The Journal of Nonlinear Sciences and Applications

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1 J. Nonlnear Sc. Appl. 1 (28), no. 2, The Journal of Nonlnear Scences and Applcatons BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS THEODORE K. BONI 1, DIABATE NABONGO 2, AND ROGER B. SERY 3 Abstract. In ths paper, we consder the followng ntal-boundary value problem u tt (x, t) = εlu(x, t) + b(t)f(u(x, t)) u(x, t) = on (, T ), n (, T ), u(x, ) = n, u t (x, ) = n, where ε s a postve parameter, b C 1 (R + ), b(t) >, b (t), t R +, f(s) s a postve, ncreasng and convex functon for nonnegatve values of s. Under some assumptons, we show that, f ε s small enough, then the soluton u of the above problem blows up n a fnte tme, and ts blow-up tme tends to that of the soluton of the followng dfferental equaton { α (t) = b(t)f(α(t)), t >, α() =, α () =. Fnally, we gve some numercal results to llustrate our analyss. 1. Introducton Let be a bounded doman n R N wth smooth boundary. Consder the followng ntal-boundary value problem u tt (x, t) = εlu(x, t) + b(t)f(u(x, t)) n (, T ), (1.1) u(x, t) = on (, T ), (1.2) u(x, ) = n, (1.3) Date: Receved: May 28; Accepted: 23 July 28. Correspondng author. 2 Mathematcs Subject Classfcaton. Prmary 35B4, 35B5; Secondary 35K6. Key words and phrases. Nonlnear wave equaton, blow-up, convergence, numercal blow-up tme. 91

2 92 THEODORE K. BONI, DIABATE NABONGO, AND ROGER B. SERY u t (x, ) = n, (1.4) where ε s a postve parameter, b C 1 (R + ), b(t) >, b (t), t R +. The operator L s defned as follows N ( Lu = a j (x) u ), x x j,j=1 where a j : R, a j C 1 (), a j = a j, 1, j N, and there exsts a constant C > such that N a j (x)ξ ξ j C ξ 2 x ξ = (ξ 1,..., ξ N ) R N,,j=1 where stands for the Eucldean norm of R N. Here (, T ) s the maxmal tme nterval of exstence of the soluton u. The tme T may be fnte or nfnte. When T s nfnte, we say that the soluton u exsts globally. When T s fnte, then the soluton u develops a sngularty n a fnte tme, namely, lm u(, t) =, t T where u(, t) = sup x u(x, t). In ths last case, we say that the soluton u blows up n a fnte tme, and the tme T s called the blow-up tme of the soluton u. Solutons of nonlnear wave equatons whch blow up n a fnte tme have been the subject of nvestgaton of many authors (see [5], [7] [9], [11] [13], [17] [19], and the references cted theren). By standard methods, local exstence, unqueness, blow-up and global exstence have been treated. In ths paper, we are nterested n the asymptotc behavor of the blow-up tme when ε s small enough. Our work was motvated by the paper of Fredman and Lacey n [6], where they have consdered the followng ntal-boundary value problem u t (x, t) = ε u(x, t) + f(u(x, t)) n (, T ), u(x, t) = on (, T ), u(x, ) = u (x) n, where s the Laplacan, f : [, ) (, ) s a C 1 convex, ncreasng functon, ds <, u f(s) (x) s a contnuous functon n. Under some addtonal condtons on the ntal data, they have shown that the soluton of the above problem blows up n a fnte tme, and ts blow-up tme tends to that of the soluton λ(t) of the followng dfferental equaton λ (t) = f(λ(t)), λ() = M, (1.5) as ε goes to zero, where M = sup x u (x). The proof developed n [6] s based on the constructon of upper and lower solutons, and t s dffcult to extend the method n [6] to the problem descrbed n (1.1) (1.4). In ths paper, we prove smlar results. More precsely, frstly, we show that when ε s small enough, then the soluton u of (1.1) (1.4) blows up

