Blow-up in the Parabolic Problems under Nonlinear Boundary ...
JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
733
Blow-up in the Parabolic Problems under Nonlinear Boundary Conditions Jin Li School of Mathematics and Statistic, Hexi University, Zhangye, Gansu 734000, PR China Email: [email protected]
Abstract—In this paper, I consider nonlinear parabolic problems under nonlinear boundary conditions. I establish respectively the conditions on nonlinearities to guarantee that u ( x, t ) exists globally or blows up at some finite time. If blow-up occurs, an upper bound for the blow-up time is derived, under somewhat more restrictive conditions, lower bounds for the blow-up time are also derived. Index Terms—Nonlinear Boundary Conditions; Blow-Up; Heat Equation
I.
INTRODUCTION
In this paper, we investigate the blow-up phenomenon of the classical solution of the following initial-boundary value problem
ut
N
(a
i , j 1
N
a
i , j 1
ij
ij
( x)uxi ) x j f (u ), ( x, t ) (0, t ) (1)
( x)uxi n j g (u ), ( x, t ) (0, t ),
(2)
u( x,0) u0 ( x) 0,
(3)
x ,
where n is the unit outward normal on the boundary assumed sufficiently smooth, is a bounded starshaped region in R N , N 2 , t is the blow-up time if blow-up occurs, or else t , and (aij ( x)) N N is a differentiable positive definite matrix. It is well known that the data f and g may greatly affect the behavior of
u ( x, t ) with the development of time. From the physical standpoint, f is the heat source function, g is the heatconduction function transmitting into interior of from the boundary of . The study of the blow-up phenomena in parabolic problems has received a great deal of attention in the last decades (we refer the reader especially to the books of Straughan [1] and Quittner-Souplet [2], the survey papers of Levine [3] and Galaktionov [4] and the references therein). Therefore, nowadays a variety of methods are known and used in the study of various questions regarding the blow-up phenomena in parabolic problems. But, most of the methods used to show that solutions blow-up provide only an upper bound for the blow-up time, while in applications, due to the explosive nature of © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.3.733-738
the solutions, it is more important to determine the lower bounds on the blow-up time. To our knowledge, there seems to have been relatively little work devoted to obtaining lower bounds on blow-up time if blow-up occurs. In [5], Payne and Schaefer used a differential inequality technique to obtain a lower bound on blow-up time for solutions of the semilinear heat equation
ut u f (u)
(4)
under homogeneous Dirichlet boundary conditions, where suitable constraints were imposed on $f$ which allowed, for instance, and f (u) u p , p 1, f (u) 2cosh( u 1), 0. A second method based on a comparison principle was also presented there. They also consider the initial-boundary value problem for the semilinear heat equation (4) under a Robin boundary condition [6]. Payne and Schaefer [7] considered
ut u.
(5)
Under suitable conditions on the nonlinearities, they determined a lower bound of the blow-up time when blow-up occurs. In addition, a sufficient condition which implies that blow-up does occur was determined. In [8] Payne, Philippin and Vernier Piro considered
ut u f (u)
(6)
and established conditions on nonlinearities to guarantee that u ( x, t ) exists for all time t 0 or blows up at some finite time t . Moreover, an upper bound for t was derived. Under somewhat more restrictive conditions, a lower bound for t was derived. Recently, Philippin and Vernier [9] investigated
ut ·(| u |2 p u), ( x, t ) (0, t )
(7)
and showed that blow-up occurs at some finite time under certain conditions on the nonlinearities and the data, upper and lower bounds for the blow-up time were obtained when blow-up occurs. Further extensions were accomplished ([10-20] and the references therein).
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JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
Motivated by the above work, we intend to study the global existence and the blow-up phenomena for problem (1)-(3). The main contribution of this paper are: (a) the problems considered in this paper are nonlinear equations with inhomogeneous Neumann boundary dissipations, these problems possess representative; (b) we give the reason and process of the definition of auxiliary functional; (c) since the models are general, the estimates are concise and precise. The present work is organized as follows. In Section 2, I show the conditions on the nonlinearities which ensure that the solution blows up at some finite time and obtain an upper bound of the blow-up time. Section 3 is devoted to showing a lower bound of blow-up time under some assumptions. In Section 4, I establish the conditions on the nonlinearities to guarantee that u ( x, t ) exists globally.In the final section, I briefly show how to obtain an analogous result for a domain R2 and how to extend this work to systems of equations. II.
