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Interactions among multi-nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux
Nonlinear Analysis 57 (2004) 519 – 530
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Interactions among multi-nonlinearities in a nonlinear di#usion system with absorptions and nonlinear boundary 'ux Sining Zheng∗ , Xianfa Song Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China Received 14 August 2003; accepted 29 February 2004
Abstract This paper deals with a nonlinear di#usion system with inner absorptions and coupled nonlinear boundary conditions. Two parameters, which solve an introduced matrix equation covering all the eight nonlinear exponents of the system, are used to get a simple description for the interactions among the multi-nonlinearities in the system. Blow-up or not of the solutions is determined by the signs of the two parameters. In order to obtain the critical properties of the solutions for the critical sign of the parameters, some further analysis on the absorption coe4cients and the geometry of is given in addition to the discussion for the eight nonlinear exponents in the system. ? 2004 Elsevier Ltd. All rights reserved. MSC: primary 35K55; 35B33 Keywords: Nonlinear di#usion; Nonlinear absorption; Nonlinear boundary 'ux; Blow-up; Global boundedness; Critical exponents
1. Introduction In this paper we consider the following nonlinear di#usion equations with nonlinear absorptions and coupled nonlinear boundary 'ux: ut =
vt =
(x; t) ∈ × (0; T );
Supported by National Natural Science Foundation of China.
∗
Corresponding author. E-mail address: [email protected] (S. Zheng).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.02.026
(1.1)
520
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
@u = epv+2 u ; @
@v = equ+ 2 v ; @
u(x; 0) = u0 (x);
(x; t) ∈ @ × (0; T );
(1.2)
C x ∈ ;
(1.3)
v(x; 0) = v0 (x);
where ⊂ RN is a bounded domain with smooth boundary @; constants m; n; p; q, a1 ; a2 ¿ 0, i ; i ¿ 0 (i = 1; 2); u0 and v0 are nonnegative functions satisfying the comC Eqs. (1.1)–(1.3) can be used to describe, e.g., heat propagapatible conditions on . tions in mixed solid nonlinear media with nonlinear di#usions, nonlinear absorptions, nonlinear convection, and nonlinear boundary 'ux. In fact, by setting w = eu and z = ev , system (1.1)–(1.3) is equivalent to some porous medius equations of the form m wt =
@z = wq z 2 +1 @ z(x; 0) = ev0
on @ × (0; T );
C on :
(1.6) (1.7)
Since w = eu ¿ 0, z = ev ¿ 0, we know that there is no substantial degeneracy in (1.4)–(1.7), although the di#usion here is nonlinear. Therefore, there do exist locally classical solutions to (1.1)–(1.3) [10,15], and the classical comparison principle is valid for the system. One can refer [14,21] for further information to the model considered. There are three kinds of nonlinear mechanisms in model (1.1)–(1.3), i.e., nonlinear di#usion, nonlinear absorption, and nonlinear boundary 'ux. We are interested in the interactions among them. Critical exponents (or su4cient and necessary conditions to global existence of solutions) for reaction–di#usion equations, together with estimates on blow-up rates, sets and proIles, are of interesting and worth to study [3,5,6]. There have been a lot of works relative to these subjects, e.g., [4,7–9,17,19,20,22,23]. Recently, Andreu et al. [2] studied the single equation problem ut = <(|u|m−1 u) − |u|p u @|u|m−1 u = |u|q−1 u @ u(x; 0) = u0 (x)
in × (0; T );
on @ × (0; T );
C on :
(1.8) (1.9) (1.10)
Under the assumption of q ¿ m, the following results were obtained [2]: (i) If p ¡ 2q − m, there exist solutions of (1.8)–(1.10) blowing up in Inite time for large initial data. (ii) If p ¿ 2q − m, there exists a weak solution globally bounded for every initial datum in L∞ (). (iii) If p = 2q − m, there exist solutions blowing up in Inite time for large initial data if ¡ q=m, while there exists a globally bounded weak solution for initial datum in L∞ () and ¿ q=m. One can see, e.g., [1,11,12,18] for similar problems of single degenerate parabolic equations.
