H 1 -perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-rod Wave Equation

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1 Mediterr. j. math ), / , DOI /s c 006 Birkhäuser Verlag Basel/Switzerland Mediterranean Journal of Mathematics H 1 -perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-rod Wave Equation Mostafa Bendahmane, Giuseppe Maria Coclite and Kenneth H. Karlsen Dedicated to the memory of Professor Aldo Cossu Abstract. We consider a weakly dissipative hyperelastic-rod wave equation or weakly dissipative Camassa Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. We fix a smooth solution and establish the existence of a strongly continuous semigroup of global weak solutions for any initial perturbation from H 1 ). In particular, the supersonic solitary shock waves [8] are included in the analysis. Mathematics Subject Classification 000). 35G5, 35L05, 35A05. Keywords. Shallow water equation, integrable equation, hyperbolic equation, weak solution, existence, uniqueness, stability. 1. Introduction and Statement of the Main esults Consider the equation t u txxu 3 +3u x u + δ xxu = γ x u xxu + u xxxu 3 ), t > 0, x. 1.1) In the case γ =1,δ= 0 it is known as the Camassa-Holm equation and describes unidirectional shallow water waves above a flat bottom: u represents the fluid velocity [1], [1]. The Camassa Holm equation possesses a bi-hamiltonian structure and thus an infinite number of conservation laws) [11], [1] and is completely integrable [1]. From a mathematical point of view the Camassa Holm equation is well studied, see [3] for a complete list of references. In particular, we recall that The research of K.H. Karlsen is supported by an Outstanding Young Investigators Award from the esearch Council of Norway. The current address of G.M. Coclite is Department of Mathematics, University of Bari, Via E. Orabona 4, 7015 Bari, Italy.

2 40 M. Bendahmane, G.M. Coclite and K.H. Karlsen Mediterr. j. math. existence and uniqueness results for global weak solutions have been proved by Constantin and Escher [4], Constantin and Molinet [5], and Xin and Zhang [18], [19], see also Danchin [9], [10]. When δ = 0,itistermedhyperelastic-rod wave equation and describes the finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. The constant γ>0depends on the material constants and the prestress of the rod [6], [7], [8]. The additional weakly dissipative term δ xxu is introduced in [16]. We coin 1.1) the weakly dissipative hyperelastic-rod wave equation. In [3] the authors consider the case δ = 0 and prove the existence of a strongly continuous semigroup of solutions belonging to L + ; H 1 )). On the other hand in [8] it is showed that for δ = 0 and any constants 0 <γ<3, c>0 there exists a ζ such that the following peakon like function is a traveling wave solution of 1.1) Ut, x) = c 1 1 ) + c 3 ) γ γ 1 e x ct ζ / γ, 1.) called supersonic solitary shock wave. It is clear that the analysis in [3] does not cover this kind of solutions which do not belong to L + ; H 1 ))!). In this paper we extend the result of [3] to cover also 1.). oughly speaking, the idea is to look at 1.) as a L + ; H 1 ))-perturbation of a constant state. Indeed we can decompose U in the following way: U = U 1 + U, U 1 := c 1 1 ), U t, x) := c γ 3 γ 1 ) e x ct ζ / γ, where U 1 is a classical solution to 1.1) and U is a perturbation that lies in the space L + ; H 1 )). To be more precise: let ϕ = ϕt, x) be a solution of 1.1) such that ϕ C 3 [0, + ) ), ϕ, t ϕ, x ϕ, txϕ, xxϕ, 3 xϕ L + ), 1.3) this is the case if ϕ is periodic or constant) and v 0 H 1 ), γ > 0, δ. 1.4) We want to study the wellposedness of the Cauchy problem { t u txxu 3 +3u x u + δ xxu = γ x u xxu + u xxxu ) 3, t > 0, x, u0,x)=ϕ0,x)+v 0 x), x. system 1.5) Observe that, at least formally, 1.5) is equivalent to the elliptic-hyperbolic t u + γu x u + x P =0, t > 0, x, xx P + P = 3 γ u + γ x u ) + δ x u, t > 0, x, u0,x)=ϕ0,x)+v 0 x), x. 1.6)