A CRITERION FOR BLOW-UP
In this section, we establish conditions on the data of problem (1)-(3) under which u ( x, t ) will blow up at finite
and
(t ) 2 ut g (u ) dS 2 ut f (u ) dx
N N a ij ( x)u xi u x j dx 2 a ij ( x)u xi u x j t dx i , j 1 i , j 1 t
2 ut g (u ) dS 2 ut f (u ) dx 2 ut g (u ) dS
2 ut f (u )dx 2 ut (a ij ( x)u xi ) x j dx
i , j 1
N
2 ut a ij ( x)u xi n j dS 2 u dx
2
i , j 1
f ( ) 2(1 ) F ( ), 0,
(8)
g ( ) 2(1 )G( ), 0,
(9)
0
0
(1 ) 0
integrating (12) over [0, t ] , and noting (0) 0
(t )(1 ) (t ) (0)(1 ) (0) : M 0, that is
Moreover we assume (0) 0 with N
a
ij
i , j 1
( ) 1 (t )(t ) 2M (1 ). (15) Noting (8), (t ) 0 and (0) 0 , we deduce
(t ) 0. From (15) and (16), it follows that
Proof of Theorem 2.1. Using Green formula and the hypotheses stated in Theorem 2.1, we get N
(t ) 2 uut dx 2 u[ (a ij ( x)u xi ) x j f (u )]dx
1 , (0) 2M (1 )
(18)
N
a
i , j 1
ij
( x )u xi u x j dx
which implies (t ) as
((0)) (0) 2 (1 ) M 2 (1 )(0) (by the definition of M ). Therefore for 0 , t T
i , j 1
2 ug (u )dS 2 u f (u )dx 2
(17)
(t )
2[2(1 ) G (u )dS
0 (t ) 2M (1 )t , that is
where (t ) u 2 dx . Where 0 , we have t .
(10)
N
a
i , j 1
ij
( x)u xi u x j dx
2(1 ) F (u )dx] 2(1 )(t ),
© 2014 ACADEMY PUBLISHER
(16)
(0) T ( 0), 2 (1 )(0)
(14)
If 0 , (14) can be written as
( x)uxi ux j dx 2 F (u )dx.
Then u ( x, t ) blows up at time t T with
(13)
By (8) and (13), we obtain
(t ) 2(1 )(t ) 2M (1 )1 (t ).
In (8), (9), and are constants satisfying the condition 0 .
(12)
(by (0) 0 ), we get
(t ) M 1 (t ).
F ( ) : f ( )d , g ( ) : g ( )d.
(t ) 2 G(u)dS
Multiplying the above inequality by 2 , we deduce
with
t
which with (0) 0 imply (t ) 0 for all t (0, t ) . Using (10), (11) and Holder inequality, we obtain 2 1 2 (t )(t ) ((t )) 2 uut dx 2(1 ) 1 2 2 1 u 2 dx u dx (t )(t ). t 1 1
time t and derive under these conditions an upper bound T for t . Theorem 2.1. Let u ( x, t ) be the solution of (1)-(3) and assume the following conditions on the data
(11)
N
t T
0) . 2 (1 )(0)
(19)
JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
735
If 0 , we have
(t ) (0)e2 Mt ,
(20)
Proof of Theorem 3.1. Since (aij ( x)) N N is positive definite matrix, then there exists a constant 0 such that N
valid for t 0 , implying that t . The proof of Theorem 2.1 is completed. III.
LOWER BOUND FOR BLOW-UP TIME
In this section, under the assumption that R3 is a convex bounded star-sharped domain in two orthogonal directions, we establish a lower bound for the blow-up time t . Now we state the result as follows. Theorem 3.1. Assumed that R3 is a bounded star-sharped convex domain in two orthogonal directions. Let u ( x, t ) be the nonnegative solution of problem (1)
and u ( x, t ) blows up at t , moreover the nonnegative f and g satisfy the conditions
1
0 g ( s) k2 s
2
, s 0,
for k1 0, k2 0, 1 . Define
(22)
Then (t ) satisfies inequality (t ) ( ) for some computable function ( ) . It follows that t is bounded below by d t . (0) ( ) In order to prove this theorem, we first give the following lemma. Lemma 3.1. Let be a bounded star-sharped region in R N , N 2 . Then for any nonnegative C 1 function u and r 0 , we have N rd r r r 1 u ds u dx u | u | dx,
0
0
div u r x Nu r ru r 1 ( x u)
over , and using divergence theorem, we get r r r 1 u ( x n)ds Nu dx ru ( x u)dx
i
i , j 1
j
N
f (u )]dx 2 (2 1) u 2 2 a ij ( x)u xi u x j dx
2 u
g (u )ds 2 u
2 1
i , j 1
2 1
f (u )dx
(23)
2 (2 1) u 2 2 | u |2 dx
5 2
where we have used successively the differential equation (1), the divergence theorem, the boundary condition (2) and the assumption (21). Next, Application of Lemma 2.1 leads to the inequality 5 5 5 1 3 5 d 2 2 2 u ds u dx u | u | dx. 0 2 0 Furthermore, since | u |2 2u 2 2 | u |2 , we replace (23) by
(t ) 5 d 2 0
2 (2 1)
u
5 1 2
|u |2 dx 2 k1[
3
0
5
u 2 dx (24)
| u | dx] 2 k2 u dx. 3
We now use the Schwarz inequality on the two integrals in the bracket in (12) and then the arithmeticgeometric mean inequality to arrive at
3 k1
0
u dx [ 2n
5kd 2 |u | dx 2 0
3 k1
(25)
5 2 k1d 2 k2 ] u 3 dx 2 0
0
for some positive which is to be determined. To bound the latter integral, we use an integral inequality which was derived in [18, (15)-(23)] and is restricted to N 3 dimensions, namely,
By the definitions of 0 and d , it follows that r
N u dx r u r
r 1
| x || u | dx
N u r dx rd u r 1 | u | dx,
which implies the desire conclusion.