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
521
In a previous work of the authors [16], the following single parabolic equations of general forms with absorption were studied: ut = <(u) − f(u) @u = g(u) @
in × (0; T );
(1.11)
on @ × (0; T );
(1.12)
C on ;
(1.13)
u(x; 0) = u0 (x)
where (s) ¿ 0 for s ¿ s0 ¿ 0. For special (u) = a1 emu , f(u) = a2 e1 u , g(u) = a3 e2 u , system (1.11)–(1.13) becomes ut = a1
in × (0; T );
on @ × (0; T );
(1.14) (1.15)
on C
(1.16)
with the following results [16]. Proposition 1. The solutions of (1:14)–(1:16) are global if 1 ¿ m + 22 , and will blow up in 5nite time if either 1 ¡ m + 22 , or 1 = m + 22 with m(m + p)a1 a23 ¿ a2 for large initial value. In this paper, we are interested in the interactions among the multi-nonlinearities in (1.1)–(1.3). To describe the critical exponent for (1.1)–(1.3), we need the following matrix equation: 2 − (1 − m)=2 1 p 1 = ; (1.17) 1 q
2 − ( 1 − n)=2 2 namely 1 = 2 =
p − 2 + ( 1 − n)=2 ; pq − ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ) q − 2 + (1 − m)=2 : pq − ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 )
(1.18)
Similar matrix equations were used for some semilinear reaction–di#usion systems (see, e.g., [24]). Clearly, all the eight exponents m, n, p, q, i , i (i = 1; 2) from the nonlinear terms of (1.1)–(1.3) are included in the matrix equation (1.17). Let ’0 be the Irst eigenfunction of <’ + ’ = 0
in ;
’=0
on @
(1.19)
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
with the Irst eigenvalue 0 , normalized by ’0 ∞ = 1. Then ’0 ¿ 0 in , and satisIes @’0 = − @’0 6 max|∇’0 | = c2 (1.20) c1 6 @ @ @ @ C for some positive constants c1 , c2 . Clearly, there exist " ¿ 0 and c3 ¿ 0 such that |∇’0 | ¿ c1 =2
for x ∈ {x ∈ : dist(x; @) 6 "}
(1.21)
and ’0 ¿ c3
for x ∈ {x ∈ : dist(x; @) ¿ "}:
(1.22)
We now state the main results of this paper. Theorem 1. Assume 1= 1 ; 1= 2 ¿ 0. If either 2 6 (1 − m)=2, 2 6 ( 1 − n)=2, or C then the solutions of (1:1)–(1:3) will
mC 2 min(0 (1 − m)c32 ; (1 + m)c12 ) ; 4(1 − m)2 c22
a2 6
nC 2 min(0 ( 1 − n)c32 ; ( 1 + n)c12 ) 4( 1 − n)2 c22
with C = min((1 − m)=2; ( 1 − n)=2), then the solutions blow up in 5nite time for large initial data. Remark 1. Although 1= 1 ; 1= 2 ¡ 0 (or 1= 1 ; 1= 2 ¿ 0) is equivalent to 1 ; 2 ¡ 0 (or 1 ; 2 ¿ 0), which corresponds to global existence (or blow-up) of solutions to (1.1)–(1.3), we think (1= 1 ; 1= 2 )=(0; 0) in stead of 1 = 2 =0 itself as the critical sign of the parameters. We learn from Theorem 3 that, in the critical case (1= 1 ; 1= 2 )=(0; 0), the blow-up or not of the solutions of (1.1)–(1.3) depends on 0 , ci (i = 1; 2; 3) as well as a1 ; a2 (the absorption coe4cients). Obviously, 0 and ci (i=1; 2; 3) are determined by the geometry (i.e. the size and the shape) of . It is well known that 0 is conversely
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
523
proportional to the squared diameter of . Roughly speaking, e.g., subcase (ii) of Theorem 3 says that in order to obtain Inite blow-up of the solutions for the critical (1= 1 ; 1= 2 ) = (0; 0) case, we need smaller , which corresponds to larger 0 , or smaller ai (i = 1; 2) (the absorption coe4cients) in the model. This paper is organized as follows. In Section 2, we will give the proof of Theorem 1 for blowing up of solutions. In Section 3, we will prove Theorem 2 concerning the global boundedness. Section 4 deals with the critical case to prove Theorem 3.
2. Proof of Theorem 1 This section deals with the blow-up case of 1= 1 ; 1= 2 ¿ 0. It is easy to see from (1.18) that the condition 1= 1 , 1= 2 ¿ 0 requires either p − 2 + ( 1 − n)=2, q − 2 + (1 − m)=2 ¿ 0, or p − 2 + ( 1 − n)=2, q − 2 + (1 − m)=2 ¡ 0. There are two cases for them: (i) 2 ¿ (1 − m)=2 or 2 ¿ ( 1 − n)=2. In this case if p − 2 + ( 1 − n)=2, q − 2 + (1 − m)=2 ¿ 0, then pq ¿ (2 − (1 − m)=2)( 2 − ( 1 − n)=2); if p − 2 + $; q − 2 + % ¡ 0, then pq ¡ (2 − (1 − m)=2)( 2 − ( 1 − n)=2), and hence 1 ; 2 ¿ 0. (ii) 2 6 (1 − m)=2, 2 6 ( 1 − n)=2 with pq ¿ ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ). Proof of Theorem 1. At Irst consider the subcase of 2 6 (1 − m)=2, 2 6 ( 1 − n)=2. Construct u(x; t) = log v(x; t) = log
[’A(1 −m)=2
A ; + (1 − ct)K ]2=(1 −m)
[’B( 1 −n)=2
B ; + (1 − ct)L ]2=( 1 −n)
1 (x; t) ∈ C × [0; ); c 1 (x; t) ∈ C × [0; ); c