3 Vol ) Weakly Dissipative Hyperelastic-rod Wave Equation 41 Motivated by this remark, we shall use the following definition of weak solution and, in the same spirit of [3, Definition 1.1], we define the admissible perturbations. Definition 1.1. We call u :[0, + ) a weak solution of the Cauchy problem 1.5) if i) u C[0, + ) ); ii) u ϕ L 0,T); H 1 ) ),T>0; iii) u satisfies 1.6) in the sense of distributions; iv) u0,x)=ϕ0,x)+v 0 x), for every x. If,inaddition,foreachT>0there exists a positive constant K T depending only on v 0 H 1 ), ϕ,γ,t,such that ) 4 x ut, x) ϕt, x) γt + K T, t, x) 0,T), 1.7) then we say that u ϕ is an admissible perturbation of 1.1). Our results are collected in the following theorem: Theorem 1.. There exists a strongly continuous semigroup of solutions associated to the Cauchy problem 1.5). More precisely, there exists a map S :t, γ, δ, v 0 ) [0, + ) 0, + ) H 1 ) S t γ,δ,v 0 ) ) C[0, + ) ), with the following properties: i) for each v 0 H 1 ), γ>0, δ the map ut, x) =S t γ,δ,v 0 )x) is a weak solution of 1.5) and u ϕ is an admissible perturbation of 1.1); ii) it is stable with respect to the initial condition and the coefficient in the following sense, if then v 0,n v 0 in H 1 ), γ n γ, δ n δin, 1.8) Sγ n,δ n,v 0,n ) ϕ Sγ,δ,v 0 ) ϕinl [0,T]; H 1 )), 1.9) for every {v 0,n } n N H 1 ), {γ n } n N 0, + ), {δ n } n N, v 0 H 1 ), γ>0, δ, T>0. Moreover, the following statements hold: iii) the estimate 1.7) is valid with γ K T := γ xx ϕ L + ) + γ 3 ) ϕ L + ) e ρt v 0 H1 ) + max { 3 γ, γ } + 3 γ 4 e ρt v 0 H 1 ) 1.10) + 5γ xϕ L + ) +δ γ ) 1/,

4 4 M. Bendahmane, G.M. Coclite and K.H. Karlsen Mediterr. j. math. ρ := 3+γ x ϕ L + ) + γ 3 xxx ϕ L + ) + δ, 1.11) for T>0; iv) there results x Sγ,δ,v 0 ) L p loc + ), 1.1) with 1 p<3. Our argument is based on the analysis of the evolution of the perturbation v := u ϕ. From 1.5) we get the following equation for v t v txx 3 v +3v xv +3ϕ x v +3v x ϕ + δ xx v = γ x v xxv + v xxxv 3 + x v xxϕ + x ϕ xx v + v 3 xxx ϕ + ϕ 3 xxx v), v0, ) =v 0, that is formally equivalent to the elliptic-hyperbolic system t v + γv x v + γv x ϕ + γϕ x v + x P =0, xx P + P = 3 γ v + γ x v ) +3 γ)ϕv + γ x ϕ x v + δ x v, v0, ) =v 0. Since the argument is very similar to the one in [3] we simply sketch it. 1.13) 1.14). Viscous Approximations: Existence and A Priori Estimates We prove the existence of a weak solution to the Cauchy problem 1.13) and, equivalently, to 1.5)) by proving compactness of a sequence of smooth solutions {v ε } ε>0 solving the following viscous problems see []): t v ε + γv ε x v ε + γv ε x ϕ + γϕ x v ε + x P ε = ε xxv ε, xx P ε + P ε = 3 γ vε + γ ) x v ε +3 γ)ϕvε.1) +γ x ϕ x v ε + δ x v ε, v ε 0, ) =v ε,0, which is equivalent to the following fourth order one: t v ε txx 3 v ε +3v ε x v ε +3ϕ x v ε +3v ε x ϕ + δ xx v ε = γ x v ε xx v ε + v ε xxx 3 v ε + x v ε xx ϕ + xϕ xx v ) ε +γ ϕ xxxv 3 ε + v ε xxxϕ ) 3 + ε xxv ε ε xxxxv 4 ε, v ε 0, ) =v ε,0. Formally, sending ε 0 in.),.1) yields 1.13), 1.14), respectively..)