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0 u ds u ( x n)ds
d : max |x|. Proof. Since is a bounded star-sharped region, we know 0 0 . Integrating the identity
r
N
(t ) 2 u 2 1ut dx 2 u 2 1[ (a ij ( x)u x ) x
2 (2 1)
0 : min (x n),
for a.e.. x and all R N . Differentiating (22) we obtain
(t )
where
( x)i j | |2
(21)
(t ) u 2 dx.
ij
2 k1 u ds 2 k2 u 3 dx
0 f ( s) k1 s 1 , s 0,
a
i , j 1
u 3 dx 3
d 1 0
3 4
[
3 2 0
u 2 dx
u dx |u 2
Using the inequalities
1 2
2
| dx
]. 1 2
3 2
(26)
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JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
(a b)
j 1 j
1 j
2 (a
j 1 j
a
j 1 j
non-increasing
)
(27)
a r bq ra qb, r q 1,
(28)
1
3 4
3
[
3
2 3 3 2 32 d 1 4 2 0 0
|u
2
| dx
3 2
3 3 3 1 3 d 1 3 3 4 21 0 2 2 2 2 3 4 1 [ 3 3 4 4 0
] (29) 3 4
|u |2 ]
3
d 2 3 k1 5 2 k1 f 3 2 1 0, 2 0 0 0 we have
3 2
al
u 0( gm R ) , f and h are non-negative functions, with f (0) 0 , t is the blow-up time if blow-up occurs,
u 2u , u,ij xi xi x j
will be used throughout this work, and summation from 1 to N is understood for repeated indices. Theorem 4.1. Assume that the nonnegative functions f and g satisfy
s 1 0 h( s ) k1 d , s 0, 0 g ( ) p
s 1 f ( s) k2 g ( s) d , s 0 0 g ( ) for k1 0, k2 0, p 1, q 1, and 2q p 1 Then the (nonnegative) solution u ( x, t ) of problem (31)-(33) does not blow up, so that u ( x, t ) exists for all time t 0 . Proof of Theorem 4.1. Set
3
(t ) C1 C2 2 C3 3 ,
(t ) v 2 (u ( x, t ))dx,
where
C1
for
q
for some positive that is to be determined. Collecting terms in (25) and (29) and choosing and so that 2 (2 1) 5k1d 2 0 1 4
g (u) gm
with
or else t . The notations u,i :
for positive a and positive b , it follows that u 3 dx 2 2 3
function,
3 k1
v( s) :
0
0
3 3 3 k 5 2 k1d 1 C2 3 4 21 0 2 2 0 0 1 3 3 k 3 5 2 k1d d 1 C3 2 2 3 4 . 1 2 0 0 0 4 Under the assumption that . blows up at finite time
t , we obtain a lower bound for t by integration of (30), i.e., d t . 3 0 3 2 C1 C2 C3 The proof of Theorem 3.1 is completed.
(t ) 2
2 v
v ( g (u )u,i )i f (u) dx g (u )
| u |2 g v | u |2 f (u )v u ds 2 dx n g (u )
(31)
u h(u ), ( x, t ) (0, t ) n
(32)
where we have used successively the differential equation (31), the divergence theorem, the fact that g 0 and the hypotheses stated in Theorem 31. Application of Lemma 3.1 leads to the inequality
In this section, we establish the conditions on the nonlinearities of the following initial-boundary value problem to guarantee that u ( x, t ) exists globally.
ut ( g (u)u,i ),i f (u), ( x, t ) (0, t )
(q 1)d
0
0
v q 1dx (36)
v q | v | dx
Inserting (36) into (35), we get
(t )
2k1 N
0
v q 1dx
2(q 1)k1d
2 g m |v | dx 2k2 v
(33)
u where is the outward normal derivative of u on n the boundary assumed sufficiently smooth, is a bounded star-shaped region in R N , N 2, g is a positive
N
v q 1ds
2
u( x,0) u0 ( x) 0, x ,
(35)
2k1 v q 1ds 2 g m |v |2 dx 2k2 v p 1dx,
CRITERION FOR GLOBAL EXISTENCE
© 2014 ACADEMY PUBLISHER
(34)
1 d. g ( )
Differentiating (34), we obtain 3 2
IV.
s
Choosing
0 p 1
dx.
k1d (q 1) , we have 2 0 g m
v q | v | dx (37)
JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
737
1 |v |2 dx 2 2 (38) 0 g m k d (q 1) 2q 2 1 v dx | v | dx . 4 0 g m k1d (q 1)
v q | v | dx
v 2 q dx
Inserting (38) into (37), we have
(t )
2k1 N
0
v
q 1
dx 2 g m
2
v 2k2 v 2q
p 1
V.
Concluding remarks A result analogous to theorem 3.1 can be obtained for a bounded domain R2 by first using Schwarz's inequality to write
dx.
(39)
p 1 2q 1 . pq Using Holder inequality and Young inequality, we obtain In view of 2q p 1, we see
CONCLUDING REMARKS
u 3 dx
u 2 dx
1 2
u 4 dx
1 2
,
then deriving an integral inequality similar to (14) for 4 u dx
by the method used in going from (15) to (23) in [18]. In this manner, we obtain a first-order differential inequality of the form 3
v
v
2q
v
( q 1)( p 1) ( q 1 2 q )( q 1) p q p q
q 1
v
dx
p 1
dx
1
dx
1 v q 1dx v p 1dx
(1 ) v p 1
(t ) K1 K2 2 K3 2
1
(40)
1
v q 1dx.
