The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux

Applied Mathematics and Computation 184 (2007) 624–630 www.elsevier.com/locate/amc

The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux Yuanhong Zhi *, Chunlai Mu School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China

Abstract In this paper we are concerned with the quenching behavior of a nonlinear parabolic problem in one-dimensional space, of which the nonlinearity appears both in source term and in Neumann boundary condition. First we proved that the solution quenches only on the left boundary in finite time if the given initial datum is monotone. Then we give the estimate for the quenching rate of the solution near the quenching time.  2006 Elsevier Inc. All rights reserved. Keywords: Quenching time; Quenching rate; Nonlinear parabolic equation; Quenching set; Nonlinear boundary outflux

1. Introduction In this paper we consider the quenching behavior of a one-dimensional nonlinear parabolic problem with nonlinear boundary outflux: p

ut ¼ uxx þ ð1  uÞ ; q

0 < x < 1; 0 < t < 1;

ux ð0; tÞ ¼ u ð0; tÞ; ux ð1; tÞ ¼ 0; uðx; 0Þ ¼ u0 ðxÞ; 0 6 x 6 1;

0 < t < 1;

ð1Þ

where p, q are positive constants, the initial datum u0 : [0, 1] ! (0, 1) is smooth enough and satisfies compatibility condition on the lateral boundary. This problem can be considered as a heat conduction model that incorporates the effects of nonlinear reaction (source) and nonlinear boundary outflux (emission). As to the term of quenching, there have been mainly two ways of definitions: the former requires that the solution approaches a constant but its derivative with respect to time variable t tends to infinity, as (x, t) tends to some point in spatial-time space (for details see [1,2]). The latter, which is now prevailing, requires only the solution tends to a constant, and in some cases is equivalent to the former (for details see Refs. [3,4]). Here we take the way of the latter and give the following definition. *

Corresponding author. E-mail address: [email protected] (Y. Zhi).

0096-3003/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.06.061

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625

Definition 1. We say that the solution u(x, t) to the problem (1) quenches in finite time, if there exists 0 < T < 1, such that lim min uðx; tÞ ¼ 0 or

t!T  06x61

lim max uðx; tÞ ¼ 1:

t!T  06x61

From now on, we denote by T(0 < T < 1) the quenching time of (1). The quenching problem has already a history of 30 years. It was in 1975 that Kawarada introduced first the concept of quenching when considering a famous initial boundary problem for the parabolic equation ut = uxx + 1/(1  u) and derived many interesting results (see [1,2,5] and references therein). In [1] Acker and Walter have considerably sharpened and generalized Kawarada’s results. Then Levine generalized these results in [6]. As far as the problem here is concerned, there have been a great number of papers on the quenching behavior of nonlinear parabolic equations which enjoy more or less similar formulation to (1). In [4,7–10] the authors have dealt with a homogeneous equation with nonlinear boundary conditions. Others considered nonlinear equation with nonlinear boundary conditions. For example, in [11], Deng and Xu considered a problem with nonlinear boundary outflux at one side: ðum Þt ¼ uxx ;

0 < x < 1; t > 0;

ux ð0; tÞ ¼ 0;

ux ð1; tÞ ¼ ub ð1; tÞ;

uðx; 0Þ ¼ u0 ðxÞ;

ð2Þ

t > 0;

0 6 x 6 1;

where 0 < b, m < 1. They showed that u quenches in finite time T (which is usually called the quenching time) and the only quenching point is x = 1, and they also gave the quenching rate near the quenching time T, which e > 0, such that C 6 uð1; tÞðT  tÞ1=ðmþ2bþ1Þ 6 C. e There are in other word says that there exists constants C; C also authors [12,13] who considered nonlinear equations with homogeneous boundary conditions. In [6], Levine considered thoroughly the following problem which greatly generalized the results of Kawarada [2] and Acker and Walter [1]. b

ut ¼ uxx þ eð1  uÞ ;

0 < x < 1; 0 < t < T ;

uð0; tÞ ¼ uð1; tÞ ¼ 0; 0 6 t 6 T ; uðx; 0Þ ¼ u0 ðxÞ; u0 < 1; 0 6 x 6 1; and gave the criteria for the quenching, nonquenching and beyond quenching of the solution. Some authors discussed such quenching problems with which the nonlinearity appears both in source (or sink) and in boundary conditions (see [14] and references therein). Zhao [14] considered a parabolic quenching problem as follows: ut ¼ Du þ up ; x 2 X; t > 0; ou ¼ uq ; x 2 oX; t > 0; om uðx; 0Þ ¼ u0 ðxÞ; x 2 X;

