Blow up and global existence of solutions for multidimensional nonlinear diffusion equations coupled

Mathematical Computation June 2014, Volume 3, Issue 2, PP.26-29

Blow-up and Global Existence of Solutions for Multidimensional Nonlinear Diffusion Equations Coupled by Nonlinear Boundary Sources Haihua Zhou 1, #, Shiqiu Liu 2 1. Jiangxi Normal University, Nanchang 330022, P .R. China 2. Nanyang Technological University, Singapore #Email: haihuazhou_jxnu@163.com.

Abstract This paper is concerned with the long time behavior of solutions to a prototype of nonlinear diffusion equations coupled by the nonlinear boundary sources on the exterior domain of the unit ball in R N . It is shown that there exist both blow-up solutions with large initial data and global solutions with small initial data for the problem considered.

Keywords: Global Existence; Blow up; Exterior Domain

1 INTRODUCTION In this paper, we study a prototypes of nonlinear diffusion equations on the exterior domain of the unit ball in R N , i.e.,

ut  div(| u | p 2 u),

vt  div(| v |q 2 v),

| u | p 2 u   v ( x, t ),

| v |q 2 v   u  ( x, t ),

u( x,0)  u0 ( x),

x  R N \ B1 (0), t  0,

(1.1)

x B1 (0), t  0,

(1.2)

x  R N \ B1 (0),

v( x,0)  v0 ( x),

(1.3)

where p, q  2, ,   0, N  2, B1 (0) is the unit ball in R N with boundary B1 (0), is the inward normal vector on B1 (0) , and u0 ( x), v0 ( x) are nonnegative, suitably smooth and bounded functions with compact supports.

As well known that the equations in (1.1) are Non-Newtionian filtration equations, they degenerate at the points where u  0. As a prototype of nonlinear diffusion equations, the local existence of solutions to these equations have been studied, see [1,6,9] and the references therein. In this paper we mainly investigate the large time behavior of solutions, such as the global existence in time and blow-up in a finite time. For the problem of linear diffusion or single equation, the study on the large time behavior of solutions has been widely developed, see the papers [2,3,4,5,7,8] and the reference therein.

In this paper, we prove that there exist both blow-up solutions and global solutions for the problem (1.1)-(1.3).

Furthermore, by virtue of the radial symmetry of the exterior domain of the unit ball, we can extend our result to the following more general equations  (| x | u )  div(| x | | u | p 2 u), t 1

1

 (| x | v)  div(| x | | v |q 2 v), t 2

with 1  p  N , 2  q  N , N  1.

We will give our main results and their proofs in the next section.

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2

x  R N \ B1 (0), t  0

(1.4)

2 MAIN RESULTS AND PROOFS In this section, we first introduce our results, then we give the proofs. Our main results are as follows.

Theorem 2.1 If   ( p  1)(q  1), N  max { p, q}, then the solutions of the problem (1.1)-(1.3) with large initial data blow up in a finite time and the solutions with small initial data exist global in time.

Theorem 2.2 Assume 1  p  N , 2  q  N , N  1. If   ( p  1)(q  1), then there exist both blow-up and global existence solutions for the problem (1.4), (1.2) and (1.3).

Before we give the proofs of Theorem 2.1 and Theorem 2.2 , we consider the radial problem

u  u p 2 u 1 u p 2 u  (| | )  (| | ), r  1, t  0, t r r r r r r v  v q 2 v 2 v p 2 v  (| | )  (| | ), r  1, t  0, t r r r r r r u p 2 u v v t  0, | | (1, t )  v (1, t ),  | | p 2 (1, t )  u  (1, t ), r r r r r 1 u(r,0)  u0 (r ), v(r ,0)  v0 (r ),

(2.1) (2.2) (2.3) (2.4)

Where r | x |, p, q  2, ,   0, N  1, 1  p  1, 2  q  1, and u0 (r ), v0 (r ) are nonnegative, nontrivial functions with compact supports. The solution (u, v) of the system (2.1)-(2.4) with 1  2  N  1 is also the solution of the system (1.1)-(1.3) if u0 ( x), v0 ( x) are radially symmetrical.

For the system (2.1)-(2.4), we have the following results.

Proposition 2.1 If   ( p  1)(q  1), then the nonnegative solution of the system (2.1)-(2.4) with large initial data blows up in a finite time.

Proposition 2.2 If   ( p  1)(q  1), then every nonnegative nontrivial solution of the problem (2.1)-(2.4) with small initial data exists globally.

Now, we prove Proposition 2.1, Proposition 2.2.

Proof of Proposition 2.1 The proposition is proved by constructing a kind of blow-up lower solutions of the system (2.1)-(2.4). For r  1,0  t  T , let

u(r, t )  (T  t )k g1 ( ),



v(r, t )  (T  t )k g2 ( ),



1

2

r 1 (T  t )l r 1

,

(2.5)

,

(2.6)

1

(T  t )l

2

Where k1 , k2 , l1 , l2 are nonnegative constants satisfying k1  1  ( p  1)k1  pl1 ,

( p  1)(k1  l1 )   k2 ,

(2.7)

k2  1  (q  1)k2  ql2 ,

(q  1)(k2  l2 )   k1 ,

(2.8)

and

g1 ( )  A1 (a1   )( p 1)/( p 2) ,

g2 ( )  A2 (a2   )(q 1)/( q 2) ,

with a1 , a2 , A1 , A2 to be determined. Then the functions (u, v) is the lower solution of the system (2.1)-(2.4) with u0 (r )  u(r ,0), v0 (r )  v(r ,0), if k1 g1 ( )  l1 g1 ( )  (| g1 | p 2 g1 )( ) 

1 (T  t )l

k2 g2 ( )  l2 g2 ( )  (| g2 |q 2 g2 )( ) 

 | g1 | p 2 g1 (0)  g2 (0),

1

r

| g1 | p 2 g1 ( ),

2 (T  t )l

 | g2 |q 2

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2

| g2 |q 2 g2 ( ),

r  g2 (0)  g1 (0).

