BAO Ai-guo,WU Guo-rong

(Department of Mathematics and Statistics,School of Science,Inner Mongolia Agricultural University,Inner Mongolia Huhhot 010018 China)

Abstract:In this paper,using a delicate application of general Sobolev inequality,we establish the lower bound for the blow-up time of the solution to a quasi-linear parabolic problem,which improves the result of Theorem 2.1,Theorem 3.1 in[1],and the model(4.1)in[2].

Keywords:Quasi-linear parabolic equation;Initial-boundary value problem;lower bound for blow-up time

1 Introduction

In this paper,we will establish the lower bound for the blow-up time of the solution to the following problems:

Herea>0,m>1,p≥0 andq≥0,Ω⊂Rn(n≥3)is a smooth bounded domain,νis the outward norm vector.The initial datau0(x)is a continuous nonnegative function and satisfies the compatible conditions.In[3],LI and XIE proved that the solution to(1.1)exists globally ifp+qmand the initial datau0(x)is sufficiently large.

The direct motivation of this paper comes from the papers[1]and[2].In[1],the authors estimated the lower bounds for the blow-up time of solution to(1.1)subject to Dirichlet boundary condition and Neumann boundary condition in 3-dimension space.In[2],the authors only established the lower bounds for the blow-up time of the solution to(1.1)subject to Dirichlet boundary condition with smooth bounded Ω⊂Rnandn≥3.Naturally,we hope to obtain the lower bound for the blow-up time of the solution to(1.1)subject to Dirichlet boundary condition and Neumann boundary condition with smooth bounded Ω⊂Rnandn≥3.Inspired by Payne-Schaefer’s idea and following the AN and SONG’s methods in[4],we will use a delicate application of general Sobolev inequality to deal with both(1.1)subject to Neumann boundary condition and(1.1)subject to Dirichlet boundary condition.There are many results about the estimates of the lower bounds for blow-up time of the solution to parabolic equation.We can refer to[5-13]and the references therein to get more information.

Our main result in this paper can be stated as follows:

Theorem 1.1Assume thatuis the blow-up solution of(1.1),which will blow-up at timet=t∗.Then the lower bound for the blow-up time of the solution is

We will give the details to proof of Theorem 1.1 in the next section.

2 Lower Bound for the Blow-Up Time

In this section,using a delicate application of general Sobolev inequality,we will establish the lower bound for the blow-up time of the solution to(1.1).

Proof of Theorem 1.1.define