(u )t
Combining (39) with (38), we have
(t ) L1 vq 1dx L2 v p 1dx ,
2 Nk1
2 g m 2
0
0,
v p 1dx
pq
q 1 p 1
| | p 1 .
(42)
Inserting (41) into (42), we have
(t ) L1
v p 1dx
q 1 p 1
[| |
pq p 1
L2 L1
v p 1dx
p q p 1
].
(43)
(t ) v 2 dx
v
p 1
dx
q 1 p 1
(| |
pq p 1
v p 1dx
2 p 1
(44)
L2 || L1
( p q )(1 p ) 2( p 1)
p q 2
).
2
(t ) L2q p | | q 1
(48)
ij
i , j 1
( x)(u ) xi n j g (u )
(49)
(50)
where 1, 2,..., n, u (u1 ,..., un ) and (aij ( x)) N N is differentiable positive definite matrixes. We shall use comma i notation to denote partial differentiation with respect to xi and the summation convention on repeated indices, where the Greek indices sum over 1, 2,..., n . We impose the following constraint on the nonlinear terms and the boundary data in (31) 1
2
,
p 1 p 1
for positive constants B1 , B2 and 1 , and assume the existence of solutions u , 1, 2,..., n, of the system (31), one or more of which becomes unbounded in finite time t . ACKNOWLEDGMENT
(46)
so that (t ) remains bounded for all time under the conditions stated in Theorem 4.1. This completes the proof of Theorem 4.1. © 2014 ACADEMY PUBLISHER
) x j f (u )
(45)
We conclude from (45) that (t ) is decreasing in each time interval on which we have 2
xi
u f (u ) B2 (u u )1
It follows from (43) and (44) that (t ) L1
N
u g (u ) B1 (u u )
Using Hoder inequality, we have
i , j 1
u ( x,0) u 0 ( x) 0
L2 2k2 2 g m 2 (1 ) for 0 small enough. v q 1dx
ij
a
with
L1
N
(a ( x)(u )
(41)
1
for computable constants K1 , K 2 and K 3 , from which a lower bound on the blow-up time follows. The results in this paper can also be extended to the following initial-boundary value problem for the nonlinear system of $n$ parabolic equations which are coupled through the nonlinear terms and the boundary data
The author is grateful to the referees for their valuable suggestions and comments on the original manuscript. REFERENCES [1] B. Straughan, Explosive instabilities in (Springer-Verlag, Berlin, 1998).
mechanics
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[2] R. Quittner and P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states (Birkhauser, Basel, 2007). [3] H. A. Levine, Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics. The method of unbounded Fourier coefficients, Math. Ann. 214 (1975), 205-220. [4] V. A. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Cont. Dyn. Syst. 8(2) (2002), 399-433. [5] L. E. Payne, P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl. 328 (2007) 1196-1205. [6] L. E. Payne, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Appl. Anal. 87 (2008) 699707. [7] L. E. Payne, P. W. Schaefer, Bounds for the blow-up time for heat equation under nonlinear boundary condition, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009) 1289-1296. [8] L. E. Payne, G. A. Philippin, S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I, Z. Angew. Math. Phys. 61 (2010) 999-1007. [9] L. E. Payne, G. A. Philippin, S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II, Nonlinear Anal. 73(2010) 971-978. [10] L. E. Payne, G. A. Philippin, P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal. 69 (2008) 3495-3502. [11] L. E. Payne, G. A. Philippin, P. W. Schaefer, Bounds for blow-up time in nonlinear parabolic problems, J. Math. Anal. Appl. 338 (2008) 438-447 [12] L. E. Payne, P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl. 328 (2007) 1196-1205.
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JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
[13] C. Enache, Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary conditions, Appl. Math. Lett. 24 (2011) 288-292. [14] E. Enache, Lower bounds for blow-up time in some nonlinear parabolic problems under Neumann boundary conditions, Glasgow Math. J. Available CJ0201 DOI: 10.1017/50017089511000139. [15] L. E. Payne, G. A. Philippin, Blow-up in a class of nonlinear parabolic problems with time-dependent coefficients under Robin type boundary conditions, Appl. Anal. DOI: 10.1080/00036811.2011.598865. [16] L. E. Payne, J. C. Song, Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl. 354, 394-396 (2009) [17] L. E. Payne, P. W. Schaefer, Blow- up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math. 48, 193-202 (2008) [18] L. E. Payne and P. W. Schaefer. Lower bounds for blowup time in parabolic problems under Neumann conditions. Appl. Anal. 85 (2006), 1301-1311 [19] W. Yan and W Xiao, A new fuzzy SVM based on the posterior probalility weighting menbership, J. Computers, 7(2012) 1385-1393. [20] X. Shi and D. Feng, LSP: A locality –aware strip prefetching scheme for striped disk array systems with concurrent accesses. J. Cumptuters, 7(2012) 1303-1312.
Jin Li was born in Gansu on January, 1957. Associate professor. The main research direction is Nonlinear partial differential equations and Applied analysis.
