Introduction

We consider the Cauchy problem for the porous media equation

$$\left\{\begin{array}{lllr} u_t={\mit\Delta}u^m&in&R^N\times(0,T)\\ u(x,0)=u_0&on&R^N,\\ \end{array}\right.\\ \leqno{(1.1)}$$

where $m>1$ and $u_0\geq0$. By a solution of (1.1) we mean a function $u(x,t)$ such that

$$u(x,t){\geq}0{\enskip}{\rm in}{\enskip}R^N{\times}(0,T),$$

$$\int_0^T\int_{R^N}\left[{u(x,t)}^2+\left\vert{\nabla}_xu(x,t)\right\vert^2\right]dxdt<\infty$$

and

$$\int_0^T\int_{R^N}\left(u{\phi}_t-{\nabla}_xu^m\cdot{\nabla}_x\phi\right)dxdt+\int_{R^N}u_0(x)\phi(x,0)dx=0$$

for any continuously differentiable function ${\phi}(x,t)$ with compact support in $R^N\times[0,T)$. The problem(1.1) has been studied by many authors. For a detailed account of (1.1) we refer to the survey of Kalashnikov [6]. The existence and the uniqueness of solutions of (1.1) are due to [8] and [9] under some assumption on $u_0$.

We are concerned to the regularity property for $u$. The local Hölder continuity of $u$ was shown by Caffarelli and Friedman [3]. Aronson and Bénilan [1] proved that ${\mit\Delta}u^m$ belongs to $L^1_{loc}(R^N\times(0,T))$. The method of their proof is to obtain the inequarity

$$u_t{\enskip}{\geq}-\frac{1}{t}\left(m-1+\frac{2}{N}\right)^{-1}{\enskip}in{\enskip}R^N\times(0,T),$$

which is in the distribution sence. Soon after Bénilan [2] proved that

$${\mit\Delta}u^m{\in}L^p_{loc}(R^N\times(0,T)),$$

if $1<p<1+\frac{1}{m}$. Here $p$ needs to be less than 2 in virtue of $m>1$. When $p=2$, there is the following result by one of the authors [4]:

$${\partial}_{x_i}{\partial}_{x_j}u^m{\in}L^2(R^N\times(0,T)),{\qquad}i,j=1,\cdots,N,$$

if $1<m<\frac{3N}{3N-2}$. When $u$ is spherically symmetric under some assumption on $u_0$, this result was improved in [5] as follows:

$the{\enskip}condition{\enskip}1<m<\frac{3N}{3N-2}{\enskip}can{\enskip}be{\enskip}weakend{\enskip}with{\enskip}1<m<3,{\enskip}for{\enskip}N=1,2,3.$

We consider the well-known Barenblatt solution:

$$w(x,t)=(t+{\tau})^{-k}\left(\left[a^2-\frac{k(m-1)}{2Nm}\frac... ...^2}{(t+{\tau})^{\frac{2k}{N}}}\right]_+\right)^{\frac{1}{m-1}},$$

where $a$, $\tau>0$ and $k=\left(m-\frac{N-2}{N}\right)^{-1}$. We rewrite $w$ simply with $(t+{\tau})^{-k}\left(\left[g\right]_+\right)^{\frac{1}{m-1}}$. Then obviously

$${\partial}_{x_i}{\partial}_{x_j}w^m=\left(\left[g\right]_+\ri... ...-1}}P_{ij}+\left(\left[g\right]_+\right)^{\frac{1}{m-1}}Q_{ij},$$

where $P_{ij}$ and $Q_{ij}$ are smooth functions. Hence we see that

$${\partial}_{x_i}{\partial}_{x_j}w^m{\in}L^p(R^N\times(0,T)),{\enskip}{\rm if}{\enskip}p\left(\frac{2-m}{m-1}\right)>-1.$$

Let $u$ be the spherically symmetric solution of (1.1). Then from the above we conjecture that for any given $p>2$

$${\qquad}{\partial}_{x_i}{\partial}_{x_j}u^m{\in}L^p(R^N\times(0,T)),{\quad}i,j=1,{\cdots},N,$$

if $m$ is close to 1. In this paper our first aim is to verify this conjecture (see Theorem 1 ). Secondly we give precisely the admissive value of $m$ in Theorem 1, when $N=1$ (see Theorem 2). The tool is to prepare some $L^p$ - estimates with a weight by the integreation by parts.