ð3Þ

and showed that the quenching can only occur on the boundary under some conditions upon the initial datum. He also gave the quenching rate estimates which is (T  t)1/(2(q + 1)) if T denotes the quenching time. Dai [15] uxx considered a nonlinear parabolic problem ut  ð1þu 2 Þp ¼ gðuÞ, in QT :¼ (a, a) · (0, T), with which nonlinearity x

appears both in the principal part and in the force term, but with homogeneous Dirichlet datum on the parabolic boundary. He proved the existence of critical length aw and gave the quenching rate estimates. Others also considered the quenching behavior of the solution to the parabolic equations with time delay (see [16] and references therein). Observe that in our problem (1) the nonlinear source term may become singular if u(x, t) ! 1 as ðx; tÞ ! ð^x; ^tÞ, where ð^x; ^tÞ is a point in [0, 1] · (0, 1). On the other hand, the outflux uq(0, t) may also become singular in some finite time. If these two cases may happen, it is certainly hard to deal with, for we must judge which term becomes first singular, (1  u)p or uq(0, t), or simultaneously. At present few author has considered this problem. We will by imposing some conditions upon the initial datum preclude the

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formation of the singularity for the source term, so that it suffice for us to consider only the case of the formation of quenching on the boundary for the solution to problem (1). And then we able to determine quenching set and to derive quenching rate estimates. Now we give a list of our main results of this paper. Theorem 1. Assume that the initial datum satisfies u000 þ ð1  u0 ðxÞÞ

p

ð4Þ

60

for 0 < x < 1. Then there exists a finite time T, such that the solution u to the problem (1) quenches in this time. The following theorem is about quenching set, that is the set:  n o  ^x 2 ½0; 1 lim uð^x; tÞ ¼ 0 or lim uð^x; tÞ ¼ 1 : t!T

t!T

Theorem 2. Assume (4) remains true. Then the quenching can only take place on the boundary of interval (0, 1). Remark 1. By noticing condition (4) and using the Maximum Principle we see ut(x, t) 6 0 in [0, 1] · (0, T), which leads immediately to the impossibility of the formation of the singularity in time T for the source term. By Theorem 2 and the Maximum Principle we have Corollary 1. Assume that the initial datum satisfies u00 ðxÞ P 0; u000 ðxÞ þ ð1  u0 ðxÞÞ

p

60

ð5Þ

in (0, 1). Then the quenching can only occur at the point x = 0 in finite time T. Remark 2. From Corollary 1, we see that in Definition 1 the case of lim max uðx; tÞ ¼ 1

t!T  06x61

will not occur because of our choice of the initial datum. Remark 3. Actually the conditions of (5) is proper because we can easily construct such a function (initial datum) satisfying (5). For example, if we set p = 1, then we can easily check that pffiffiffi pffiffiffi u0 ðxÞ :¼ ðð 2  1Þ=2Þ1=2 ðx þ ð 2  1Þ=2Þ1=2 satisfies the conditions of (5). The two Theorems below give the lower and upper bound for the solution u near the quenching time T. At first, by Walter’s Method of Inequalities (see [1]), we proved the existence of a lower bound for the quenching rate. Theorem 3. There exists a positive constant C1 such that 1

uð0; tÞ P C 1 ðT  tÞ2ðqþ1Þ

as t close to T ;

provided the initial datum satisfies u00 ðxÞ P 0 and u000 ðxÞ þ ð1  u0 ðxÞÞp 6 0 in ð0; 1Þ. In order to obtain the upper bound for the quenching rate of the solution of the problem under considering, it is in some cases useful to transform the quenching problem to a corresponding blow-up problem (for details

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627

see [14]). But to our problem (1), we can not use this method of transformation directly, because if we set u ¼ w1 , then clearly u solves the following: wt ¼ wxx 

2 w2þp 2 ðwx Þ  p; w ðw  1Þ

wx ð0; tÞ ¼ w2þq ð0; tÞ; wðx; 0Þ ¼ ðu0 ðxÞÞ1 ;