5 Vol ) Weakly Dissipative Hyperelastic-rod Wave Equation 43 We shall assume that v ε,0 H ), v ε,0 H 1 ) v 0 H 1 ), ε > 0, and v ε,0 v 0 in H 1 )..3) The starting point of our analysis is the following wellposedness result for.1) see [, Theorem.3]). Lemma.1. Assume 1.3), 1.4) and.3), letε>0. There exists a unique smooth solution v ε C [0, + ); H ) ) to the Cauchy problem.1). The next step in our analysis is to derive the following a priori estimates: Lemma.. Assume 1.3), 1.4) and.3), andletε>0. Then the following estimates hold: j) Energy Conservation) for each t 0 v ε t, ) t +ε H 1 ) x v ε τ, ) dτ H 1 ) eρt v 0 H 1 ),.4) 0 where ρ is defined in 1.11); jj) Oleinik type Estimate [15]) for any 0 <t<t and x, x v ε t, x) 4 γt + K T,.5) where K T is defined in 1.10); jjj) Higher Integrability Estimate) for every 0 α<1, T>0, anda, b, a< b, there exists a positive constant C T depending only on v 0 H1 ), ϕ, α, T, a and b, butindependentonε, such that T b 0 a x v ε t, x) +α dtdx C T..6) emark.3. Due to [13, Theorem 8.5],.3) and.4), we have for each t 0 v ε t, ) L ) 1 v ε t, ) H 1 ) eρt v 0 H 1 )..7) Proof of Lemma.. We begin with j). Multiplying.) by v ε,integratingon, and integrating by parts we get 1 d v dt ε + x v ε ) ) dx + ε x v ε ) + xxv ε ) ) dx ρ v ε + x v ε ) ) dx, where ρ is defined in 1.11). Hence.4) is a consequence of.3) and the Gronwall Lemma. We continue by proving jj). Introduce the notation q ε := x v ε. From.1) we get the following equation for q ε : t q ε + γ q ε + γv ε x q ε + γv ε xx ϕ + γ xϕq ε δq ε +γϕ x q ε + γ 3 vε +γ 3)ϕv ε + P ε ε xxq ε =0..8)

6 44 M. Bendahmane, G.M. Coclite and K.H. Karlsen Mediterr. j. math. Using the fact that e x / is the Green s function of the operator 1 xx, P ε t, x) = 1 e x y 3 γ vε t, y)+γ x v ε t, y) ) ) +3 γ)ϕt, y)v ε t, y)+γ x ϕt, y) x v ε t, y) dy..9) It follows from.4) and.7) see [3, Proof of Lemma 3.1]) that γv ε xxϕ + γ 3 vε L +3 γ)ϕv ε + P ε L T,.10) 0,T ) ) for some constant L T > 0. Then, from.8), t q ε + γ q ε + γv ε x q ε + γ x ϕq ε δq ε + γϕ x q ε ε xx q ε L T. Since we conclude γ ξ +γ x ϕ δ)ξ γ 4 ξ γ xϕ δ), ξ, γ t q ε + γ 4 q ε + γv ε x q ε + γϕ x q ε ε xx q ε L T +γ x ϕ L + ) +δ γ =: LT. Employing the comparison principle for parabolic equations, we get where h solves Since the map.11) q ε t, x) ht), 0 t T, x,.1) dh dt + γ 4 h = L T, h0) = x v ε,0 L )..13) Ht) := 4 γt + 4 L T γ, t > 0, is a super-solution of.13) in the interval [0,T], due to the comparison principle for ordinary differential equations, we get ht) Ht) for all 0 <t T. Therefore,.5) is proved. Finally, we consider jjj). The argument is very similar to the one of [3, Lemma 4.1]. Pick a cut-off function χ C ) such that { 1, if x [a, b], 0 χ 1, χx) = 0, if x,a 1] [b +1, + ), consider the map θξ) := ξ ξ +1 ) α, ξ, then multiply.8) by χθ q ε ), integrate over 0,T) and use.4).

