Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

XUE Yingzhen

School of Business,Xi’an International University,Xi’an 710077,China

Abstract. In this paper, we establish the lower bounds estimate of the blow up time for solutions to the nonlocal cross-coupled porous medium equations with nonlocal source terms under Dirichlet and Neumann boundary conditions. The results are obtained by using some differential inequality technique.

Key Words:Lower bounds;Blow up time;Nonlocal source terms;Dirichlet and Neumann boundary conditions.

1 Introduction

In this article,we consider the lower bound of blow up time for solutions of the nonlocal cross-coupled porous medium equations

and continuous bounded initial values

under Dirichlet boundary condition

or Neumann boundary condition

where Ω ∈R3is a bounded region of∂Ω with a smooth boundary, and satisfies thatp>m>1,q>n>1,υis the unit external normal vector in the external normal direction of∂Ω. There are many research achievements on the lower bound estimation of blow up time for the solution of a single porous media equation,see,e.g.,[1-3]. Liu,et al. [1]studied the following nonlocal porous equation with Dirichlet boundary conditions

They have obtained the lower bound of the blow up time of the solution which

and homogeneous Neumann boundary conditions,the lower bound of the blow up time of the solution which

Liu[2]considered Eq. (1.6)with Robin boundary conditions,they have obtained the lower bound of the blow up time of the solution which

Fang and Chai[3]studied Eq. (1.6)with Neumann boundary conditions

the lower bound of the blow up time of the solution which

whenl>0 of(1.5). The lower bound of the blow up time of the solution which

whenl<0 of Eq.(1.5).

To sum up, most of the existing results focus on a single equation,However,studies on the lower bound of blow up time for the equation set(1.1)-(1.5)have not been found.Blow-up at a finite time and lower bound of blowing up time of solution for the parabolic equations are studied in[4,5]. The lower bound of blowing up time for solutions of other similar equations or equation set is shown in[6-9].

Inspired by[4-5], this paper studies the lower bound estimation of blowing up time for the solutions of the porous media equation set(1.1)-(1.5)with non-local source crosscoupling andm>1,n>1 with relevant formulas and some basic inequalities in[9].

2 The some inequalities

This part introduces some important inequalities used in this paper.

Lemma 2.1.(Membrane inequality)

where λ is the first eigenvalue of∆ω+λω=0,ω>0,x∈Ω,and ω=0,x∈∂Ω.

Lemma 2.2.([9])LetΩbe the bounded star region in RN,and N≥2. Then

Lemma 2.3.(Special Young inequality)Let γ be an arbitrary constant,and0

3 Lower bound of blow up time under Dirichlet boundary conditions

The lower bound of blow up time for solutions of equations under Dirichlet boundary conditions is discussed below.

Theorem 3.1.Defines auxiliary functions

for s>max{1,p,q,m−1,n−1,3p−2,3q−2}.If(u,v)is a non-negative classical solution of equation set(1.1)-(1.5)and blow up occurs in the sense of measure J(t)at time t⋆,then the lower bound of t⋆is

where,The normal number K1,K2,β1is given in the following proof.

Proof.Note that

When the equations takeDirichletboundary conditions of Eq.(1.4),Eq.(3.2)becomes

where

First,Hölder inequality is used to estimate the second term ofin Eq. (3.3),and it is obtained that

according to Young inequality and equation above,it is obtained that

Second,Hölder inequality is used to estimate the first term on the right side of Eq. (3.4),and it is obtained that

from the second term on the right side of inequality sign of(3.5)and Hölder inequality,we can know

from the first term on the right side of inequality sign of(3.5)and Hölder inequality,we can know

using the following Sobolev inequality([10]):

where,the second term of Eq.(3.7)can be simplified to

by synthesizing Eqs.(3.7)and(3.8),we have

based on Lemma 2.1,Eq.(3.9)becomes

Combining Eqs.(3.6)and(3.10),Eq.(3.5)becomes

where

By synthesizing Eqs.(3.4)-(3.11),of Eq.(3.3)becomes

Using the fundamental inequality

Eq.(3.12)becomes

The same derivation method is used to estimate theJ′

2(t)term in Eq. (3.3):

where

In order to deal with the gradient terms in Eqs. (3.13)and(3.14),we setandFinally,by synthesizing Eqs.(3.13)and(3.14),we obtain

Integrating(3.16)from 0 tot⋆,we obtain

This completes the proof of the theorem.

4 Lower bound of blow up time under Neumann boundary conditions

The lower bound of blow up time for solutions of equations under the Neumann boundary conditions is discussed below.

4.1 The case of l>0

Theorem 4.1.Define the same measure as(3.1)and the same condition as s. If(u,v)is a nonnegative classical solution to the equation set(1.1)-(1.2)with(1.3)and(1.5),then the lower bound of t⋆is

where,the normal number K1,K2,K3,K4,β1,β2is given in the followingproof.

Proof.Lemma 2.2 is used to estimate two boundary terms in Eq. (3.2),then

where

From the second term on the right hand side of Eq.(4.1)and using Hölder inequality and Lemma 2.3,we can know

wherer1is an arbitrary constant.

The Hölder inequality is used to estimate the second term on the right hand side of Eq.(4.2). We then have

Substituting(4.2)and(4.3)into Eq.(4.1),we get

where

Similarly,another boundary term in Eq.(3.2)is estimated as follows

where,r2is an arbitrary constant.

Substituting(3.16),(4.4)and(4.5)into Eq. (3.2),we get

where

The Eq.(4.6)becomes

Integrating(4.7)from 0 tot⋆,we obtain

4.2 The case of l ≤0

Ifl≤0,then,according to Eq.(3.2),we obtain

That is, the same measure relation is obtained with Eq. (3.3). Therefore, whenl≤0,the lower bound of blow up time of the equation set (1.1)-(1.2) with (1.3) and (1.5) is consistent with that of Eq. (3.17).

Acknowledgments

This work is supported by Natural Science Basic Research Project of Shaanxi Province(2019JM-534);Soft Science Project of Shaanxi Province(2019KRM169);Planned Projects of the 13th Five-year Plan for Education Science of Shaanxi Province(SGH18H544);Project on Higher Education Teaching Reform of Xi’an International University (2019B36), and the Youth Innovation Team of Shaanxi Universities. The author would like to deeply thank all the reviewers for their insightful and constructive comments.