0 < x < 1; 0 < t < T ; ð6Þ

wx ð1; tÞ ¼ 0; 0 < t < T ; 0 6 x 6 1;

thus a nonlinear gradient damping term and a nonlinear sink appear in the equation, which make it fairly difficult to derive the lower bound for the blow-up rate of the solution to (6). Therefore we preferred to work with problem (1) directly, and derived following Theorem concerning the upper bound for the quenching rate. Theorem 4. Assume the initial datum satisfies u00 ðxÞ P 0, u000 ðxÞ þ ð1  u0 ðxÞÞp 6 0. Then there exists a constant C 2 > 0, such that 1

uð0; tÞ 6 C 2 ðT  tÞ2ðqþ1Þ as t close to T : Combining Theorems 3 and 4 we have Corollary 2. Assume the initial datum satisfies u00 ðxÞ P 0, u000 ðxÞ þ ð1  u0 ðxÞÞp 6 0. Then, near the quenching time T, the solution u(x, t) to problem (1) has following quenching rate estimate: 1

uð0; tÞ  ðT  tÞ2ðqþ1Þ : Remark 4. From Theorem 4 we have that the upper bound for the quenching rate remains the same as that of [4] in the substitute of b there with q here, but the initial datum there was assumed to be smoother than here. By comparing problem (1) with (2) for the case of m = 1 and (3), we have from Corollary 2 that although the nonlinearity appears also in the source (1  u)p the quenching rates still remain the same as that of [11] and of [14]. Therefore nonlinearity of source term has in fact no effect upon the quenching behavior of problem (1). The rest of this paper is organized as follows. In the next section we prove Theorems 1 and 2. The proofs of Theorems 3 and 4 are given in Section 3. 2. The proof of Theorems 1 and 2 In this section, we prove that the quenching can always occur in finite time T, provided the initial datum satisfies (4), and that the quenching set is the singleton {x = 0}. Proof of Theorem 1. Assume on the contrary u cannot quenche at all times, we will derive a contradiction. Set Z 1 c :¼ uq ð0Þ  ð1  u0 ðxÞÞp dx 0 0

then clearly c > 0 according to (4). R1 Now introduce a mass function: F ðtÞ ¼ 0 uðx; tÞdx; 0 6 t < T . Then Z 1 q p F 0 ðtÞ ¼ ðuð0; tÞÞ þ ð1  uðx; tÞÞ dx 6 c 0

considering the condition (4) and the definition of c. Thus integrating from 0 to t leads to F ðtÞ 6

Z

1

u0 ðxÞdx  ct;

0

which contradicts our assumption. The proof of Theorem 1 is completed.

h

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Y. Zhi, C. Mu / Applied Mathematics and Computation 184 (2007) 624–630

To prove Theorem 2, we need a lower estimate for u(x, t) near T, that is Lemma 1. If (4) is true, then for all 0 < s < T, there exists a constant C3 > 0, such that uðx; tÞ P C 3 ðT  tÞ

1=ðqþ1Þ

on Q [ PQ;

ð7Þ

where Q :¼ (0, 1) · (s, T), PQ is the parabolic boundary of Q. Proof. Let K(x, t) :¼ ut + uq in Q,  > 0 to be specified later. Then K t  K xx  pð1  uÞ

p1

K ¼ quq1 ð1  uÞ

p1

2

 qðq þ 1Þuq2 ðux Þ 6 0 in Q:

K(x, s) 6 0 if  is small enough; Kx(1, t) = 0, Kx(0, t) = quq1K,s 6 t < T. Thus by using the Maximum Principle we get KjQ[PQ 6 0, which implies ut P uq

in Q [ PQ:

Integrating for t to T we derive 1=ðqþ1Þ

uðx; tÞ P ½ð1 þ qÞ

ðT  tÞ

The proof of Lemma 1 is finished.