733
Blow-up in the Parabolic Problems under Nonlinear Boundary Conditions Jin Li School of Mathematics and Statistic, Hexi University, Zhangye, Gansu 734000, PR China Email: [email protected]
Abstract—In this paper, I consider nonlinear parabolic problems under nonlinear boundary conditions. I establish respectively the conditions on nonlinearities to guarantee that u ( x, t ) exists globally or blows up at some finite time. If blow-up occurs, an upper bound for the blow-up time is derived, under somewhat more restrictive conditions, lower bounds for the blow-up time are also derived. Index Terms—Nonlinear Boundary Conditions; Blow-Up; Heat Equation
I.
INTRODUCTION
In this paper, we investigate the blow-up phenomenon of the classical solution of the following initial-boundary value problem
ut
N
(a
i , j 1
N
a
i , j 1
ij
ij
( x)uxi ) x j f (u ), ( x, t ) (0, t ) (1)
( x)uxi n j g (u ), ( x, t ) (0, t ),
(2)
u( x,0) u0 ( x) 0,
(3)
x ,
where n is the unit outward normal on the boundary assumed sufficiently smooth, is a bounded starshaped region in R N , N 2 , t is the blow-up time if blow-up occurs, or else t , and (aij ( x)) N N is a differentiable positive definite matrix. It is well known that the data f and g may greatly affect the behavior of
u ( x, t ) with the development of time. From the physical standpoint, f is the heat source function, g is the heatconduction function transmitting into interior of from the boundary of . The study of the blow-up phenomena in parabolic problems has received a great deal of attention in the last decades (we refer the reader especially to the books of Straughan [1] and Quittner-Souplet [2], the survey papers of Levine [3] and Galaktionov [4] and the references therein). Therefore, nowadays a variety of methods are known and used in the study of various questions regarding the blow-up phenomena in parabolic problems. But, most of the methods used to show that solutions blow-up provide only an upper bound for the blow-up time, while in applications, due to the explosive nature of © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.3.733-738
the solutions, it is more important to determine the lower bounds on the blow-up time. To our knowledge, there seems to have been relatively little work devoted to obtaining lower bounds on blow-up time if blow-up occurs. In [5], Payne and Schaefer used a differential inequality technique to obtain a lower bound on blow-up time for solutions of the semilinear heat equation
ut u f (u)
(4)
under homogeneous Dirichlet boundary conditions, where suitable constraints were imposed on $f$ which allowed, for instance, and f (u) u p , p 1, f (u) 2cosh( u 1), 0. A second method based on a comparison principle was also presented there. They also consider the initial-boundary value problem for the semilinear heat equation (4) under a Robin boundary condition [6]. Payne and Schaefer [7] considered
ut u.
(5)
Under suitable conditions on the nonlinearities, they determined a lower bound of the blow-up time when blow-up occurs. In addition, a sufficient condition which implies that blow-up does occur was determined. In [8] Payne, Philippin and Vernier Piro considered
ut u f (u)
(6)
and established conditions on nonlinearities to guarantee that u ( x, t ) exists for all time t 0 or blows up at some finite time t . Moreover, an upper bound for t was derived. Under somewhat more restrictive conditions, a lower bound for t was derived. Recently, Philippin and Vernier [9] investigated
ut ·(| u |2 p u), ( x, t ) (0, t )
(7)
and showed that blow-up occurs at some finite time under certain conditions on the nonlinearities and the data, upper and lower bounds for the blow-up time were obtained when blow-up occurs. Further extensions were accomplished ([10-20] and the references therein).
734
JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
Motivated by the above work, we intend to study the global existence and the blow-up phenomena for problem (1)-(3). The main contribution of this paper are: (a) the problems considered in this paper are nonlinear equations with inhomogeneous Neumann boundary dissipations, these problems possess representative; (b) we give the reason and process of the definition of auxiliary functional; (c) since the models are general, the estimates are concise and precise. The present work is organized as follows. In Section 2, I show the conditions on the nonlinearities which ensure that the solution blows up at some finite time and obtain an upper bound of the blow-up time. Section 3 is devoted to showing a lower bound of blow-up time under some assumptions. In Section 4, I establish the conditions on the nonlinearities to guarantee that u ( x, t ) exists globally.In the final section, I briefly show how to obtain an analogous result for a domain R2 and how to extend this work to systems of equations. II.
