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Mathematics > Analysis of PDEs

Title: Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems

Abstract: We study the local continuity and asymptotic behavior of solutions, $\bold{u}=(u^1,\cdots, u^k)$, of degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$} \end{equation*} describing the degenerate diffusion of the populations density vector, $\bold{u}$, of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. We adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto for the local continuity of solutions, $u^i$. Under some regularity condition, we also prove that the population density function, $\bold{u}$, of $i$-th species with the population $M_i$ converges to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ in the space of differentiable functions of all order where $\mathcal{B}_M$ is the Barenblatt profile of the Porous Medium Equation with $L^1$ mass $M=M_1+\cdots+M_k$ while $U$ converges to $\mathcal{B}_M$. As a consequence, each $u^i$ becomes a concave function after a finite time.
Comments: 32p
Subjects: Analysis of PDEs (math.AP)
Cite as: arXiv:1803.06465 [math.AP]
  (or arXiv:1803.06465v1 [math.AP] for this version)

Submission history

From: Sunghoon Kim [view email]
[v1] Sat, 17 Mar 2018 05:06:32 GMT (28kb)