where ’ = M1 ’0 , ’0 is the normalized Irst eigenfunction of (1.19), M1 ; A; B; c; K; L are positive constants to be determined. Noticing ’0 = 0 on @, we have by using (1.20) that @u 2A(1 −m)=2 M1 c2 ; 6 @ (1 − m)(1 − ct)K @v 2B( 1 −n)=2 M1 c2 ; 6 @ ( 1 − n)(1 − ct)L
epv+2 u = equ+ 2 v =
A 2 B p (1 −
ct)22 K=(1 −m)+2pL=( 1 −n) Aq B 2
(1 − ct)2qK=(1 −m)+2 2 L=( 1 −n)
;
(2.1)
(2.2)
hold on @×(0; T ). The assumption 1= 1 ; 1= 2 ¿ 0 with 2 6 (1 −m)=2 and 2 6 ( 1 − n)=2 implies
1 − n 1 − m − 2 − 2 ; pq ¿ 2 2
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
and hence there exist positive A; B; K; L large such that 1=p 1=(( 1 −n)=2− 2 ) 2M1 c2
1 − n ((1 −m)=2−2 )=p ¡B¡ Aq=(( 1 −n)=2− 2 ) ; A 1 − m 2M1 c2
1 − n(1 − m)=2 − 2
1 − nq K; K ¡L¡ 1 − m( 1 − n)=2 − 2 1 − mp which are equivalent to 2M1 c2 ( 1 −n)=2 B ¡ Aq B 2 ;
1 − n
2M1 c2 (1 −m)=2 ¡ B p A 2 ; A 1 − m K(1 − m − 22 ) 2Lp ¡ ; 1 − m
1 − n
2Kq L( 1 − n − 2 2 ) : ¡ 1 − m
1 − n
By the relationships above, we obtain from (2.1) and (2.2) that @u 6 epv+2 u ; @
@v 6 equ+ 2 v @
on @ × (0; T ):
(2.3)
As for the estimates in × (0; T ), a simple computation shows ut =
Kc(1 − ct)K−1 ; + (1 − ct)K
’A(1 −m)=2
e1 u =
A 1 ; [’A(1 −m)=2 + (1 − ct)K ]21 =(1 −m)
2mA(1 +m)=2 1 ’ (1 − m)[’A(1 −m)=2 + (1 − ct)K ](1 +m)=(1 −m) +
2m(1 + m)A1 |∇’|2 : (1 − m)2 [’A(1 −m)=2 + (1 − ct)K ]21 =(1 −m)
Notice |∇’| = |∇(M1 ’0 )| ¿ M1 c1 =2 ’ = M1 ’0 ¿ M1 c3
for x ∈ {x ∈ : dist(x; @) 6 "};
for x ∈ {x ∈ : dist(x; @) ¿ "}:
Choose M1 and c satisfying m1 M12 c32 ¿ a1 (1 − m);
m(1 + m)M12 c12 ¿ 4a1 (1 − m)2
and c ¡ a1 =K(M1 A(1 −m)=2 + 1)(1 +m)=(1 −m) with K ¿ 1. Thus, we have u t 6
in × (0; T ):
(2.4)
Similarly, we can get vt 6
in × (0; T ):
(2.5)
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525
C If in addition the initial data are large such that u0 (x) ¿ u(x; 0), v0 (x) ¿ v(x; 0) on , then together (2.3)–(2.5), (u; v) is a pair of sub-solution of (1.1)–(1.3), which means that the solutions of (1.1)–(1.3) will blow up in Inite time. For the subcase of 2 ¿ (1 − m)=2 or 2 ¿ ( 1 − n)=2, denote w = ut and z = vt . Then w and z satisfy wt = m<(emu w) − a1 1 e1 u w;
vt = n<(env z) − a2 1 e 1 v z
@w = epv+2 u (2 w + pz); @
@v = equ+ 2 v (qw + 2 z) @
w(x; 0) =
z(x; 0) =
in × (0; T ); (2.6)
on @ × (0; T ); C on :
(2.7) (2.8)
Due to the positivity of exponential functions in system (2.6)–(2.8), we know by using the classical comparison principle (see, e.g., [13, Lemma 5.1, p. 480]) with the assumption
in × (0; T );
(2.9)
@U = epv0 e2 U @
on @ × (0; T );
(2.10)
U (x; 0) = u0 (x)
C on :
(2.11)
It follows from Proposition 1 that the solutions of (2.9)–(2.11) will blow up in Inite time for large initial value. We can choose (U; v0 ) as a pair of sub-solution of (1.1)–(1.3) to get the blow-up conclusion to (1.1)–(1.3) immediately. The proof is complete. Due to the results of Theorem 1, we will always assume 2 6 (1 −m)=2, 2 6 ( 1 − n)=2 in the sequels.
3. Proof of Theorem 2 This section treats the case 1= 1 ; 1= 2 ¡ 0 with 2 6 (1 − m)=2 and 2 6 ( 1 − n)=2. Proof of Theorem 2. Construct time-independent functions u(x; C t) = log
A ; 2 − (1 − ’)A%
v(x; C t) = log
B ; 2 − (1 − ’)B$
(x; t) ∈ C × [0; T );
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
where % = (1 − m)=2, $ = ( 1 − n)=2, ’ = "1 ’0 , ’0 is the normalized Irst eigenfunction of (1.19), and "1 ; A; B are positive constants to be determined. Then
mAm+% 1 ’(1 − ’)A
%
−1
+ mAm+% (A(1 −m)=2 − 1)|∇’|2 (1 − ’)A (2 − (1 − ’)A% )m+1
%
−2
%
m(m + 1)A1 (1 − ’)2A −2 |∇’|2 + ; (2 − (1 − ’)A% )m+2 and hence
"1 = min
a1 a2 ; ; 1 ; (m1 + m(m + 2)c42 )21 (n1 + n(n + 2)c42 )2 1
then
in :
(3.1)
Moreover, we have @uC ¿ A(1 −m)=2 "1 c1 ; @
C 2 uC epv+ = A2 B p
@vC ¿ B( 1 −n)=2 "1 c1 ; @
C 2 vC equ+
= Aq B 2
on @ × (0; T ); on @ × (0; T ):
The assumption 1= 1 ; 1= 2 ¡ 0 with 2 6 (1 − m)=2 and 2 6 ( 1 − n)=2 implies pq ¡ ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ). So, there exist A; B large such that q=(( 1 −n)=2− 2 ) ((1 −m)=2−2 )=p 1 1 q=(( 1 −n)=2− 2 ) A ¡B¡ A((1 −m)=2−2 )=p "1 c1 "1 c1 or equivalently, Aq B 2 ¡ "1 c1 B( 1 −n)=2 ;
A2 Bp ¡ "1 c1 A(1 −m)=2 ;
by which we can obtain @uC C 2 uC ; ¿ epv+ @
@vC C 2 vC ¿ equ+
@
on @:
(3.2)
C Moreover, we can choose A; B large enough such that uC ¿ u0 (x), vC ¿ v0 (x) on . Together with (3.1), (3.2), we know that the time-independent (u; C v) C is a pair of super solution of (1.1)–(1.3), and hence the solutions of (1.1)–(1.3) are globally bounded.