A CRITERION FOR BLOW-UP
In this section, we establish conditions on the data of problem (1)-(3) under which u ( x, t ) will blow up at finite
and
(t ) 2 ut g (u ) dS 2 ut f (u ) dx
N N a ij ( x)u xi u x j dx 2 a ij ( x)u xi u x j t dx i , j 1 i , j 1 t
2 ut g (u ) dS 2 ut f (u ) dx 2 ut g (u ) dS
2 ut f (u )dx 2 ut (a ij ( x)u xi ) x j dx
i , j 1
N
2 ut a ij ( x)u xi n j dS 2 u dx
2
i , j 1
f ( ) 2(1 ) F ( ), 0,
(8)
g ( ) 2(1 )G( ), 0,
(9)
0
0
(1 ) 0
integrating (12) over [0, t ] , and noting (0) 0
(t )(1 ) (t ) (0)(1 ) (0) : M 0, that is
Moreover we assume (0) 0 with N
a
ij
i , j 1
( ) 1 (t )(t ) 2M (1 ). (15) Noting (8), (t ) 0 and (0) 0 , we deduce
(t ) 0. From (15) and (16), it follows that
Proof of Theorem 2.1. Using Green formula and the hypotheses stated in Theorem 2.1, we get N
(t ) 2 uut dx 2 u[ (a ij ( x)u xi ) x j f (u )]dx
1 , (0) 2M (1 )
(18)
N
a
i , j 1
ij
( x )u xi u x j dx
which implies (t ) as
((0)) (0) 2 (1 ) M 2 (1 )(0) (by the definition of M ). Therefore for 0 , t T
i , j 1
2 ug (u )dS 2 u f (u )dx 2
(17)
(t )
2[2(1 ) G (u )dS
0 (t ) 2M (1 )t , that is
where (t ) u 2 dx . Where 0 , we have t .
(10)
N
a
i , j 1
ij
( x)u xi u x j dx
2(1 ) F (u )dx] 2(1 )(t ),
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(16)
(0) T ( 0), 2 (1 )(0)
(14)
If 0 , (14) can be written as
( x)uxi ux j dx 2 F (u )dx.
Then u ( x, t ) blows up at time t T with
(13)
By (8) and (13), we obtain
(t ) 2(1 )(t ) 2M (1 )1 (t ).
In (8), (9), and are constants satisfying the condition 0 .
(12)
(by (0) 0 ), we get
(t ) M 1 (t ).
F ( ) : f ( )d , g ( ) : g ( )d.
(t ) 2 G(u)dS
Multiplying the above inequality by 2 , we deduce
with
t
which with (0) 0 imply (t ) 0 for all t (0, t ) . Using (10), (11) and Holder inequality, we obtain 2 1 2 (t )(t ) ((t )) 2 uut dx 2(1 ) 1 2 2 1 u 2 dx u dx (t )(t ). t 1 1
time t and derive under these conditions an upper bound T for t . Theorem 2.1. Let u ( x, t ) be the solution of (1)-(3) and assume the following conditions on the data
(11)
N
t T
0) . 2 (1 )(0)
(19)
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If 0 , we have
(t ) (0)e2 Mt ,
(20)
Proof of Theorem 3.1. Since (aij ( x)) N N is positive definite matrix, then there exists a constant 0 such that N
valid for t 0 , implying that t . The proof of Theorem 2.1 is completed. III.
LOWER BOUND FOR BLOW-UP TIME
In this section, under the assumption that R3 is a convex bounded star-sharped domain in two orthogonal directions, we establish a lower bound for the blow-up time t . Now we state the result as follows. Theorem 3.1. Assumed that R3 is a bounded star-sharped convex domain in two orthogonal directions. Let u ( x, t ) be the nonnegative solution of problem (1)
and u ( x, t ) blows up at t , moreover the nonnegative f and g satisfy the conditions
1
0 g ( s) k2 s
2
, s 0,
for k1 0, k2 0, 1 . Define
(22)
Then (t ) satisfies inequality (t ) ( ) for some computable function ( ) . It follows that t is bounded below by d t . (0) ( ) In order to prove this theorem, we first give the following lemma. Lemma 3.1. Let be a bounded star-sharped region in R N , N 2 . Then for any nonnegative C 1 function u and r 0 , we have N rd r r r 1 u ds u dx u | u | dx,
0
0
div u r x Nu r ru r 1 ( x u)
over , and using divergence theorem, we get r r r 1 u ( x n)ds Nu dx ru ( x u)dx
i
i , j 1
j
N
f (u )]dx 2 (2 1) u 2 2 a ij ( x)u xi u x j dx
2 u
g (u )ds 2 u
2 1
i , j 1
2 1
f (u )dx
(23)
2 (2 1) u 2 2 | u |2 dx
5 2
where we have used successively the differential equation (1), the divergence theorem, the boundary condition (2) and the assumption (21). Next, Application of Lemma 2.1 leads to the inequality 5 5 5 1 3 5 d 2 2 2 u ds u dx u | u | dx. 0 2 0 Furthermore, since | u |2 2u 2 2 | u |2 , we replace (23) by
(t ) 5 d 2 0
2 (2 1)
u
5 1 2
|u |2 dx 2 k1[
3
0
5
u 2 dx (24)
| u | dx] 2 k2 u dx. 3
We now use the Schwarz inequality on the two integrals in the bracket in (12) and then the arithmeticgeometric mean inequality to arrive at
3 k1
0
u dx [ 2n
5kd 2 |u | dx 2 0
3 k1
(25)
5 2 k1d 2 k2 ] u 3 dx 2 0
0
for some positive which is to be determined. To bound the latter integral, we use an integral inequality which was derived in [18, (15)-(23)] and is restricted to N 3 dimensions, namely,
By the definitions of 0 and d , it follows that r
N u dx r u r
r 1
| x || u | dx
N u r dx rd u r 1 | u | dx,
which implies the desire conclusion.