7 Vol ) Weakly Dissipative Hyperelastic-rod Wave Equation Compactness Lemma 3.1. The family {P ε } ε>0 is uniformly bounded in L [0,T); W 1, )) and L [0,T); H 1 )) for each T>0. Proof. The argument is the same of [3, Lemma 5.1]: use the integral representation of P ε.9) and then employ.7). Lemma 3.. There exists a sequence {ε j } j N tending to zero and a function v L [0,T); H 1 )) H 1 [0,T] ), foreacht 0, such that v εj v weakly in Hloc 1 [0,T] ), foreacht 0, 3.1) v εj v strongly in L loc[0, + ) ). 3.) Proof. Fix T>0. Observe that, from.1), t v ε = ε xx v ε γv ε x v ε γv ε x ϕ γϕ x v ε x P ε, hence, by.7),.4), Lemma 3.1, and the Hölder inequality, {v ε } ε>0 is uniformly bounded in H 1 [0,T] ) L [0,T); H 1 )), and 3.1) follows. Finally, since H 1 ) L loc ) L loc ), 3.) is a consequence of [17, Theorem 5]. Lemma 3.3. The family {P ε } ε>0 is uniformly bounded in W 1,1 loc [0,T) ) for any T>0. In particular, there exists a sequence {ε j } j N tendingtozeroandafunction P L [0,T); W 1, )) such that for each 1 p<+ P εj P strongly in L p loc [0, + ) ). 3.3) Proof. The argument is analogous to the one of [3, Lemma 5.3]. Using the integral representation.9) of P ε and then employing.7) we get the uniform boundedness of { t P ε }ε>0 in L1 loc [0, + ) ). Then, due to Lemma 3.1, {P ε} ε>0 is bounded in W 1,1 loc [0,T) ). Finally, by using again Lemma 3.1, we have the existence of a pointwise converging subsequence which is uniformly bounded in L [0,T) ). Clearly, this implies 3.3). Lemma 3.4. There exist a sequence {ε j } j N tendingtozeroandtwofunctions q L p loc [0, + ) ), q L r loc [0, + ) ) such that q εj q in L p loc [0, + ) ), q ε j q in L loc [0, + ); L )), 3.4) q ε j q in L r loc[0, + ) ), 3.5) for each 1 <p<3 and 1 <r<3/. Moreover, and q t, x) q t, x) for almost every t, x) [0, + ) 3.6) x v = q in the sense of distributions on [0, + ). 3.7) Proof. Formulas 3.4) and 3.5) are direct consequences of Lemma.1 and.6). Inequality 3.6) is true thanks to the weak convergence in 3.5). Finally, 3.7) is a consequence of the definition of q ε, Lemma 3., and 3.4).

8 46 M. Bendahmane, G.M. Coclite and K.H. Karlsen Mediterr. j. math. In the remainig part of the paper, for notational convenience, we replace the sequences {v εj } j N, {q εj } j N, {P εj } j N by {v ε } ε>0, {q ε } ε>0, {P ε } ε>0, respectively. In view of 3.4), we conclude that for any η C 1 ) withη bounded, Lipschitz continuous on and any 1 p<3wehave ηq ε ) ηq) inl p loc [0, + ) ), ηq ε) ηq) inl loc[0, + ); L )). 3.8) Multiplying the equation in.8) by η q ε ), we get t ηq ε )+γ x vɛ + ϕ)ηq ε ) ) ε xxηq ε ) εη q ε ) x ηq ε )) γq ε ηq ε ) + γv ε xxϕη q ε )+γ x ϕ q ε η q ε ) ηq ε ) ) δq ε η q ε ) 3.9) + γ 3 v εη q ε )+ γ q εη q ε )+γ 3)ϕv ε η q ε )+P ε η q ε )=0. Lemma 3.5. For any convex η C 1 ) with η bounded, Lipschitz continuous on, we have t ηq)+γ x v + ϕ)ηq) ) γqηq) + γv xx ϕη q)+γ x ϕ qη q) ηq) ) δqη q) 3.10) + γ 3 v η q)+ γ q η q)+γ 3)ϕvη q)+pη q) 0, in the sense of distributions on [0, + ). Hereqηq), q η q) and η q)q denote the weak limits of q ε ηq ε ), qε η q ε ) and η q ε )q ε in L r loc [0, + ) ), 1 <r<3/, respectively. Proof. By convexity εη q ε ) 0. Hence, due to 3.), 3.4), and 3.5), sending ε 0 in 3.9) yields 3.10). emark 3.6. From 3.4) and 3.5), it is clear that q = q + + q = q + + q, q =q + ) +q ), q = q + ) + q ), almost everywhere in [0, + ), whereξ + := ξχ [0,+ ) ξ), ξ := ξχ,0] ξ), ξ. Moreover, by.5) and 3.4), q ε t, x),qt, x) 4 γt + K T, 0 <t<t, x. 3.11) Lemma 3.7. There holds ) γ t q + γ x v + ϕ)q q + γv xx ϕ + γ 3 v +γ 3)ϕv + P δq =0, 3.1) in the sense of distributions on [0, + ). Proof. Using 3.), 3.3), 3.4), and 3.5), the result 3.1) follows from ε 0in.8). The next lemma contains a renormalized formulation of 3.1).

9 Vol ) Weakly Dissipative Hyperelastic-rod Wave Equation 47 Lemma 3.8 [3, Lemma 5.8]). For any η C 1 ) with η L ), ) q t ηq)+γ x v + ϕ)ηq) γqηq) γ q) η q) + γv xxϕη q)+γ x ϕ qη q) ηq) ) δqηq) 3.13) + γ 3 v η q)+γ 3)ϕvη q)+pη q) =0, in the sense of distributions on [0, + ). Following [3, Section 6] and [18], we improve the weak convergence of q ε in 3.4) to strong convergence and then we have an existence result for 1.5)). The idea is to derive a transport equation for the evolution of the defect measure q q ) t, ) 0, so that if it is zero initially then it will continue to be zero at all later times t>0. The proof is complicated by the fact that we do not have a uniform bound on q ε from below but merely 3.11) and that in.6) we have only α<1. Lemma 3.9. Assume 1.3) and.3). Then for each t 0 ) q + ) t, x) q + ) t, x) dx t e λt e λs Ss, x)[q + s, x) q + s, x)] dsdx, where λ := γ x ϕ L + ) + δ, Ss, x) := vs, x) xx ) γ ϕs, x)+3 v s, x)+3 γ)ϕs, x)vs, x) P s, x). Proof. Let T>0, >K T see.5)). Subtract 3.13) from 3.10) by using the renormalization ξ /, if ξ> η + ξ) := ξ /, if 0 ξ, 0, if ξ<0 0, if ξ>0 η ξ) := ξ /, if ξ 0. ξ /, if ξ< By reasoning as in [3, Lemma 6.4] we get d q + ) dt q + ) ) dx λ q + ) q + ) ) dx + St, x)[q + q + ] dx, for 4/γ K T )) <t<t. First we apply the Gronwall Lemma to the previous inequality on the interval 4/γ K T )),T):

10 48 M. Bendahmane, G.M. Coclite and K.H. Karlsen Mediterr. j. math. q + ) T,x) q + T,x)) ) dx q + ) 4 γ K T ),x) ) 4 q + γ K T ),x)) dx T + e λt s) St, x)[q + q + ] dx. 4 γ K T ) Then, it is enough to send + and to use ) η + q)t, x) η+ qt, x)) dx =0, > ) lim t 0+ see [3, Lemma 6.]). Lemma For any t 0 and any >0, ] [η q)t, x) η q)t, x) dx t eλt e λs γ γ x ϕ +δ) + q)χ, ) q) dsdx 0 t eλt e λs γ γ x ϕ +δ) + q)χ, ) q) dsdx 0 t + γeλt e λs q + ) q + ) ) dsdx 0 t ) +γe λt e λs η q) η q) dsdx 0 t [ ] +e λt e λs Ss, x) η ) q) η ) q) dsdx ) Proof. The argument is very similar to the one of [3, Lemma 6.3]. We begin by subtracting 3.13) from 3.10), using the renormalization η. Then we integrate on and use the Gronwall Lemma and see [3, Lemma 6.]) ) η q)t, x) η qt, x)) dx =0, > ) lim t 0+ Lemma There holds q = q almost everywhere in [0, + ). Proof. We follow the argument of [3, Lemma 6.6]. We add 3.14) and 3.16). Using the concavity of ξ + ξ)χ, ) ξ), the Gronwall Lemma, 3.15), 3.17) and lim q t, x) dx = lim q t, x) dx = x v 0 ) dx, t 0+ t 0+ we conclude that 1 [ q + ) q + ) ] + [ η q) η q) ] ) t, x) dx =0, for each 0 <t<t.