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
527
4. Proof of Theorem 3 Finally, we discuss the critical case of (1= 1 ; 1= 2 ) = (0; 0), or equivalently, pq = ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ). Proof of Theorem 3. Consider the subcase (i) at Irst. Construct time-independent functions u(x; C t) = log
c1−1
A ; + 1 − c1−1 (1 − ’0 )A%
v(x; C t) = log
c1−1
B + 1 − c1−1 (1 − ’0 )B$
for (x; t) ∈ C × [0; T ), where ’0 is the normalized Irst eigenfunction of (1.19), A; B are positive constants to be determined later. Due to the assumption a1 ¿ mc1−2−1 (c1−1 + 1)1 (1 c1 + (m + 2)c42 ); a2 ¿ nc1−2− 1 (c1−1 + 1) 1 (1 c1 + (n + 2)c42 ); it is easy to get that
in × (0; T ):
(4.1)
Moreover, we have @uC ¿ A(1 −m)=2 ; @
C 2 uC epv+ = A2 B p
@vC ¿ B( 1 −n)=2 ; @
C 2 vC equ+
= Aq B 2
on @ × (0; T ); on @ × (0; T ):
Since pq = ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ) implies q=(( 1 − n)=2 − 2 ) = ((1 − m)=2 − 2 )=p, we know A((1 −m)=2−2 )=p = B = Aq=(( 1 −n)=2− 2 ) or equivalently, A(1 −m)=2 = A2 Bp ;
B( 1 −n)=2 = Aq B 2
hold for any A ¿ 0. Thus, we have @uC C 2 uC ; ¿ epv+ @
@vC C 2 vC ¿ equ+
@
on @ × (0; T ):
(4.2)
Moreover, we can choose A; B large enough such that uC ¿ u0 (x);
vC ¿ v0 (x)
C on :
(4.3)
Inequalities (4.1)–(4.3) show that the time-independent (u; C v) C is a pair of super solution of (1.1)–(1.3), and hence the solutions of (1.1)–(1.3) are globally bounded.
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
Subcase (ii). Construct u(x; t) = log
A ; [’A(1 −m)=2 + (1 − ct)K ]2=(1 −m)
v(x; t) = log
B ; [’B( 1 −n)=2 + (1 − ct)L ]2=( 1 −n)
1 (x; t) ∈ C × [0; ); c 1 (x; t) ∈ C × [0; ); c
where ’ = ’0 c2−1 min((1 − m)=2; ( 1 − n)=2), ’0 is the normalized Irst eigenfunction of (1.19), and A; B; c; K; L are positive constants to be determined. By using the assumption a1 6
mC 2 min(1 (1 − m)c32 ; (1 + m)c12 ) ; 4(1 − m)2 c22
a2 6
nC 2 min(1 ( 1 − n)c32 ; ( 1 + n)c12 ) 4( 1 − n)2 c22
with C = min((1 − m)=2; ( 1 − n)=2), we can obtain that u t 6
vt 6
in × (0; T ):
(4.4)
We have moreover on the boundary @ that @u A(1 −m)=2 6 ; @ (1 − ct)K
epv+2 u =
@v B( 1 −n)=2 ; 6 @ (1 − ct)L
equ+ 2 v =
A 2 B p (1 −
ct)22 K=(1 −m)+2pL=( 1 −n) Aq B 2
(1 − ct)2qK=(1 −m)+2 2 L=( 1 −n)
;
:
As pq = ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ), there exist positive A; B; K; L large such that A((1 −m)=2−2 )=p = B = Aq=(( 1 −n)=2− 2 ) ; q
1 − n (1 − m)=2 − 2
1 − n K; K =L= 1 − m p 1 − m ( 1 − n)=2 − 2 or equivalently, A(1 −m)=2 = Bp A2 ;
B 1 −n=2 = Aq B 2 ;
(1 − m)=2 − 2 p K= L; 1 − m
1 − n
( 1 − n)=2 − 2 q L= K:
1 − n 1 − m
Thereby we can obtain @u 6 epv+2 u ; @
@v 6 equ+ 2 v @
on @ × (0; T ):
(4.5)
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
529
Now let the initial data be large such that u0 (x) ¿ u(x; 0);
v0 (x) ¿ v(x; 0)
C on :
(4.6)
In summary of (4.4)–(4.6), we know that (u; v) is a pair of sub-solution of (1.1)–(1.3), and hence the solutions of (1.1)–(1.3) will blow up in Inite time with such large initial data. The proof is complete.
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[21] M.X. Wang, S. Wang, Quasilinear reaction–di#usion systems with nonlinear boundary conditions, J. Math. Anal. Appl. 231 (1999) 21–33. [22] M.X. Wang, Y.M. Wang, Reaction di#usion systems with nonlinear boundary conditions, Aci. China Ser. A 39 (1996) 834–840. [23] M.X. Wang, Y.H. Wu, Existence and nonexistence of global positive solutions to nonlinear di#usion problems with nonlinear absorption through the boundary, Trans. Amer. Math. Soc. 349 (1997) 955–971. [24] S.N. Zheng, Nonexistence of positive solutions to a semilinear elliptic system and blow up estimates for a reaction–di#usion system, J. Math. Anal. Appl. 232 (1999) 293–311.
www.elsevier.com/locate/na
Interactions among multi-nonlinearities in a nonlinear di#usion system with absorptions and nonlinear boundary 'ux Sining Zheng∗ , Xianfa Song Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China Received 14 August 2003; accepted 29 February 2004
Abstract This paper deals with a nonlinear di#usion system with inner absorptions and coupled nonlinear boundary conditions. Two parameters, which solve an introduced matrix equation covering all the eight nonlinear exponents of the system, are used to get a simple description for the interactions among the multi-nonlinearities in the system. Blow-up or not of the solutions is determined by the signs of the two parameters. In order to obtain the critical properties of the solutions for the critical sign of the parameters, some further analysis on the absorption coe4cients and the geometry of is given in addition to the discussion for the eight nonlinear exponents in the system. ? 2004 Elsevier Ltd. All rights reserved. MSC: primary 35K55; 35B33 Keywords: Nonlinear di#usion; Nonlinear absorption; Nonlinear boundary 'ux; Blow-up; Global boundedness; Critical exponents
1. Introduction In this paper we consider the following nonlinear di#usion equations with nonlinear absorptions and coupled nonlinear boundary 'ux: ut =
vt =
(x; t) ∈ × (0; T );
Supported by National Natural Science Foundation of China.
∗
Corresponding author. E-mail address: [email protected] (S. Zheng).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.02.026
(1.1)
520
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
@u = epv+2 u ; @
@v = equ+ 2 v ; @
u(x; 0) = u0 (x);
(x; t) ∈ @ × (0; T );
(1.2)
C x ∈ ;
(1.3)
v(x; 0) = v0 (x);
where ⊂ RN is a bounded domain with smooth boundary @; constants m; n; p; q, a1 ; a2 ¿ 0, i ; i ¿ 0 (i = 1; 2); u0 and v0 are nonnegative functions satisfying the comC Eqs. (1.1)–(1.3) can be used to describe, e.g., heat propagapatible conditions on . tions in mixed solid nonlinear media with nonlinear di#usions, nonlinear absorptions, nonlinear convection, and nonlinear boundary 'ux. In fact, by setting w = eu and z = ev , system (1.1)–(1.3) is equivalent to some porous medius equations of the form m wt =
@z = wq z 2 +1 @ z(x; 0) = ev0
on @ × (0; T );
C on :
(1.6) (1.7)
Since w = eu ¿ 0, z = ev ¿ 0, we know that there is no substantial degeneracy in (1.4)–(1.7), although the di#usion here is nonlinear. Therefore, there do exist locally classical solutions to (1.1)–(1.3) [10,15], and the classical comparison principle is valid for the system. One can refer [14,21] for further information to the model considered. There are three kinds of nonlinear mechanisms in model (1.1)–(1.3), i.e., nonlinear di#usion, nonlinear absorption, and nonlinear boundary 'ux. We are interested in the interactions among them. Critical exponents (or su4cient and necessary conditions to global existence of solutions) for reaction–di#usion equations, together with estimates on blow-up rates, sets and proIles, are of interesting and worth to study [3,5,6]. There have been a lot of works relative to these subjects, e.g., [4,7–9,17,19,20,22,23]. Recently, Andreu et al. [2] studied the single equation problem ut = <(|u|m−1 u) − |u|p u @|u|m−1 u = |u|q−1 u @ u(x; 0) = u0 (x)
in × (0; T );
on @ × (0; T );
C on :
(1.8) (1.9) (1.10)
Under the assumption of q ¿ m, the following results were obtained [2]: (i) If p ¡ 2q − m, there exist solutions of (1.8)–(1.10) blowing up in Inite time for large initial data. (ii) If p ¿ 2q − m, there exists a weak solution globally bounded for every initial datum in L∞ (). (iii) If p = 2q − m, there exist solutions blowing up in Inite time for large initial data if ¡ q=m, while there exists a globally bounded weak solution for initial datum in L∞ () and ¿ q=m. One can see, e.g., [1,11,12,18] for similar problems of single degenerate parabolic equations.
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
521
In a previous work of the authors [16], the following single parabolic equations of general forms with absorption were studied: ut = <(u) − f(u) @u = g(u) @
in × (0; T );
(1.11)
on @ × (0; T );
(1.12)
C on ;
(1.13)
u(x; 0) = u0 (x)
where (s) ¿ 0 for s ¿ s0 ¿ 0. For special (u) = a1 emu , f(u) = a2 e1 u , g(u) = a3 e2 u , system (1.11)–(1.13) becomes ut = a1
in × (0; T );
on @ × (0; T );
(1.14) (1.15)
on C
(1.16)
with the following results [16]. Proposition 1. The solutions of (1:14)–(1:16) are global if 1 ¿ m + 22 , and will blow up in 5nite time if either 1 ¡ m + 22 , or 1 = m + 22 with m(m + p)a1 a23 ¿ a2 for large initial value. In this paper, we are interested in the interactions among the multi-nonlinearities in (1.1)–(1.3). To describe the critical exponent for (1.1)–(1.3), we need the following matrix equation: 2 − (1 − m)=2 1 p 1 = ; (1.17) 1 q
2 − ( 1 − n)=2 2 namely 1 = 2 =
p − 2 + ( 1 − n)=2 ; pq − ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ) q − 2 + (1 − m)=2 : pq − ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 )
(1.18)
Similar matrix equations were used for some semilinear reaction–di#usion systems (see, e.g., [24]). Clearly, all the eight exponents m, n, p, q, i , i (i = 1; 2) from the nonlinear terms of (1.1)–(1.3) are included in the matrix equation (1.17). Let ’0 be the Irst eigenfunction of <’ + ’ = 0
in ;
’=0
on @
(1.19)
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
with the Irst eigenvalue 0 , normalized by ’0 ∞ = 1. Then ’0 ¿ 0 in , and satisIes @’0 = − @’0 6 max|∇’0 | = c2 (1.20) c1 6 @ @ @ @ C for some positive constants c1 , c2 . Clearly, there exist " ¿ 0 and c3 ¿ 0 such that |∇’0 | ¿ c1 =2
for x ∈ {x ∈ : dist(x; @) 6 "}
(1.21)
and ’0 ¿ c3
for x ∈ {x ∈ : dist(x; @) ¿ "}:
(1.22)
We now state the main results of this paper. Theorem 1. Assume 1= 1 ; 1= 2 ¿ 0. If either 2 6 (1 − m)=2, 2 6 ( 1 − n)=2, or C then the solutions of (1:1)–(1:3) will
mC 2 min(0 (1 − m)c32 ; (1 + m)c12 ) ; 4(1 − m)2 c22
a2 6
nC 2 min(0 ( 1 − n)c32 ; ( 1 + n)c12 ) 4( 1 − n)2 c22
with C = min((1 − m)=2; ( 1 − n)=2), then the solutions blow up in 5nite time for large initial data. Remark 1. Although 1= 1 ; 1= 2 ¡ 0 (or 1= 1 ; 1= 2 ¿ 0) is equivalent to 1 ; 2 ¡ 0 (or 1 ; 2 ¿ 0), which corresponds to global existence (or blow-up) of solutions to (1.1)–(1.3), we think (1= 1 ; 1= 2 )=(0; 0) in stead of 1 = 2 =0 itself as the critical sign of the parameters. We learn from Theorem 3 that, in the critical case (1= 1 ; 1= 2 )=(0; 0), the blow-up or not of the solutions of (1.1)–(1.3) depends on 0 , ci (i = 1; 2; 3) as well as a1 ; a2 (the absorption coe4cients). Obviously, 0 and ci (i=1; 2; 3) are determined by the geometry (i.e. the size and the shape) of . It is well known that 0 is conversely
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
523
proportional to the squared diameter of . Roughly speaking, e.g., subcase (ii) of Theorem 3 says that in order to obtain Inite blow-up of the solutions for the critical (1= 1 ; 1= 2 ) = (0; 0) case, we need smaller , which corresponds to larger 0 , or smaller ai (i = 1; 2) (the absorption coe4cients) in the model. This paper is organized as follows. In Section 2, we will give the proof of Theorem 1 for blowing up of solutions. In Section 3, we will prove Theorem 2 concerning the global boundedness. Section 4 deals with the critical case to prove Theorem 3.
2. Proof of Theorem 1 This section deals with the blow-up case of 1= 1 ; 1= 2 ¿ 0. It is easy to see from (1.18) that the condition 1= 1 , 1= 2 ¿ 0 requires either p − 2 + ( 1 − n)=2, q − 2 + (1 − m)=2 ¿ 0, or p − 2 + ( 1 − n)=2, q − 2 + (1 − m)=2 ¡ 0. There are two cases for them: (i) 2 ¿ (1 − m)=2 or 2 ¿ ( 1 − n)=2. In this case if p − 2 + ( 1 − n)=2, q − 2 + (1 − m)=2 ¿ 0, then pq ¿ (2 − (1 − m)=2)( 2 − ( 1 − n)=2); if p − 2 + $; q − 2 + % ¡ 0, then pq ¡ (2 − (1 − m)=2)( 2 − ( 1 − n)=2), and hence 1 ; 2 ¿ 0. (ii) 2 6 (1 − m)=2, 2 6 ( 1 − n)=2 with pq ¿ ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ). Proof of Theorem 1. At Irst consider the subcase of 2 6 (1 − m)=2, 2 6 ( 1 − n)=2. Construct u(x; t) = log v(x; t) = log
[’A(1 −m)=2
A ; + (1 − ct)K ]2=(1 −m)
[’B( 1 −n)=2
B ; + (1 − ct)L ]2=( 1 −n)
1 (x; t) ∈ C × [0; ); c 1 (x; t) ∈ C × [0; ); c
