# Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time

###### Abstract

Let , , , and . For any , we will prove the existence and uniqueness (for ) of radially symmetric singular solution of the elliptic equation , , in , satisfying . When is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as . We also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence we will prove the extinction behaviour of the solution of the fast diffusion equation in near the extinction time .

Key words: extinction behaviour, fast diffusion equation, self-similar solution, higher order asymptotic behaviour

AMS 2010 Mathematics Subject Classification: Primary 35B40, 35K65 Secondary 35J70

## 1 Introduction

The equation

(1.1) |

appears in many physical models. When , (1.1) is the porous medium equation which models the flow of gases or liquid through porous media. When , (1.1) is the heat equation. When , (1.1) is the fast diffusion equation. When , , and is a metric on which evolves by the Yamabe flow,

where is the scalar curvature of the metric , then satisfies [DKS], [PS],

It is because of the importance of the equation (1.1) and its relation to the Yamabe flow, there are a lot of research on this equation recently by P. Daskalopoulos, J. King, M. del Pino, N. Sesum, M. Sáez, [DKS], [DPS], [DS1], [DS2], [PS], S.Y. Hsu [Hs1–3], K.M. Hui [Hu1], [Hu2], M. Fila, J.L. Vazquez, M. Winkler, E. Yanagida, [FVWY], [FW], A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J.L. Vazquez, [BBDGV], [BDGV], etc. We refer the reader to the survey paper [A] by D.G. Aronson and the books [DK], [V2], by P. Daskalopoulos, C.E. Kenig, and J.L. Vazquez on the recent progress on this equation.

As observed by J.L. Vazquez [V1], M.A. Herrero and M. Pierre [HP], and others [Hs2], [Hu1], there is a big difference on the behaviour of solution of (1.1) for the case , , and the case , . For example for any , , when , , there exists a unique global positive smooth solution of (1.1) in with initial value [HP]. On the other hand when , , there exists , , and such that the solution of

(1.2) |

extincts at time [DS1]. Since the asymptotic behaviour of the solution of (1.2) near the extinction time is usually similar to the asymptotic behaviour of the self-similar solution of (1.1), in order to understand the behaviour of the solution of (1.2) near the extinction time we will first study various properties of the self-similar solutions of (1.1) in this paper.

Let , , , and . For any , by Theorem 1.1 of [Hs1] there exists a unique radially symmetric solution of the equation

(1.3) |

in that satisfies . By [Hs3], satisfies

(1.4) |

Note that when , the function

(1.5) |

is a solution of (1.1) in for any . On the other hand if , , and , then the metric

(1.6) |

on is a Yamabe shrinking soliton [DS2]. Conversely as proved by P. Daskalopoulos and N. Sesum [DS2] any Yamabe shrinking soliton on complete locally conformally flat manifold is of the form (1.6) where is a solution of (1.3) in for some with for some constant .

Let ,

and , , be the two roots of the equation

(1.7) |

given by

(1.8) |

where

Now if , then

Hence are real roots of (1.7) when , , and . Note that

(1.9) |

and when and , then , , and (1.7) is equivalent to

In [DKS] P. Daskalopoulos, J. King and N. Sesum, proved that when , , , and , the radially symmetric solution of (1.3) in with satisfies

(1.10) |

for some constants ,

(1.11) |

and where if and , and otherwise. In this paper we will extend this second order asymptotic result to the case , and . For any , , and , we will also extend Theorem 1.2 of [DKS] and prove the existence and uniqueness of radially symmetric singular solution of (1.3) in that satisfies

(1.12) |

We also obtain higher order decay rate of as . Let

Then is a solution of (1.3) in . In the papers [DS1], [BBDGV], etc. P. Daskalopoulos and N. Sesum, A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J.L. Vazquez, etc. obtain the asymptotic behaviour of the solution of (1.2) near the extinction time for , , when the initial value is sandwiched between two Barenblatt solutions. In this paper we will extend their results to initial values that satisfies other growth conditions.

More precisely we obtain the following main results in this paper.

###### Theorem 1.1.

###### Corollary 1.2.

###### Theorem 1.3.

###### Theorem 1.4.

By direct computation we also have the following inversion formula for the solution of (1.3).

###### Theorem 1.5.

Let , , , . Let be a radially symmetric solution of (1.3) in and . Then satisfies

where , . If for some constant , then . Moreover . for all

###### Theorem 1.6.

###### Theorem 1.7.

The plan of the paper is as follows. In section two we will prove Theorem 1.1. We will prove Theorem 1.3 and Theorem 1.4 in section three and four respectively. Finally we will sketch the proof of Theorem 1.6 and Theorem 1.7 in section five.

Unless stated otherwise we will assume that , , , , and , for the rest of the paper. For any , we let . For any and domain , we say that is a solution of (1.1) in if is a smooth positive solution of (1.1) in . For any we say that is a solution of (1.1) in with initial value if is a solution of (1.1) in with in as .

## 2 Existence of blow-up solutions

In this section we will prove Theorem 1.1. We first start with a technical lemma.

###### Lemma 2.1.

Proof: We first claim that there exists a sequence of positive numbers , as , such that

(2.3) |

In order to proof the claim we choose a sequence of positive numbers such that as . Then by (1.12) and the mean value theorem for any there exists such that

for some constant . Hence the sequence has a subsequece which we may assume without loss of generality to be the sequence itself such that converges to some constant as . Then by (1.12) and L’Hospital’s Rule,

and (2.3) follows. By (1.3) satisfies

(2.4) |

Integrating (2) over ,

(2.5) |

Since , . Putting in (2) and letting , by (1.9), (1.12), and (2.3), we get that satisfies (2.1) if and satisfies (2.2) if .

Similarly we have the following lemma.

###### Lemma 2.2.

###### Lemma 2.3.

Let and . Then

Proof: We will use a modification of the proof of Lemma 2.3 of [HuK] to proof the lemma. Since , by (1.12) there exists a constant such that for any . Let be the maximal interval such that

(2.6) |

Suppose . Then and . If , then by Lemma 2.1 both and satisfy (2.1). Hence if , by (1.9), (2.6), and Lemma 2.1,

If , by Lemma 2.1 both and satisfy (2.2). Hence by (2.6),

Hence for any ,

and contradiction arises. Thus and the lemma follows.

By direct computation satisfies (2.1) and (2). Since as and as , by Lemma 2.1, Lemma 2.2, and an argument similar to the proof of Lemma 2.3 we have the following result.

###### Lemma 2.4.

Let . Then

and

(2.7) |

###### Theorem 2.5.

Proof: We choose a monotone decreasing sequence for all such that as . Let

Then satisfies (1.3) in and

(2.8) |

Similarly by interchanging the role of and in the above argument we get for all . Hence on and the theorem follows.

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1: By Theorem 2.5 we only need to prove existence of radially symmetric solution of (1.3) in or solution of (2) that satisfies (1.12), (1.13) and (1.14), when . Let .

Claim 1: For any , there exists a radially symmetric solution of (1.3) in which satisfies

(2.9) |

In order to prove this claim we first observe that by the standard O.D.E. theory there exist and a solution of (2) in which satisfies (2.9). Let be the maximal interval of existence of solution of (2) which satisfies (2.9). Let , , and

By (2.9), . As observed in [Hs1], satisfies

in . Hence

(2.10) | ||||

(2.11) |