11 Vol ) Weakly Dissipative Hyperelastic-rod Wave Equation 49 By the Fatou Lemma, emark 3.6, and 3.6), sending + yields 0 q q ) t, x) dx 0, 0 <t<t, and, since the argument holds for each T>0, we are done. 4. ProofofTheorem1. In this last section we prove Theorem 1.. The first step consists in the proof of the existence of solutions for 1.5). Lemma 4.1. Assume 1.3) and.3). Then there exists an admissible weak solution of 1.5), satisfying iii) and iv) of Theorem 1.. Proof. The conditions i), ii), iv) of Definition 1.1 are satisfied, due to.3),.4) and Lemma 3.. We have to verify ii). Due to Lemma 3.11, we have q ε q strongly in L loc [0, + ) ). 4.1) Clearly 3.), 3.3), and 4.1) imply that v is a distributional solution of 1.14). Therefore u := v+ϕ is a weak solution of 1.5) and v is an admissible perturbation of 1.1). Finally, iii) and iv) of Theorem 1. are consequences of.5) and.6), respectively. The second step is the existence of the semigroup. Lemma 4.. There exists a strongly continuous semigroup of solutions associated to the Cauchy problem 1.5) S :[0, + ) 0, + ) H 1 ) C[0, + ) ), namely, for each v 0 H 1 ), γ>0, δ the map ut, x) =S t γ,δ,v 0 )x) is an admissible weak solution and u ϕ and admissible perturbation of 1.5). Moreover, iii) and iv) of Theorem 1. are satisfied. Clearly, this lemma is a direct consequence of the following one and of the ones in the previous sections. Lemma 4.3. Let {ε n } n N, {µ n } n N 0, + ) and v, w L [0,T]; H 1 )) H 1 [0,T] ), foreacht 0, be such that ε n,µ n 0 and v εn v, v µn w, strongly in L [0,T]; H 1 )), T > 0. Then, v = w. Proof. Let t>0. From [, Theorem 3.1], we have that v ε t, ) v µ t, ) H1 ) At, ε + µ) v 0,ε v 0,µ H1 ) + Bt, ε + µ) ε µ, with At, ε + µ) =O e t/ε+µ)), Bt, ε + µ) =O e t/ε+µ)),

12 430 M. Bendahmane, G.M. Coclite and K.H. Karlsen Mediterr. j. math. for each ε, µ > 0. Hence, v εn t, ) v µn t, ) H 1 ) c 1 e t/εn+µn) ε n µ n + v 0,εn v 0,µn H 1 )), n N, for some constant c 1 > 0. Choosing suitable subsequences as in [3, Lemma 7.] we get v = w. The third and last step is the stability of the semigroup. Lemma 4.4. The semigroup S defined on [0, + ) 0, + ) H 1 ) satisfies the stability property ii) of Theorem 1.. Proof. Fix ε>0 and denote by S ε the semigroup associated to the viscous problem.1) and S := S ϕ. Choose{v 0,n } n N H 1 ), {γ n } n N 0, + ), {δ n } n N, v 0 H 1 ), γ>0, δ satisfying 1.8). The initial data satisfy v 0,ε,n,v 0,ε H ) and.3). Finally, write v ε,n := S ε γ n,δ n,v 0,n ), v n := Sγ n,δ n,v 0,n ), v := Sγ,δ,v 0 ). Let t>0. Due to Lemmas 3. and 4.1, v n t, ) vt, ) H 1 ) = lim ε 0 v ε,n t, ) v ε t, ) H 1 ). By using [, Theorem 3.1], we have that v ε,n t, ) v ε t, ) H 1 ) At, ε) v 0,n v 0 H 1 ) + Bt, ε) γ n γ + δ n δ ), with At, ε) =Oe T/ε ), Bt, ε) =Oe T/ε ), t [0,T]. Now, by using the same argument as in [3, Lemma 8.1] we prove the claim. Proof of Theorem 1.. It is a direct consequence of Lemmas 4. and 4.4. eferences [1]. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons. Phys. ev. Lett ), [] G.M. Coclite, H. Holden and K.H. Karlsen, Wellposedness of solutions of a parabolicelliptic system. Discrete Contin. Dynam. Systems ), [3] G.M. Coclite, H. Holden and K.H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation. SIAM J. Math. Anal. To appear. [4] A. Constantin and J. Escher, Global weak solutions for a shallow water equation. Indiana Univ. Math. J ), [5] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation. Comm. Math. Phys ), [6] H.-H. Dai, Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods. Wave Motion ), [7] H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney ivlin rod. Acta Mech ),

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14 43 M. Bendahmane, G.M. Coclite and K.H. Karlsen Mediterr. j. math. Kenneth H. Karlsen Centre of Mathematics for Applications CMA) University of Oslo P.O. Box 1053, Blindern N 0316 Oslo Norway kennethk@math.uio.no UL: eceived: October 1, 005 Accepted: April 30, 006