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0 u ds u ( x n)ds
d : max |x|. Proof. Since is a bounded star-sharped region, we know 0 0 . Integrating the identity
r
N
(t ) 2 u 2 1ut dx 2 u 2 1[ (a ij ( x)u x ) x
2 (2 1)
0 : min (x n),
for a.e.. x and all R N . Differentiating (22) we obtain
(t )
where
( x)i j | |2
(21)
(t ) u 2 dx.
ij
2 k1 u ds 2 k2 u 3 dx
0 f ( s) k1 s 1 , s 0,
a
i , j 1
u 3 dx 3
d 1 0
3 4
[
3 2 0
u 2 dx
u dx |u 2
Using the inequalities
1 2
2
| dx
]. 1 2
3 2
(26)
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JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
(a b)
j 1 j
1 j
2 (a
j 1 j
a
j 1 j
non-increasing
)
(27)
a r bq ra qb, r q 1,
(28)
1
3 4
3
[
3
2 3 3 2 32 d 1 4 2 0 0
|u
2
| dx
3 2
3 3 3 1 3 d 1 3 3 4 21 0 2 2 2 2 3 4 1 [ 3 3 4 4 0
] (29) 3 4
|u |2 ]
3
d 2 3 k1 5 2 k1 f 3 2 1 0, 2 0 0 0 we have
3 2
al
u 0( gm R ) , f and h are non-negative functions, with f (0) 0 , t is the blow-up time if blow-up occurs,
u 2u , u,ij xi xi x j
will be used throughout this work, and summation from 1 to N is understood for repeated indices. Theorem 4.1. Assume that the nonnegative functions f and g satisfy
s 1 0 h( s ) k1 d , s 0, 0 g ( ) p
s 1 f ( s) k2 g ( s) d , s 0 0 g ( ) for k1 0, k2 0, p 1, q 1, and 2q p 1 Then the (nonnegative) solution u ( x, t ) of problem (31)-(33) does not blow up, so that u ( x, t ) exists for all time t 0 . Proof of Theorem 4.1. Set
3
(t ) C1 C2 2 C3 3 ,
(t ) v 2 (u ( x, t ))dx,
where
C1
for
q
for some positive that is to be determined. Collecting terms in (25) and (29) and choosing and so that 2 (2 1) 5k1d 2 0 1 4
g (u) gm
with
or else t . The notations u,i :
for positive a and positive b , it follows that u 3 dx 2 2 3
function,
3 k1
v( s) :
0
0
3 3 3 k 5 2 k1d 1 C2 3 4 21 0 2 2 0 0 1 3 3 k 3 5 2 k1d d 1 C3 2 2 3 4 . 1 2 0 0 0 4 Under the assumption that . blows up at finite time
t , we obtain a lower bound for t by integration of (30), i.e., d t . 3 0 3 2 C1 C2 C3 The proof of Theorem 3.1 is completed.
(t ) 2
2 v
v ( g (u )u,i )i f (u) dx g (u )
| u |2 g v | u |2 f (u )v u ds 2 dx n g (u )
(31)
u h(u ), ( x, t ) (0, t ) n
(32)
where we have used successively the differential equation (31), the divergence theorem, the fact that g 0 and the hypotheses stated in Theorem 31. Application of Lemma 3.1 leads to the inequality
In this section, we establish the conditions on the nonlinearities of the following initial-boundary value problem to guarantee that u ( x, t ) exists globally.
ut ( g (u)u,i ),i f (u), ( x, t ) (0, t )
(q 1)d
0
0
v q 1dx (36)
v q | v | dx
Inserting (36) into (35), we get
(t )
2k1 N
0
v q 1dx
2(q 1)k1d
2 g m |v | dx 2k2 v
(33)
u where is the outward normal derivative of u on n the boundary assumed sufficiently smooth, is a bounded star-shaped region in R N , N 2, g is a positive
N
v q 1ds
2
u( x,0) u0 ( x) 0, x ,
(35)
2k1 v q 1ds 2 g m |v |2 dx 2k2 v p 1dx,
CRITERION FOR GLOBAL EXISTENCE
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(34)
1 d. g ( )
Differentiating (34), we obtain 3 2
IV.
s
Choosing
0 p 1
dx.
k1d (q 1) , we have 2 0 g m
v q | v | dx (37)
JOURNAL OF NETWORKS, VOL. 9, NO. 3, MARCH 2014
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1 |v |2 dx 2 2 (38) 0 g m k d (q 1) 2q 2 1 v dx | v | dx . 4 0 g m k1d (q 1)
v q | v | dx
v 2 q dx
Inserting (38) into (37), we have
(t )
2k1 N
0
v
q 1
dx 2 g m
2
v 2k2 v 2q
p 1
V.
Concluding remarks A result analogous to theorem 3.1 can be obtained for a bounded domain R2 by first using Schwarz's inequality to write
dx.
(39)
p 1 2q 1 . pq Using Holder inequality and Young inequality, we obtain In view of 2q p 1, we see
CONCLUDING REMARKS
u 3 dx
u 2 dx
1 2
u 4 dx
1 2
,
then deriving an integral inequality similar to (14) for 4 u dx
by the method used in going from (15) to (23) in [18]. In this manner, we obtain a first-order differential inequality of the form 3
v
v
2q
v
( q 1)( p 1) ( q 1 2 q )( q 1) p q p q
q 1
v
dx
p 1
dx
1
dx
1 v q 1dx v p 1dx
(1 ) v p 1
(t ) K1 K2 2 K3 2
1
(40)
1
v q 1dx.