where ’ = M1 ’0 , ’0 is the normalized Irst eigenfunction of (1.19), M1 ; A; B; c; K; L are positive constants to be determined. Noticing ’0 = 0 on @, we have by using (1.20) that @u 2A(1 −m)=2 M1 c2 ; 6 @ (1 − m)(1 − ct)K @v 2B( 1 −n)=2 M1 c2 ; 6 @ ( 1 − n)(1 − ct)L
epv+2 u = equ+ 2 v =
A 2 B p (1 −
ct)22 K=(1 −m)+2pL=( 1 −n) Aq B 2
(1 − ct)2qK=(1 −m)+2 2 L=( 1 −n)
;
(2.1)
(2.2)
hold on @×(0; T ). The assumption 1= 1 ; 1= 2 ¿ 0 with 2 6 (1 −m)=2 and 2 6 ( 1 − n)=2 implies
1 − n 1 − m − 2 − 2 ; pq ¿ 2 2
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
and hence there exist positive A; B; K; L large such that 1=p 1=(( 1 −n)=2− 2 ) 2M1 c2
1 − n ((1 −m)=2−2 )=p ¡B¡ Aq=(( 1 −n)=2− 2 ) ; A 1 − m 2M1 c2
1 − n(1 − m)=2 − 2
1 − nq K; K ¡L¡ 1 − m( 1 − n)=2 − 2 1 − mp which are equivalent to 2M1 c2 ( 1 −n)=2 B ¡ Aq B 2 ;
1 − n
2M1 c2 (1 −m)=2 ¡ B p A 2 ; A 1 − m K(1 − m − 22 ) 2Lp ¡ ; 1 − m
1 − n
2Kq L( 1 − n − 2 2 ) : ¡ 1 − m
1 − n
By the relationships above, we obtain from (2.1) and (2.2) that @u 6 epv+2 u ; @
@v 6 equ+ 2 v @
on @ × (0; T ):
(2.3)
As for the estimates in × (0; T ), a simple computation shows ut =
Kc(1 − ct)K−1 ; + (1 − ct)K
’A(1 −m)=2
e1 u =
A 1 ; [’A(1 −m)=2 + (1 − ct)K ]21 =(1 −m)
2mA(1 +m)=2 1 ’ (1 − m)[’A(1 −m)=2 + (1 − ct)K ](1 +m)=(1 −m) +
2m(1 + m)A1 |∇’|2 : (1 − m)2 [’A(1 −m)=2 + (1 − ct)K ]21 =(1 −m)
Notice |∇’| = |∇(M1 ’0 )| ¿ M1 c1 =2 ’ = M1 ’0 ¿ M1 c3
for x ∈ {x ∈ : dist(x; @) 6 "};
for x ∈ {x ∈ : dist(x; @) ¿ "}:
Choose M1 and c satisfying m1 M12 c32 ¿ a1 (1 − m);
m(1 + m)M12 c12 ¿ 4a1 (1 − m)2
and c ¡ a1 =K(M1 A(1 −m)=2 + 1)(1 +m)=(1 −m) with K ¿ 1. Thus, we have u t 6
in × (0; T ):
(2.4)
Similarly, we can get vt 6
in × (0; T ):
(2.5)
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C If in addition the initial data are large such that u0 (x) ¿ u(x; 0), v0 (x) ¿ v(x; 0) on , then together (2.3)–(2.5), (u; v) is a pair of sub-solution of (1.1)–(1.3), which means that the solutions of (1.1)–(1.3) will blow up in Inite time. For the subcase of 2 ¿ (1 − m)=2 or 2 ¿ ( 1 − n)=2, denote w = ut and z = vt . Then w and z satisfy wt = m<(emu w) − a1 1 e1 u w;
vt = n<(env z) − a2 1 e 1 v z
@w = epv+2 u (2 w + pz); @
@v = equ+ 2 v (qw + 2 z) @
w(x; 0) =
z(x; 0) =
in × (0; T ); (2.6)
on @ × (0; T ); C on :
(2.7) (2.8)
Due to the positivity of exponential functions in system (2.6)–(2.8), we know by using the classical comparison principle (see, e.g., [13, Lemma 5.1, p. 480]) with the assumption
in × (0; T );
(2.9)
@U = epv0 e2 U @
on @ × (0; T );
(2.10)
U (x; 0) = u0 (x)
C on :
(2.11)
It follows from Proposition 1 that the solutions of (2.9)–(2.11) will blow up in Inite time for large initial value. We can choose (U; v0 ) as a pair of sub-solution of (1.1)–(1.3) to get the blow-up conclusion to (1.1)–(1.3) immediately. The proof is complete. Due to the results of Theorem 1, we will always assume 2 6 (1 −m)=2, 2 6 ( 1 − n)=2 in the sequels.
3. Proof of Theorem 2 This section treats the case 1= 1 ; 1= 2 ¡ 0 with 2 6 (1 − m)=2 and 2 6 ( 1 − n)=2. Proof of Theorem 2. Construct time-independent functions u(x; C t) = log
A ; 2 − (1 − ’)A%
v(x; C t) = log
B ; 2 − (1 − ’)B$
(x; t) ∈ C × [0; T );
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
where % = (1 − m)=2, $ = ( 1 − n)=2, ’ = "1 ’0 , ’0 is the normalized Irst eigenfunction of (1.19), and "1 ; A; B are positive constants to be determined. Then
mAm+% 1 ’(1 − ’)A
%
−1
+ mAm+% (A(1 −m)=2 − 1)|∇’|2 (1 − ’)A (2 − (1 − ’)A% )m+1
%
−2
%
m(m + 1)A1 (1 − ’)2A −2 |∇’|2 + ; (2 − (1 − ’)A% )m+2 and hence
"1 = min
a1 a2 ; ; 1 ; (m1 + m(m + 2)c42 )21 (n1 + n(n + 2)c42 )2 1
then
in :
(3.1)
Moreover, we have @uC ¿ A(1 −m)=2 "1 c1 ; @
C 2 uC epv+ = A2 B p
@vC ¿ B( 1 −n)=2 "1 c1 ; @
C 2 vC equ+
= Aq B 2
on @ × (0; T ); on @ × (0; T ):
The assumption 1= 1 ; 1= 2 ¡ 0 with 2 6 (1 − m)=2 and 2 6 ( 1 − n)=2 implies pq ¡ ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ). So, there exist A; B large such that q=(( 1 −n)=2− 2 ) ((1 −m)=2−2 )=p 1 1 q=(( 1 −n)=2− 2 ) A ¡B¡ A((1 −m)=2−2 )=p "1 c1 "1 c1 or equivalently, Aq B 2 ¡ "1 c1 B( 1 −n)=2 ;
A2 Bp ¡ "1 c1 A(1 −m)=2 ;
by which we can obtain @uC C 2 uC ; ¿ epv+ @
@vC C 2 vC ¿ equ+
@
on @:
(3.2)
C Moreover, we can choose A; B large enough such that uC ¿ u0 (x), vC ¿ v0 (x) on . Together with (3.1), (3.2), we know that the time-independent (u; C v) C is a pair of super solution of (1.1)–(1.3), and hence the solutions of (1.1)–(1.3) are globally bounded.
