Blow-up for solutions of some linear wave equations with mixed nonlinear boundary conditions

Applied Mathematics and Computation 123 (2001) 133±140 www.elsevier.com/locate/amc

Blow-up for solutions of some linear wave equations with mixed nonlinear boundary conditions A.M. Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA

Abstract This paper deals with the blow-up in solutions of some linear wave equations with mixed nonlinear boundary conditions. The ®nite time blow-up in solutions are caused from the nonlinear boundary conditions even if the initial data are smooth and well de®ned for all times. The pointwise blow-up in exact solution and blow-up in energy solution are investigated. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Adomian decomposition method; Wave equation; Blow-up

1. Introduction The partial di€erential equations, linear or nonlinear, admitting solutions which blow-up in a ®nite time, have been the subject of extensive analytical and numerical studies [3±11]. The behavior of the solution as the blow-up is approached has been the subject of intensive analytical, numerical, and asymptotic studies [5]. The literature on blow-up of solutions has been studied from various points of views. Some sucient conditions for the blow-up of solutions in ®nite time have been established from these investigations. The studies have given valuable insights into some of the initial stages of combustion [5] as well as describing cellular aggregation and self-focusing in

E-mail address: [email protected] (A.M. Wazwaz). 0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 0 ) 0 0 0 6 9 - 2

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A.M. Wazwaz / Appl. Math. Comput. 123 (2001) 133±140

lasers. It was found by Budd et al. [5] among others that many nonlinear evolution partial di€erential equations, that describe combustion models, have solutions which form strong singularities or blow-up in a ®nite time. A point x0 is called a blow-up point if there exists sequences …xn ; tn † such that xn ! x0 ; tn ! T ; u…xn ; tn † ! 1

as n ! 1:

…1†

In [6], the wave equation with nonlinear damping and source terms, on a bounded domain X of Rn : utt

Du ‡ aut jut j

n 1

p 1

ˆ bujuj

;

x 2 X; t > 0

…2†

with the conditions u…x; t† ˆ 0; x 2 oX; t > 0; u…x; 0† ˆ u0 …x†; ut …x; 0† ˆ u1 …x†;

x 2 X;

…3†

where a; b > 0, p; m > 1 has been studied. The interaction between the damping and source terms [10,11] was examined and a global existence for p > m and a blow-up result for p < m were established. Moreover, it was formally shown that the source term causes a ®nite time blow-up for a ˆ 0, and the nonlinear damping term develops global existence for small initial data for b ˆ 0. Budd et al. [5] studied the semilinear parabolic equation for combustion theory ut ˆ uxx ‡ f …u† for …x; t† 2 … L; L†  R‡

…4†

with the Dirichlet boundary and initial equations u… L; t† ˆ u…L; t† ˆ 0; t > 0; u…x; 0† ˆ u0 …x† P 0; in … L; L†:

…5†

The function f …u† > 0 for u > 0 is superlinear term for u  1 and satis®es the well-known necessary blow-up condition Z 1 du < 1: …6† f …u† 1 The two important nonlinear functions f …u† ˆ eu ;

and

f …u† ˆ un ;

n>1

…7†

were investigated in [5]. The blow-up behavior of two reaction±di€usion problems with a quasilinear degenerate di€usion was e€ectively studied. Bandle et al. [4] conducted a useful survey on blow-up in di€usion equations. In that survey, authors collected some of the known results and algorithms and directed the attention to some open problems. Moreover, the main goal of the survey was to highlight important results and tools in the analysis of

A.M. Wazwaz / Appl. Math. Comput. 123 (2001) 133±140

135

the blow-up.Recently, Messaoudi [10,11] examined the following linear partial di€erential equation utt ˆ uxx ;

x 2 I;

t>0

…8†

with the mixed nonlinear boundary conditions a

ux …0; t† ˆ ju…0; t†j u…0; t†; u…x; 0† ˆ f …x†;

a

ux …1; t† ˆ ju…1; t†j u…1; t†;

ut …x; 0† ˆ g…x†;