3 BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS 93 n a fnte tme, and ts blow-up tme tends to that of the soluton α(t) of the followng dfferental equaton α (t) = b(t)f(α(t)), α() =, α () =. (1.6) A smlar result has been obtaned by N gohsse and Bon n [15] n the case of the phenomenon of quenchng (we say that a soluton quenches n a fnte tme f t reaches a fnte sngular value n a fnte tme). Our paper s wrtten n the followng manner. In the next secton, under some assumptons, we show that the soluton u of (1.1) (1.4) blows up n a fnte tme, and ts blow-up tme goes to that of the soluton α(t) of the dfferental equaton defned n (1.6). Fnally, n the last secton, we gve some numercal results to llustrate our analyss. 2. Blow-up tmes In ths secton, under some assumptons, we show that the soluton u of (1.1) (1.4) blows up n a fnte tme, and ts blow-up tme goes to that of the soluton of the dfferental equaton defned n (1.6) when ε tends to zero. We also prove that the above result remans vald when ε s fxed, and the doman s large enough and s taken as parameter. Before startng, let us recall a well known result. Consder the followng egenvalue problem Lϕ = λϕ n, (2.1) ϕ = on, (2.2) ϕ > n. (2.3) The above problem admts a soluton (ϕ, λ) wth λ >. We can normalze ϕ so that ϕdx = 1. Our frst result s the followng. Theorem 2.1. Let A = λ ds, and assume that the soluton α(t) of the b() f(s) dfferental equaton defned n (1.6) blows up at the tme T e. If ε < A, then the soluton u of (1.1) (1.4) blows up n a fnte tme, and ts blow-up tme T satsfes the followng estmates T T e AT e 2 + o(ε). (2.4) Proof. Snce (, T ) s the maxmal tme nterval on whch the soluton u exsts, our am s to show that T s fnte and satsfes the above nequaltes. Introduce the functon v(t) defned as follows v(t) = ϕ(x)u(x, t)dx for t [, T ). Takng the dervatve of v n t, and usng (1.1), we fnd that v (t) = ε ϕ(x)lu(x, t)dx + b(t) f(u(x, t))ϕ(x)dx for t (, T ).

4 94 THEODORE K. BONI, DIABATE NABONGO, AND ROGER B. SERY Accordng to Green s formula, and makng use of (2.1), we note that Lu(x, t)ϕ(x)dx = u(x, t)lϕ(x)dx = λ ϕ(x)u(x, t)dx for t (, T ), whch mples that v (t) = λεv(t) + b(t) Jensen s nequalty renders f(u(x, t))ϕ(x)dx for t (, T ). v (t) λεv(t) + b(t)f(v(t)) for t (, T ). Ths estmate may be rewrtten n the followng manner ( v (t) b(t)f(v(t)) 1 λεv(t) ) for t (, T ). b(t)f(v(t)) We observe that b(t) b() for t (, T ), and sup t t f(t) dσ t dσ = sup f(σ) t f(σ), because f(s) s an ncreasng functon for nonnegatve values of s. Usng these observatons, we arrve at Set v (t) (1 εa)b(t)f(v(t)) for t (, T ). (2.5) ( ) t w(t) = v 1 εa for t [, 1 εat ). A straghtforward computaton reveals that ( ) t w (t) b f(w(t)) for t [, 1 εat ). 1 εa Snce b(s) s nondecreasng for nonnegatve values of s, we dscover that It s not hard to check that w (t) b(t)f(w(t)) for t [, 1 εat ). (2.6) Integrate the nequalty (2.6) over (, t) to obtan w (t) t w() = for w () =. (2.7) b(s)f(w(s))ds for t [, 1 εat ). (2.8) Recall that α(t) s the soluton of the followng dfferental equaton whch mples that α (t) = α (t) = b(t)f(α(t)), t [, T e ), α() =, α () =, t [, T e ), t b(s)f(α(s))ds for t [, T e ). (2.9)

