also find an 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity.