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# Introduction

We consider the Cauchy problem for the porous media equation

where and . By a solution of (1.1) we mean a function such that

and

for any continuously differentiable function with compact support in . 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 .

We are concerned to the regularity property for . The local Hölder continuity of was shown by Caffarelli and Friedman [3]. Aronson and Bénilan [1] proved that belongs to . The method of their proof is to obtain the inequarity

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

if . Here needs to be less than 2 in virtue of . When , there is the following result by one of the authors [4]:

if . When is spherically symmetric under some assumption on , this result was improved in [5] as follows:

We consider the well-known Barenblatt solution:

where , and . We rewrite simply with . Then obviously

where and are smooth functions. Hence we see that

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

if 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 in Theorem 1, when (see Theorem 2). The tool is to prepare some - estimates with a weight by the integreation by parts.

Next: Bibliography Up: On Regularity for Weak Previous: On Regularity for Weak
Nobuki Takayama 2002-04-24