1=ðqþ1Þ

in Q [ PQ:

h

Proof of Theorem 2. By introducing an auxiliary function H ðx; tÞ ¼ uðx; tÞ  C 4 U0 ðxÞ  C 5 ðT  tÞ

1=ðqþ1Þ

in Q;

where C4, C5 are positive constants to be determined, U0 is the normalized principal eigenfunction correspond2 ing to the principal eigenvalue k0 of the operator  oxo 2 in interval (0, 1), we have: H t ¼ H xx  C 4 k0 þ ð1  uÞp þ C 5

q 1 ðT  tÞqþ1 P H xx 1þq

in Q;

1 if C4, C5 are chosen so small that C 4 6 C 5 ðqþ1ÞT q=ðqþ1Þ . k 0 1

In addition we have H ðx; sÞ P uðx; sÞ  C 4  C 5 T qþ1 P 0, if C4,C5 are small enough, and H ðx; tÞjSQ P 0 if C5 6 C3 and by (7), where SQ is the lateral boundary of Q. From these above and the Maximum Principle, we conclude that H(x, t)j(0,1)·[s,T) P 0, which implies uðx; tÞjð0;1Þ½s;T Þ P C 4 U0 ðxÞ > 0: Then the theorem follows. h 3. The proof of Theorems 3 and 4 In order to give the proofs of Theorems 3 and 4, we need first a lower bound for ux. Lemma 2. If u00 ðxÞ P 0, then there exists a positive constant C6, such that ux ðx; tÞ P C 6 in Q1 :¼ ½0; 1=2  ½T =2; T Þ: Proof. Set hðx; tÞ :¼ ux þ lðx  34Þ, where l > 0 will be specified later. Then we have h(x, T/2) P 0 if l is small, and h(3/4, t) P 0 as T/2 6 t < T, hð0; tÞ ¼ uq ð0; tÞ  l 34 P 0 if l is small, and as well ht  hxx  pð1  uÞ

p1

hP0

in ð0; 3=4Þ  ðT =2; T Þ;

which according to Maximum Principle implies hj[0,3/4] · [T/2, T) P 0. Therefore we have ux P l/4=:C6 in Q1. The lemma is proved. h We then proceed to prove the existence of a lower bound for u(0, t) near the quenching time T.

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Proof of Theorem 3. Notice that under our assumptions on the initial datum we have ut ðx; tÞj½0;1½s;T Þ < 0

ð8Þ

for all 0 < s < T. Set W(x, t) = ut + qu1(ux)2 where  is a positive constant to be specified. After some calculation we have Wt  Wxx  pð1  uÞ

p1

W

¼ 4qu ðux Þ ut þ pqð1  uÞp1 u1 ðux Þ2  2qu3 ðux Þ4  5qu2 ð1  uÞp ðux Þ2  2qu1 ðuxx Þ2 : 2

2

Let A ¼ pqð1  uÞ

p1 1

2

p

4

u ðux Þ  2qu3 ðux Þ  5qu2 ð1  uÞ ðux Þ

2

2

¼ qu3 ðux Þ ½2ðux Þ þ 5uð1  uÞ

p

 pð1  uÞ

2

p1 2

u

¼ qu3 ðux Þ2 ½2ðux Þ2 þ uð1  uÞp1 ð5ð1  uÞ  puÞ Let Q2 :¼ (0, T  s) · (s, T), where s is close enough to T. Then by Lemma 2 AjQ2 P 0 provided s is close enough to T. Thus by Lemma 2 and (8) we have Wt  Wxx  pð1  uÞ

p1

W 6 0;

ðx; tÞ 2 Q2 ;

and Wðx; sÞ 6 0;

WðT  s; tÞ 6 0

if  is small enough, T  s 6 t < T, and Wx ð0; tÞ ¼ qð2  1Þuq1 W  22 qu3q2  2quq1 ð1  uÞp 6 qð2  1Þuq1 W; if  is small enough. Therefore by using the Maximum Principle, we have Wðx; tÞjQ2 6 0, which implies ut ð0; tÞ P q½uð0; tÞ1 ðux Þ2 ð0; tÞ ¼ qu2q1 ð0; tÞ: Now integrating from t to T implies 1

1

uð0; tÞ P ½2qðq þ 1Þ2ðqþ1Þ ðT  tÞ2ðqþ1Þ ; as t close to T. The proof of Theorem 3 is completed.

h

Finally we give the proof of Theorem 4 not by the methods of transforming (1)–(6). Proof of Theorem 4. Due to the hypothesis it is possible to chose a function / 2 C2([0, 1]) such that / P 0, / 0 6 0, /00 P 0, /(0) = 1, /(1) = 0, / 0 (0) = 0, and / 6 uq0 u00 (because u0 is compatible on the lateral boundary, we have /ð0Þ ¼ uq0 u00 ð0Þ ¼ 1). Introduce an auxiliary function J(x, t) = ux  /(x)uq(x, t). Then J ð0; tÞ ¼ ux ð0; tÞ  /ð0Þuq ð0; tÞ ¼ 0;

