The Nonexistence of the Solutions for the Non-Newtonian Filtration Equation with Absorption

OU Qitong

School of Applied Mathematics, Xiamen University of Technology,Xiamen 361024,China.

Abstract. The paper proves the nonexistence of the solution for the following Cauchy problem

Key Words: Non-Newtonian filtration equation;Cauchy problem;nonexistence.

1 Introduction

Theorem 1.1.Suppose that

2 Proof of Theorem 1.1.

Lemma 2.1.Suppose that

We have to estimate the last item of(2.6). Without loss of generality,we asumme thatuis strictly positive. Otherwise,we can replace it withu+εand then letε→0.

In Definition 1.1.,we set

For the last term on the right hand side of(2.11),an integration by parts and lettingh→0,we have

By(2.7)-(2.12),and the Young inequality

lettingh→0,we obtain

At the same time,by the H¨older inequality,we have

Using(2.14) and(2.15),we obtain

Now we discuss it in two situations.

Acknowledgement

We would like to thank the reviewers for their helpful suggestions.The paper is supported by Natural Science Foundation of Fujian province in China(No: 2019J01858).