5 BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS 95 Snce w() = v(), thanks to (2.8) and (2.9), an applcaton of the maxmum prncple gves w(t) α(t) for t [, T ), (2.1) where T = mn{t e, 1 εat }. We deduce that T T e 1 εat. (2.11) To prove ths estmate, we argue by contradcton. Suppose that T > Te 1 εat = T. From (2.1), we observe that w blows up at the tme T e, because w(t e ) α(t e ) =, whch mples that T e v(t ) = v( ) = w(t e ) =. (2.12) 1 εat Let us notce that u(, t) v(t) for t (, T ). Snce v blows up at the tme T, owng to the above estmate, t s easy to check that u also blows up at the tme T. But, ths contradcts the fact that (, T ) s the maxmal tme nterval of exstence of the soluton u. Now, ntroduce the functon U(t) defned as follows U(t) = max u(x, t) for t [, T ). (2.13) x We know that there exsts x such that U(t) = u(x, t) for t (, T ). It s not hard to see that Lu(x, t) for t (, T ). Makng use of (1.1), we see that U (t) b(t)f(u(t)), t (, T ). (2.14) Obvously, because of (1.3) and (1.4), we also have U() = and U () =. (2.15) Integratng the nequalty (2.14) from to t, we fnd that U (t) t b(s)f(u(s))ds for t (, T ). (2.16) Snce U() = α(), usng (2.9) and (2.16), an applcaton of the maxmum prncple renders where T = mn{t, T e }. We deduce that U(t) α(t) for t (, T ), (2.17) T T e. (2.18) Indeed, assume that T < T e. Takng nto account (2.17), we observe that U(T ) α(t ) <. But, ths contradcts the fact that (, T ) s the maxmal tme nterval of exstence of the soluton u. Apply Taylor s expanson to obtan 1 = εa + o(ε). (2.19) 1 εa 2 Use (2.11), (2.18) and the above relaton to complete the rest of the proof.

6 96 THEODORE K. BONI, DIABATE NABONGO, AND ROGER B. SERY Remark 2.2. If b(t) = 1, then the soluton α(t) defned n (1.6) satsfes α (t) = f(α(t)), t (, T e ), (2.2) Multply both sdes of (2.2) by α (t) to obtan α() =, α () =. (2.21) ( (α (t)) 2 ) = (F (α(t))) t for t (, T e ), (2.22) 2 where F (s) = s f(σ)dσ. Integratng the equalty (2.22) over (, t), we fnd that whch mples that (α (t)) 2 Let us notce that f the ntegral 2 = F (α(t)) for t (, T e ), α (t) = 2F (α(t)) for t (, T e ). tme T e = 1 dσ 2. In fact, we observe that F (σ) dσ s fnte, then α(t) blows up at the F (σ) dσ F (σ) = 2dt for t (, T e ). Integrate the above equalty over (, T e ) to arrve at T e = 1 dσ. (2.23) 2 F (σ) If f(s) = e s, then F (s) = e s 1. In ths case T e = 1 dσ 2 e, and ts value s σ 1 slghtly equal Remark 2.3. Assume that Drchlet boundary condton (1.2) s replaced by that of Robn, that s, u η + β(x)u = for (, T ), (2.24) where β C ( ), β(x) > on, u η = N,j=1 a j cos(ν, x ) u x j, ν s the exteror normal unt vector on. Consder the followng egenvalue problem ψ η Lψ = λψ n, (2.25) + β(x)ψ = for, (2.26) ψ(x) > n. (2.27)

7 BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS 97 We know that the above egenvalue problem admts a soluton (ψ, λ) wth λ >, and we can normalze ψ so that ψ(x)dx = 1. Introduce the functon v(t) defned as follows v(t) = u(x, t)ψ(x)dx n t [, T ). Take twce the dervatve of v n t and use (1.1), to obtan v (t) = ε Lu(x, t)ψ(x)dx + b(t) f(u(x, t))ψ(x)dx for t (, T ). By Green s formula, we have Lu(x, t)ψ(x)dx = u(x, t)lψ(x)dx + ψ(x) u(x, t) ds η u(x, t) ψ(x) η ds. Usng (2.24) (2.26), we fnd that Lu(x, t)ψ(x)dx = λ u(x, t)ψ(x)dx. Now, reasonng as n the proof of Theorem 2.1, we see that the result of Theorem 2.1 remans vald. Now, let us show that we can obtan a result as the one gven n Theorem 2.1 f the parameter ε s fxed, and the doman s large enough. Assume that the doman contans the ball B(, R) = {x R N ; x < R}. Snce B(, R), we know that the egenvalue λ defned n (2.1) obeys the followng estmates < λ λ R = D R 2, (2.28) where D s a postve constant whch depends only on the upper bound of the coeffcents of the operator L and the dmenson N. At the moment, we may state our result n the case where the doman s large enough. Theorem 2.4. Assume that dst(, ) >. Suppose that the soluton α(t) of the dfferental equaton defned n (1.6) blows up at the tme T e and let E = D dσ. If dst(, ) > εe, then the soluton u of (1.1) (1.4) blows up n b() f(σ) a fnte tme, and ts blow-up tme T obeys the followng estmates εet e T T e 2(dst(, )) + o( 1 2 (dst(, )) ). 2 Proof. As n the proof of Theorem 2.1, t s not dffcult to see that the soluton u of (1.1) (1.4) blows up n a fnte tme T whch obeys the followng estmates T e T e T, (2.29) 1 εa where A = λ dσ. b() f(σ) Thanks to (2.28), λ D, whch mples that A R 2 R = dst(, ). We deduce from (2.29) that D b()r 2 dσ f(σ) = E R 2, where T e T T e 1 εe R 2. (2.3)