x2I

t>0 …9†

for a > 0. A local existence theorem was established by Messaoudi [10,11], and for suitably chosen initial data, it was formally shown that the solution blows up in a ®nite time. In [10], it was shown that if f …x† 2 H 2 …I† and g…x† 2 H 1 …I† be given satisfying Z 1   2  jf …1†ja‡2 jf …0†ja‡2 < 0 E0 ˆ g2 …x† ‡ …f 0 …x††2 dx …10† a‡2 0 then any solution of Eq. (8) blows up in a ®nite time. For more details about the blow-up in solution and the proof of the local existence theorem, see [10,11]. In this paper, the linear partial di€erential equation (8) subject to the mixed nonlinear boundary conditions (9) will be approached di€erently, but analytically, to obtain the exact solutions. The Adomian decomposition method [1,2,12±16] will be e€ectively used to approach Eq. (8). The Adomian algorithm assumes a series solution for the unknown solution u…x; t†. Unlike the method of separation of variables that require initial and boundary conditions, the decomposition method may provide an analytic solution by using the initial conditions only. The boundary conditions can be used only to justify the obtained result. The exact solution and the energy functional will be investigated numerically to show that the peak of the solution is increasingly narrow as proven by Budd et al. [5]. 2. Analysis of the method In an operator form, Eq. (8) becomes Lu ˆ uxx ;

x 2 I;

t > 0;

…11†

where the di€erential operator L is Lˆ

o2 ot2

…12†

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A.M. Wazwaz / Appl. Math. Comput. 123 (2001) 133±140

so that L

1

is a twofold integral operator Z tZ t …
† dt dt: L 1 …
† ˆ 0

Operating with L

…13†

0

1

on both sides of (11) and using the initial conditions yields

u…x; t† ˆ f …x† ‡ tg…x† ‡ L 1 …uxx †:

…14†

The Adomian decomposition method [1,2,12±16] assumes a series solution for u…x; t† given by an in®nite sum of components u…x; t† ˆ

1 X

un …x; t†;

…15†

nˆ0

where the components un …x; t† will be determined recursively. Substituting (15) into both sides of (14) gives ! ! 1 1 X X 1 un …x; t† ˆ f …x† ‡ tg…x† ‡ L un …x; t† : …16† nˆ0

nˆ0

xx

If the series (15) is convergent, then we can determine the components un …x; t†; n P 0 recursively by using the recurrence relation u0 …x; t† ˆ f …x† ‡ tg…x†; uk‡1 …x; t† ˆ L 1 ……uk …x; t††xx †;

k P 0:

…17†

Consequently, the components are determined by u0 …x; t† ˆ f …x† ‡ tg…x†; u1 …x; t† ˆ L 1 …u0xx †; u2 …x; t† ˆ L 1 …u1xx †; u3 …x; t† ˆ L 1 …u2xx †; .. . un …x; t† ˆ L 1 …un .. .

1xx †;

…18†

It is clear that we can proceed further to determine more terms as far as we like. For numerical purposes, the solution can be enhanced dramatically by increasing the number of components calculated. The algorithm (18) deter-

A.M. Wazwaz / Appl. Math. Comput. 123 (2001) 133±140

137

mines un …x; t†, n P 0 and hence the solution in a series form follows immediately. The obtained series can be used to give an insight into the character and behavior of the solution just as in a closed form solution. The obtained series solution can be used to determine the exact solution, or it can be combined with the Pade approximants to examine the blow-up behavior in physical problems. An interesting remark can be made here. The components in (18) of the series solution are obtained by using the initial conditions only. As a result, the diculty of using the nonlinear boundary conditions has been overcome. This is a signi®cant advantage of the decomposition method. Two conclusions can be made here. The pointwise blow-up of the solution can be examined closely through the obtained series solution. However, Messaoudi [10,11], among others, introduced the following de®nition for the energy functional F …t† ˆ

1 2

Z

1 0

1 u2 …x; t† dx ‡ b…t ‡ t0 †2 ; 2

t>0

…19†

for t0 > 0 and b > 0, where both will be chosen so small. Having determined u…x; t†, the blow-up in energy solution can be examined by using (19). To give a clear overview of the discussion presented above, the example introduced in [10] will be investigated.

3. Numerical experiments Example. In this example we consider the linear wave equation: utt ˆ uxx ;

x 2 …0; 1†;

t>0

…20†

with the mixed nonlinear boundary conditions ux …x; t† ˆ ju…x; t†j

1=3

u…x; t†; 3

x† ;

u…x; 0† ˆ 27…2

x ˆ 0; 1;

ut …x; 0† ˆ 81…2

t > 0; 4

x† :

…21†

Following the discussion presented above, we obtain the recurrence relation u0 …x; t† ˆ

27 …2

x†

3

‡

81 …2

uk‡1 …x; t† ˆ L 1 …ukxx †;

x†

4

t;

k P 0:

…22†

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A.M. Wazwaz / Appl. Math. Comput. 123 (2001) 133±140

In view of (22), the ®rst few components are u0 …x; t† ˆ

27 …2

x†

3

‡

81 …2

x†

4

t;

u1 …x; t† ˆ L 1 u1xx 162 2 270 3 ˆ t ‡ t; 5 …2 x† …2 x†6 u2 …x; t† ˆ L 1 u1xx 405 4 567 5 ˆ t ‡ t; 7 8 …2 x† …2 x† u3 …x; t† ˆ L 1 u2xx 756 6 972 ˆ t ‡ t7 9 10 …2 x† …2 x†