(u )t
Combining (39) with (38), we have
(t ) L1 vq 1dx L2 v p 1dx ,
2 Nk1
2 g m 2
0
0,
v p 1dx
pq
q 1 p 1
| | p 1 .
(42)
Inserting (41) into (42), we have
(t ) L1
v p 1dx
q 1 p 1
[| |
pq p 1
L2 L1
v p 1dx
p q p 1
].
(43)
(t ) v 2 dx
v
p 1
dx
q 1 p 1
(| |
pq p 1
v p 1dx
2 p 1
(44)
L2 || L1
( p q )(1 p ) 2( p 1)
p q 2
).
2
(t ) L2q p | | q 1
(48)
ij
i , j 1
( x)(u ) xi n j g (u )
(49)
(50)
where 1, 2,..., n, u (u1 ,..., un ) and (aij ( x)) N N is differentiable positive definite matrixes. We shall use comma i notation to denote partial differentiation with respect to xi and the summation convention on repeated indices, where the Greek indices sum over 1, 2,..., n . We impose the following constraint on the nonlinear terms and the boundary data in (31) 1
2
,
p 1 p 1
for positive constants B1 , B2 and 1 , and assume the existence of solutions u , 1, 2,..., n, of the system (31), one or more of which becomes unbounded in finite time t . ACKNOWLEDGMENT
(46)
so that (t ) remains bounded for all time under the conditions stated in Theorem 4.1. This completes the proof of Theorem 4.1. © 2014 ACADEMY PUBLISHER
) x j f (u )
(45)
We conclude from (45) that (t ) is decreasing in each time interval on which we have 2
xi
u f (u ) B2 (u u )1
It follows from (43) and (44) that (t ) L1
N
u g (u ) B1 (u u )
Using Hoder inequality, we have
i , j 1
u ( x,0) u 0 ( x) 0
L2 2k2 2 g m 2 (1 ) for 0 small enough. v q 1dx
ij
a
with
L1
N
(a ( x)(u )
(41)
1
for computable constants K1 , K 2 and K 3 , from which a lower bound on the blow-up time follows. The results in this paper can also be extended to the following initial-boundary value problem for the nonlinear system of $n$ parabolic equations which are coupled through the nonlinear terms and the boundary data
The author is grateful to the referees for their valuable suggestions and comments on the original manuscript. REFERENCES [1] B. Straughan, Explosive instabilities in (Springer-Verlag, Berlin, 1998).
mechanics
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[2] R. Quittner and P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states (Birkhauser, Basel, 2007). [3] H. A. Levine, Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics. The method of unbounded Fourier coefficients, Math. Ann. 214 (1975), 205-220. [4] V. A. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Cont. Dyn. Syst. 8(2) (2002), 399-433. [5] L. E. Payne, P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl. 328 (2007) 1196-1205. [6] L. E. Payne, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Appl. Anal. 87 (2008) 699707. [7] L. E. Payne, P. W. Schaefer, Bounds for the blow-up time for heat equation under nonlinear boundary condition, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009) 1289-1296. [8] L. E. Payne, G. A. Philippin, S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I, Z. Angew. Math. Phys. 61 (2010) 999-1007. [9] L. E. Payne, G. A. Philippin, S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II, Nonlinear Anal. 73(2010) 971-978. [10] L. E. Payne, G. A. Philippin, P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal. 69 (2008) 3495-3502. [11] L. E. Payne, G. A. Philippin, P. W. Schaefer, Bounds for blow-up time in nonlinear parabolic problems, J. Math. Anal. Appl. 338 (2008) 438-447 [12] L. E. Payne, P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl. 328 (2007) 1196-1205.
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[13] C. Enache, Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary conditions, Appl. Math. Lett. 24 (2011) 288-292. [14] E. Enache, Lower bounds for blow-up time in some nonlinear parabolic problems under Neumann boundary conditions, Glasgow Math. J. Available CJ0201 DOI: 10.1017/50017089511000139. [15] L. E. Payne, G. A. Philippin, Blow-up in a class of nonlinear parabolic problems with time-dependent coefficients under Robin type boundary conditions, Appl. Anal. DOI: 10.1080/00036811.2011.598865. [16] L. E. Payne, J. C. Song, Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl. 354, 394-396 (2009) [17] L. E. Payne, P. W. Schaefer, Blow- up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math. 48, 193-202 (2008) [18] L. E. Payne and P. W. Schaefer. Lower bounds for blowup time in parabolic problems under Neumann conditions. Appl. Anal. 85 (2006), 1301-1311 [19] W. Yan and W Xiao, A new fuzzy SVM based on the posterior probalility weighting menbership, J. Computers, 7(2012) 1385-1393. [20] X. Shi and D. Feng, LSP: A locality –aware strip prefetching scheme for striped disk array systems with concurrent accesses. J. Cumptuters, 7(2012) 1303-1312.
Jin Li was born in Gansu on January, 1957. Associate professor. The main research direction is Nonlinear partial differential equations and Applied analysis.