8 98 THEODORE K. BONI, DIABATE NABONGO, AND ROGER B. SERY Apply Taylor s expanson to obtan 1 εe = εe 2R + o( 1 ). (2.31) 2 R2 R 2 Use (2.3), and the above relaton to complete the rest of the proof. A drect consequence of Theorem 2.4 s that, f the soluton α(t) of the dfferental equaton defned n (1.6) blows up at the tme T e, and = R N, then the soluton u of (1.1) (1.4) blows up at the tme T, and the followng relaton holds T = T e. 3. Numercal results In ths secton, we gve some computatonal results to confrm the theory establshed n the prevous secton. We consder the radal symmetrc soluton of (1.1) (1.4) when = B(, 1), L =, b(t) = 1, and f(u) = e u. Hence, the problem (1.1) (1.4) may be rewrtten as follows u tt = ε(u rr + N 1 u r ) + e u, r (, 1), t (, T ), (3.1) r u r (, t) =, u(1, t) =, t (, T ), (3.2) u(r, ) =, u t (r, ) =, r (, 1). (3.3) Let I be a postve nteger and let h = 1/I. Defne the grd x = h, I and approxmate the soluton u of (3.1) (3.3) by the soluton U (n) h = (U (n),..., U (n) I ) T of the followng explct scheme U (n+1) 2U (n) + U (n 1) t 2 n = εn (n) 2U 1 2U (n) h 2 + e U (n), U (n+1) 2U (n) t 2 n + U (n 1) = ε( U (n) (n) +1 2U + U (n) 1 + h 2 +e U (n), 1 I 1, (N 1) U (n) h +1 U (n) 1 ) 2h U (n) I =, U () =, U (1) =, I. In order to permt the dscrete soluton to reproduce the propertes of the contnuous one when the tme t approaches the blow-up tme T, we need to adapt the sze of the tme step so that we take t n = mn{h 2, e 1 (n) U 2 h } wth U (n) h = sup I U (n). We also approxmate the soluton u of (3.1) (3.3) by the soluton of the mplct scheme below U (n) h U (n+1) 2U (n) + U (n 1) t 2 n = εn (n+1) 2U 1 2U (n+1) h 2 + e U (n),

9 BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS 99 U (n+1) 2U (n) t 2 n + U (n 1) = ε( U (n+1) +1 2U (n+1) + U (n+1) 1 + h 2 +e U (n), 1 I 1, (N 1) U (n+1) h +1 U (n+1) 1 ) 2h U (n+1) I =, U () =, U (1) =, I. As n the case of the explct scheme, here, we also choose t n = mn{h 2, e 1 (n) U 2 h }. We need the followng defnton. Defnton 3.1. We say that the dscrete soluton U (n) h of the explct scheme or the mplct scheme blows up n a fnte tme f lm n U (n) h =, and the seres n= t n converges, where U (n) h = sup I U (n). The quantty n= t n s called the numercal blow-up tme of the dscrete soluton U (n) h. In the tables 1 and 2, n rows, we present the numercal blow-up tmes, the numbers of teratons n, the CPU tmes and the orders of the approxmatons correspondng to meshes of 16, 32, 64, 128. We take for the numercal blow-up tme T n = n 1 j= t j whch s computed at the frst tme when t n = T n+1 T n The order(s) of the method s computed from s = log((t 4h T 2h )/(T 2h T h )). log(2) Numercal experments for N=2; ε = 1 5 Table 1. Numercal blow-up tmes, numbers of teratons, CPU tmes (seconds) and orders of the approxmatons obtaned wth the explct Euler method I T n n CP U t s