…23†

and so on. Substituting (23) into (15), the solution in a series form is u…x; t† ˆ

27 3

…2 ‡

81 4

t‡

162 5

t2 ‡

270 6

t3 ‡

…2 x† x† …2 x† 756 972 t5 ‡ t6 ‡ t7 ‡


x†8 …2 x†9 …2 x†10

x† 567 …2

‡

…2

405 …2

x†

7

t4 …24†

and therefore, the exact solution u…x; t† ˆ 27…2

x

t†

3

…25†

is readily obtained. This result is in full agreement with the result obtained in [10]. It is obvious that the given boundary conditions justify the exact solution (25). To determine the energy functional, we substitute (25) into (19) to obtain ! 729 1 1 F …t† ˆ …26† 10 …1 t†5 …2 t†5 by selecting B  1. 4. Blow-Up Substituting the given initial conditions (21) in the hypothesis (10) indicates that the condition for a ®nite time blow-up in the solution (25) holds. However, a careful examination of the energy functional (26) shows clearly that a ®nite time blow-up in energy occurs at t ˆ 1. On the other hand, it is clear that as

A.M. Wazwaz / Appl. Math. Comput. 123 (2001) 133±140

139

x ! 1 and t ! 1, then the exact solution u…x; t† in (25) blows-up. This means that as x ! 1; t ! 1;

u…1; 1† ! 1:

…27†

Fig. 1 below shows the pointwise blow-up in the solution (25). It was indicated by Budd et al. [5] that as blow-up is approached, the peak of the solution u…x; t† is increasingly narrow. This can be easily observed by examining Fig. 1. However, Fig. 2 below shows the blow-up in the energy solution (26).

Fig. 1. The pointwise blow-up in the exact solution u…x; t†.

Fig. 2. The blow-up in the energy solution F …t†.

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A.M. Wazwaz / Appl. Math. Comput. 123 (2001) 133±140

5. Discussion The main concern of this work has been to study the blow-up in the energy solution and the exact solution of a linear wave equation with mixed nonlinear boundary conditions. The goal has been achieved by applying Adomian decomposition method and by using the initial conditions only. The blow-up in the solution concerning the pointwise concept and the energy phenomena was studied. The results are in full agreement with other methods used in [10,11]. It was shown by using graphs, that as blow-up is approached, the peak of the solution is increasingly narrow. References [1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994. [2] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988) 501±544. [3] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. 28 (2) (1977) 473±483. [4] C. Bandle, H. Brenner, Blowup in di€usion equations: a survey, J. Comput. Appl. Math. 97 (1998) 3±22. [5] C.J. Budd, G.J. Collins, V.A. Galaktionov, An asymptotic and numerical description of selfsimilar blow-up in quasilinear parabolic equations, J. Comput. Appl. Math. 97 (1998) 51±80. [6] V. Georgiev, G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, Int. J. Di€. Eq. 109 (2) (1994) 295±308. [7] H.A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974) 138±146. [8] W. Liu, The blow-up rate of solutions of semilinear heat equations, J. Di€. Equations 77 (1989) 104±122. [9] N.T. Long, T.N. Diem, On the nonlinear wave equation utt uxx ˆ f …x; t; u; ux ; ut † associated with the mixed homogeneous conditions, Nonlinear Anal. 29 (11) (1997) 1217±1230. [10] S.A. Messaoudi, Blow-up in solutions of a linear wave equation with mixed nonlinear boundary conditions, Technical Report #242, KFUPM, Saudi Arabia, 1999. [11] S.A. Messaoudi, Global existence and blow-up in solutions of a wave equation with mixed boundary conditions, AJSE 25 (1A) (2000) 39±44. [12] A.M. Wazwaz, A First Course in Integral Equations, World Scienti®c, Singapore, 1997. [13] A.M. Wazwaz, Analytical approximations and Pade approximants for Volterra's population model, Appl. Math. Comput. 100 (1999) 13±25. [14] A.M. Wazwaz, The modi®ed decomposition method and Pade approximants for solving the Thomas-Fermi equation, Appl. Math. Comput. 105 (1999) 11±19. [15] A.M. Wazwaz, A reliable modi®cation of Adomian's decomposition method, Appl. Math. Comput. 92 (1998) 1±7. [16] E. Yee, Application of the decomposition method to the solution of the reaction-convection di€usion equation, Appl. Math. Comput. 56 (1993) 1±27.