J ð1; tÞ ¼ 0;

and J ðx; 0Þ ¼ u0 ðxÞ  /ðxÞuq 0 ðxÞ P 0; because of our choice of /. In addition in domain (0, 1) · (0, T) we have J t  J xx ¼ utx  uxxx þ /quq1 ðut  uxx Þ þ /00 uq  2/0 quq1 ux þ qðq þ 1Þ/uq2 ðux Þ ¼ pð1  uÞ

p1

ux þ /quq1 ð1  uÞ

p

2 2

þ /0 uq  2/0 quq1 ux þ qðq þ 1Þ/uq2 ðux Þ P 0;

according to the choice of / and our assumption upon initial datum u0. Therefore with the help of the Maximum Principle we have J(x, t) P 0 in region [0, 1] · [0, T).

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Y. Zhi, C. Mu / Applied Mathematics and Computation 184 (2007) 624–630

Thus, J x ð0; tÞ ¼ limþ h!0

J ð0; tÞ  J ðh; tÞ J ðh; tÞ ¼ limþ P 0: h h h!0

On the other hand, J x ð0; tÞ ¼ ut ð0; tÞ  ð1  uð0; tÞÞ

p

 /0 ð0Þuq ð0; tÞ þ /ð0Þqu2q1 ð0; tÞ:

Therefore we claim that p

ut ð0; tÞ P ð1  uð0; tÞÞ

 qu2p1 þ /0 ð0Þuq ð0; tÞ P qu2q1 :

Integrating from t to T we obtain the desired upper bound estimate with C6 = (2q(q + 1))1/(2(q+1)). The proof of Theorem 4 is thus finished. h Acknowledgements This work was supported by the NNSF of China (10571126) and partly by the Program for New Century Excellent Talents in University. References [1] A. Acker, W. Walter, The quenching problem for nonlinear parabolicdifferential equations, Lecture Notes in Mathematics, 564, Springer-Verlag, 1976, pp. 1–12. 1 [2] H. Kawarada, On solutions of initial boundary value problem for ut ¼ uxx þ 1u , Publ. Res. Inst. Math. Sci. 10 (1975) 729–736. [3] L. Ke, S. Ning, Quenching for degenerate parabolic equations, Nonlinear Anal. TMA 34 (1998) 1123–1135. [4] M. Fila, H.A. Levine, Quenching on the boundary, Nonlinear Anal. TMA 21 (1993) 795–802. [5] H.A. Levine, J.T. Montgomery, The quenching of solutions of some nonlinear parabolic equations, SIAM J. Math. Anal. 11 (1980) 842–847. [6] H.A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Math. Pure Appl. CLV (1989) 243–260. [7] C.Y. Chan, S.I. Yuen, Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput. 121 (2001) 203–209. [8] H.A. Levine, The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions, SIAM J. Math. Appl. 14 (1983) 1139–1153. [9] H.A. Levine, G.M. Lieberman, Quenching of solutions of parabolic equations wiht nonlinear boundary conditions in several dimensions, J. Reine Angew. Math. 345 (1983) 23–38. [10] C.A. Roberts, W.E. Olmstead, Local and non-local boundary quenching, Math. Meth. Appl. Sci. 22 (1999) 1465–1484. [11] K. Deng, M.X. Xu, Quenching for a nonlinear diffusion equation with a singular boundary conditon, Z. Angew. Math. Phys. 50 (1999) 574–584. [12] J. Da´vila, M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation, Trans. Am. Math. Soc. 357 (2004) 1801–1828. [13] T. Salin, Quenching-rate estimate for a reaction diffusion equation with weakly singular reaction term, Dyn. Contin. Discrete. Impuls. Syst. Ser. A Math. Anal. 11 (2004) 469–480. [14] C.L. Zhao, Blow-up and quenching for solutions of some parabolic equations, Ph.D. thesis, Univ. of Louisiana, Lafayette, 2000. [15] Q.Y. Dai, Quenching phenomenon for quasilinear parabolic equation, Acta Math. Sin. 41 (1998) 87–96. [16] Y.P. Chen, C.H. Xie, Quenching for a nonlinear degenerate parabolic equation with time delay, Acta Math. Sin. 24 (2004) 265–274.