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
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4. Proof of Theorem 3 Finally, we discuss the critical case of (1= 1 ; 1= 2 ) = (0; 0), or equivalently, pq = ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ). Proof of Theorem 3. Consider the subcase (i) at Irst. Construct time-independent functions u(x; C t) = log
c1−1
A ; + 1 − c1−1 (1 − ’0 )A%
v(x; C t) = log
c1−1
B + 1 − c1−1 (1 − ’0 )B$
for (x; t) ∈ C × [0; T ), where ’0 is the normalized Irst eigenfunction of (1.19), A; B are positive constants to be determined later. Due to the assumption a1 ¿ mc1−2−1 (c1−1 + 1)1 (1 c1 + (m + 2)c42 ); a2 ¿ nc1−2− 1 (c1−1 + 1) 1 (1 c1 + (n + 2)c42 ); it is easy to get that
in × (0; T ):
(4.1)
Moreover, we have @uC ¿ A(1 −m)=2 ; @
C 2 uC epv+ = A2 B p
@vC ¿ B( 1 −n)=2 ; @
C 2 vC equ+
= Aq B 2
on @ × (0; T ); on @ × (0; T ):
Since pq = ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ) implies q=(( 1 − n)=2 − 2 ) = ((1 − m)=2 − 2 )=p, we know A((1 −m)=2−2 )=p = B = Aq=(( 1 −n)=2− 2 ) or equivalently, A(1 −m)=2 = A2 Bp ;
B( 1 −n)=2 = Aq B 2
hold for any A ¿ 0. Thus, we have @uC C 2 uC ; ¿ epv+ @
@vC C 2 vC ¿ equ+
@
on @ × (0; T ):
(4.2)
Moreover, we can choose A; B large enough such that uC ¿ u0 (x);
vC ¿ v0 (x)
C on :
(4.3)
Inequalities (4.1)–(4.3) show that the time-independent (u; C v) C is a pair of super solution of (1.1)–(1.3), and hence the solutions of (1.1)–(1.3) are globally bounded.
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S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
Subcase (ii). Construct u(x; t) = log
A ; [’A(1 −m)=2 + (1 − ct)K ]2=(1 −m)
v(x; t) = log
B ; [’B( 1 −n)=2 + (1 − ct)L ]2=( 1 −n)
1 (x; t) ∈ C × [0; ); c 1 (x; t) ∈ C × [0; ); c
where ’ = ’0 c2−1 min((1 − m)=2; ( 1 − n)=2), ’0 is the normalized Irst eigenfunction of (1.19), and A; B; c; K; L are positive constants to be determined. By using the assumption a1 6
mC 2 min(1 (1 − m)c32 ; (1 + m)c12 ) ; 4(1 − m)2 c22
a2 6
nC 2 min(1 ( 1 − n)c32 ; ( 1 + n)c12 ) 4( 1 − n)2 c22
with C = min((1 − m)=2; ( 1 − n)=2), we can obtain that u t 6
vt 6
in × (0; T ):
(4.4)
We have moreover on the boundary @ that @u A(1 −m)=2 6 ; @ (1 − ct)K
epv+2 u =
@v B( 1 −n)=2 ; 6 @ (1 − ct)L
equ+ 2 v =
A 2 B p (1 −
ct)22 K=(1 −m)+2pL=( 1 −n) Aq B 2
(1 − ct)2qK=(1 −m)+2 2 L=( 1 −n)
;
:
As pq = ((1 − m)=2 − 2 )(( 1 − n)=2 − 2 ), there exist positive A; B; K; L large such that A((1 −m)=2−2 )=p = B = Aq=(( 1 −n)=2− 2 ) ; q
1 − n (1 − m)=2 − 2
1 − n K; K =L= 1 − m p 1 − m ( 1 − n)=2 − 2 or equivalently, A(1 −m)=2 = Bp A2 ;
B 1 −n=2 = Aq B 2 ;
(1 − m)=2 − 2 p K= L; 1 − m
1 − n
( 1 − n)=2 − 2 q L= K:
1 − n 1 − m
Thereby we can obtain @u 6 epv+2 u ; @
@v 6 equ+ 2 v @
on @ × (0; T ):
(4.5)
S. Zheng, X. Song / Nonlinear Analysis 57 (2004) 519 – 530
529
Now let the initial data be large such that u0 (x) ¿ u(x; 0);
v0 (x) ¿ v(x; 0)
C on :
(4.6)
In summary of (4.4)–(4.6), we know that (u; v) is a pair of sub-solution of (1.1)–(1.3), and hence the solutions of (1.1)–(1.3) will blow up in Inite time with such large initial data. The proof is complete.
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