(2.9) (2.10) (2.11)

The fact that g1 ( ), g2 ( )  0 permits us to get the valid of (2.9) , (2.10) by choosing

a1  min{

2( p  1) 2( p  1) p , }, 1 ( p  2) k1 ( p  2) p

a2  min{

2(q  1) 2(q  1)q , } 2 (q  2) k2 (q  2)q

and A1  1, A2  1.

On the boundary, for enough large A1  1, let A2  A1s , here the positive constant s satisfies

( p  1) /   s   / (q  1). The existence of s can be obtained from the condition   ( p  1)(q  1).

Thus, we have

p  1 p 1 ( p 1)/( p 2) ) a1  A2 a2 ( q 1)/( q 2) , p2 q  1 q 1 ( q 1)/( q 2) A2q 1 ( ) a2  A1 a1 ( p 1)/( p 2) , q2

A1p 1 (

which implies (2.11) hold..

Therefore, the solution (u, v) of the system (2.1)-(2.4) blows up in a finite time if (u0 , v0 ) is large enough.

The proof is completed.

Proof of Proposition 2.2. We take the stationary solution as the global sup-solution of the system (2.1)-(2.4) to prove this proposition. Denote v( x, t )  A2 r1

u( x, t )  A1r1 /( p 1) , 1

2

/( q 1)

,

and

A1  [( A2  [(

1 p 1

1 p 1

 1) p 1 (  1) (

2 q 1

2 q 1

 1) ]( q 1)/[ ( p 1)( q 1)] ,

 1)q 1 ]( p 1)/[ ( p 1)( q 1)] .

By some calculations, we get

u  u p 2 u 1 u p 2 u  (| | )  (| | ), r  1, t  0, t r r r r r r  v  v q 2 v 2 v q 2 v u p  2 u  (| | )  (| | ), r  1, t  0, | | (1, t )  v (1, t ), t r r r r r r r r  u q  2 u | | (1, t )  u (1, t ), t  0. r r That is to say that (u, v) is a stationary solution of the problem (2.1)-(2.4). If the initial data satisfy

u0 ( x)  u( x,0), v0 ( x)  v( x,0), r  1, then the solution of the system (2.1)-(2.4) exists globally because of the global existence of (u, v) comparison principle.

by the

Now, we prove the main result for the system (1.1)-(1.3), i.e., Theorem 2.1.

Proof of Theorem 2.1 For the large and radially symmetric functions u(| x |,0), v(| x |,0) defined in the proof of Proposition 2.1, if (u0 , v0 ) is large enough such that u0 ( x)  u(| x |,0), v0 ( x)  v(| x |,0), then the solutions of the system (1.1)-(1.3) with   ( p  1)(q  1) blow up by the comparison principle and Proposition 2.1.

On the other hand, using the comparison principle again, we conclude that if

u0 ( x)  A1 | x |( p  N )/( p 1) ,

v0 ( x)  A2 | x |(q  N )/( q 1) ,

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x  R N \ B1 (0),

(2.12)

N  p p 1 N  q  ( q 1)/[ ( p 1)( q 1)] ) ( ) ] , p 1 q 1 N  p  N  q q 1 ( p 1)/[ ( p 1)( q 1)] A2  [( ) ( ) ] , p 1 q 1 A1  [(

then the solution (u, v) of (1.1)-(1.3) exists globally for   ( p  1)(q  1) by Proposition 2.2. This combined with Proposition 2.1 indicate the valid of Theorem 2.1.

Proof of Theorem 2.1. We take 1  1  N  1, 2  2  N  1 in the system (2.1)-(2.4), and replace (2.12) with

u0 ( x)  A1 | x |(  N  p)/( p 1) , 1

v0 ( x)  A2 | x |(  N q)/( q 1) , 2

where

  N  q  ( q 1)/[ ( p 1)( q 1)] ) p 1 ( 2 ) ] , p 1 q 1   N  p  2  N  q q 1 ( p 1)/[ ( p 1)( q 1)] A2  [( 2 ) ( ) ] . p 1 q 1 A1  [(

1  N  p

Then the proof can be finished by virtue of the same discussion in the proof of Theorem 2.1.

ACKNOWLEDGMENT This paper is supported by the National Natural Science Foundation of China (No. 11361029) and Scientific Research Fund of Jiangxi Provincial Education Department (NO.GJJ14270).

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AUTHORS 1Haihua

Zhou, (1991- ). She graduated

2Shiqiu

Liu, she graduated from College of Mathematics, Jilin

from Department of Mathematics and

University in 2013 and now she is in Division of Mathematical

Computational

Sciences, School of Physical and Mathematical Sciences

Science,

Heng-

yang

Normal University in 2012 and now she is

Nanyang Technological University, Singapore for a doctorate.

studying in College of Mathematical and Information

Science,

Jiangxi

Normal

University for her Master's degree. Email: haihuazhou_jxnu@163.com - 29 www.ivypub.org/mc