We consider the following reaction-diffusion equation:
$$
{\rm (KS)}
\left\{
\begin{array}{llll}
u_t =
\nabla \cdot \Big( \nabla u^m - u^{q-1} \nabla v \Big),
& x \in \mathbb{R}^N, \ 0<t<\infty, \nonumber
0 =
\Delta v - v + u,
& x \in \mathbb{R}^N, \ 0<t<\infty, \nonumber
u(x,0) = u_0(x),
& x \in \mathbb{R}^N,
\end{array}
\right.
$$
where $N \ge 1, \ m > 1, \ q \ge \max\{m+\frac{2}{N},2\}$.
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)]
it was shown that
in the case of $q \ge \max\{m+\frac{2}{N},2\}$,
the above problem (KS) is solvable globally in time for “small $L^{\frac{N(q-m)}{2}}$ data”.
Moreover,
the decay of the solution (u,v) in $L^p(\mathbb{R}^N)$
was proved.
In this paper, we consider
the case of “$q \ge \max\{m+\frac{2}{N},2\}$ and
small $L^{\ell}$ data” with any fixed $\ell \ge \frac{N(q-m)}{2}$
and show that
(i)
there exists a time global solution (u,v) of (KS) and
it decays to 0 as t tends to ∞ and
(ii)
a solution u of the first equation in (KS)
behaves
like the Barenblatt solution asymptotically as t tends to ∞,
where the Barenblatt solution is the exact solution (with self-similarity)
of the porous medium equation
$u_t = \Delta u^m$ with m>1.