10 1 THEODORE K. BONI, DIABATE NABONGO, AND ROGER B. SERY Table 2. Numercal blow-up tmes, numbers of teratons, CPU tmes (seconds) and orders of the approxmatons obtaned wth the mplct Euler method I T n n CP U t s References 1. L. M. Aba, J. C. López-Marcos and J. Martínez, On the blow-up tme convergence of semdscretzatons of reacton-dffuson equatons, Appl. Numer. Math., 26 (1998), T. K. Bon, Extncton for dscretzatons of some semlnear parabolc equatons, C.R.A.S, Sere I, 333 (21), T. K. Bon, On blow-up and asymptotc behavor of solutons to a nonlnear parabolc equaton of second order wth nonlnear boundary condtons, Comment. Math. Unv. Comenan, 4 (1999), H. Brezs, T. Cazenave, Y. Martel and A. Ramandrsoa, Blow-up for u t = u xx + g(u) revsted, Adv. Dff. Eq., 1 (1996), K. Deng, Nonexstence of global solutons of a nonlnear hyperbolc system, Trans. Am. Math. Soc., 349 (1997), A. Fredman and A. A. Lacey, he blow-up tme for solutons of nonlnear heat equatons wth small dffuson, SIAM J. Math. Anal., 18 (1987), , 1 7. R. T. Glassey, Blow-up Theorems for nonlnear wave equatons, Math. Z., 132 (1973), V. Georgev and G. Todorova, Exstence of a soluton of the wave equaton wth nonlnear Dampng and Source Trems, J. of Dff. Equat., 19 (1994), G. Guowang and W. Shubn, Exstence and nonexstence of global solutons for the generalzed IMBq equaton, Nonl. Anal. TMA, 36 (1999), S. Kaplan, On the growth of solutons of quas-lnear parabolc equatons, Comm. Pure Appl. Math., 16 (1963), H. A. Levne, Instablty and nonexstence of global solutons to nonlnear wave equatons of the form ρu tt = Av + F (u), Trans. Am. Math. Soc., 192 (1974), H. A. Levne, Some addtonal remarks on the nonexstence of global solutons to nonlnear wave equatons, SIAM J. Math. Anal., 5 (1974), R. C. Maccamy and V. J. Mzel, Exstence and nonexstence n the lare of solutons of quaslnear wave equatons, Arch. Ratonal Meth. Anal., 25 (1967), T. Nakagawa, Blowng up on the fnte dfference soluton to u t = u xx + u 2, Appl. Math. Optm., 2 (1976), F. K. N gohsse and T. K. Bon, Quenchng tme of some nonlnear wave equatons, To appear M. H. Protter and H. F. Wenberger, Maxmum prncples n dfferental equatons, Prentce Hall, Englewood Clffs, NJ, (1967). 17. M. Reed, Abstract nonlnear wave equatons, Lecture notes n Mathematcs 57, Sprnger- Verlag Berln, New-York, (1976). 1

11 BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS D. H. Sattnger, On global soluton of nonlnear hyperbolc equatons, Arch. Ratonal Mech. Anal., 3 (1968), A. Sang and R. Park, Remarks on quaslnear hyperbolc equatons wth dynamc boundary condtons, Math. Narchr., 198 (1999), W. Walter, Dfferental-und Integral-Unglechungen, Sprnger, Berln, (1954). 1 Insttut Natonal Polytechnque Houphouet-Bogny de Yamoussoukro, BP 193 Yamoussoukro, (Cote d Ivore). E-mal address: theokbon@yahoo.fr. 2 Unverste d Abobo-Adjame, UFR-SFA, Departement de Mathematques et Informatques, 16 BP 372 Abdjan 16, (Cote d Ivore) E-mal address: nabongo dabate@yahoo.fr 3 Insttut Natonal Polytechnque Houphouet-Bogny de Yamoussoukro, BP 193 Yamoussoukro, (Cote d Ivore). E-mal address: serybrc@yahoo.fr