Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation

Transcription

1 J. Math. Anal. Al. 3 6) Blow-u of ositive-initial-energy solutions of a nonlinear viscoelastic hyerbolic equation Sali A. Messaoudi Matheatical Sciences Deartent, KFUPM, Dhahran 36, Saudi Arabia Received 7 Deceber 4 Available online Seteber 5 Subitted by J. Lavery Abstract In this aer, we consider the nonlinear viscoelastic equation t u tt Δu + gt τ)δuτ)dτ + u t u t = u u with initial conditions and Dirichlet boundary conditions. For nonincreasing ositive functions g and for >, we rove that there are solutions with ositive initial energy that blow u in finite tie. 5 Elsevier Inc. All rights reserved. Keywords: Blow-u; Finite tie; Hyerbolic; Nonlinear daing; Positive initial energy; Viscoelastic. Introduction In this aer, we are concerned with the initial-boundary-value roble u tt Δu + t gt τ)δuτ)dτ + u t u t = u u, in, ), ux, t) =, x, t,.) ux, ) = u x), u t x, ) = u x), x. E-ail address: essaoud@kfu.edu.sa. -47X/$ see front atter 5 Elsevier Inc. All rights reserved. doi:.6/j.jaa.5.7.

2 S.A. Messaoudi / J. Math. Anal. Al. 3 6) where is a bounded doain of R n n ) with a sooth boundary, >,, and g is a ositive function. In the absence of the viscoelastic ter that is, if g = ), the equation in.) reduces to the nonlinearly daed wave equation u tt Δu + u t u t = u u. This equation has been extensively studied by any atheaticians. It is well known that in the further absence of the daing echanis u t u t, the source ter u u causes finite-tie blow-u of solutions with negative initial energy see,9). In contrast, in the absence of the source ter, the daing ter assures global existence for arbitrary initial data see 8,). The interaction between the daing and source ters was first considered by Levine, for linear daing = ). Levine showed that solutions with negative initial energy blow u in finite tie. Georgiev and Todorova 7 extended Levine s result to nonlinear daing >). In their work, the authors introduced a new ethod and deterined relations between and for which there is global existence and other relations between and for which there is finite-tie blow-u. Secifically, they showed that solutions with negative energy continue to exist globally if and blow u in finite tie if >and the initial energy is sufficiently negative. Messaoudi 5 extended the blow-u result of 7 to solutions with only negative initial energy. For related results, we refer the reader to Levine and Serrin 3, Levine and Ro Park 4, Vitillaro 9, Yang and Messaoudi and Said-Houari 8. In the resence of the viscoelastic ter g ), Cavalcanti et al. 4 studied.) for = and a localized daing echanis ax)u t ax) null on a art of the doain). They obtained an exonential rate of decay by assuing that the kernel g is of exonential decay. This work was later iroved by Cavalcanti et al. 6 and Berrii and Messaoudi using different ethods. In related work, Cavalcanti et al. 3 studied solutions of t u t ρ u tt Δu Δu tt + gt τ)δuτ)dτ γδu t =, x, t >, for ρ> and roved a global existence result for γ and an exonential decay result for γ>. This latter result was extended by Messaoudi and Tatar 6 to a situation where a source ter is coeting with the daing induced by γδu t and the integral ter. Also, Cavalcanti et al. 5 established an existence result and a decay result for viscoelastic robles with nonlinear boundary daing. Concerning nonexistence, Messaoudi 7 showed that Todorova and Georgiev s results can be extended to.) using the technique of 7 with a odification in the energy functional due to the different nature of the robles. In this article, we irove our earlier result by adoting and odifying the ethod of 9. In articular, we will show that there are solutions of.) with ositive initial energy that blow u in finite tie. We first state a local existence theore that can be established by cobining arguents of 4,7.

3 94 S.A. Messaoudi / J. Math. Anal. Al. 3 6) 9 95 Theore.. Let u,u ) H ) L ) be given. Let >, > be such that n ) ax{, }, n n 3..) Let g be a C function satisfying gs)ds = l>..3) Then roble.) has a unique local solution u C,T ); H )), u t C,T ); L ) ) L,T ) ),.4) for soe T >. Reark.. Condition.) is needed to establish the local existence result see 4,7). In fact under this condition, the nonlinearity in the source is Lischitz fro H ) to L ). Condition.3) is necessary to guarantee the hyerbolicity and well-osedness of syste.). Next we state our ain result. For this urose, we assue that g satisfies, in addition to.3), the inequalities gs), g s), gs)ds < /) /) + /)..5) Theore.. Let and be such that >, >ax{,} and.) holds. Assue further that g satisfies.3),.5). Then any solution of.) with initial data satisfying.7) below blows u in finite tie.. Proof of the blow-u result In this section we rove our ain result Theore.). For this urose we let B be the best constant of the Sobolev ebedding H L and B = B/l /.Weset α = B / ), E = ) α..) We also define Et) = u t + t ) gs)ds u + g u)t) u,.)

4 S.A. Messaoudi / J. Math. Anal. Al. 3 6) where t g v)t) = gt τ) vt) vτ) dτ. Lea.. Assue that.),.3) and.5) hold. Let u be a solution of.). Then Et)is nonincreasing, that is, E t)..3) Proof. By ultilying Eq..) by u t and integrating over we obtain { d u dx + u t dx } u dx dt t gt τ) u t t). uτ)dxdτ = u t dx,.4) for any regular solution. This result reains valid for weak solutions by a sile density arguent. For the last ter on the left side of.4) we have t gt τ) u t t). uτ)dxdτ t = gt τ) u t t). uτ) ut) dxdτ + t = + t t = d dt + d dt gt τ) u t t). ut)dxdτ gt τ) d dt d gτ) dt t t gt τ) gτ) uτ) ut) dxdτ ut) dx ) dτ uτ) ut) dxdτ ut) dxdτ

5 96 S.A. Messaoudi / J. Math. Anal. Al. 3 6) t g t τ) Inserting.5) into.4), we obtain { d u dx + u t dx dt + d dt d dt = t t gt τ) u t dx + uτ) ut) dxdτ gt) gτ) ut) dτ t } u dx uτ) ut) dxdτ g t τ) uτ) ut) dxdτ ut) dxdτ..5) gt) ut)..6) This coletes the roof. Lea.. Assue that.),.3) and.5) hold. Let u be a solution of.) with initial data satisfying E)<E, u >B / )..7) Then there exists a constant β>b / ) such that t ) / gs)ds u + g u)t) β, t,t),.8) and u B β, t,t)..9) Proof. We first note that, by.), we have Et) t t ) gs)ds u + g u)t) u ) gs)ds u + g u)t) B l u

6 B S.A. Messaoudi / J. Math. Anal. Al. 3 6) t ) gs)ds u + g u)t) t gs)ds ) u + g u)t) / = ζ B ζ = hζ ),.) where t ) / ζ = gs)ds u + g u)t). It is easy to verify that h is increasing for <ζ <α, decreasing for ζ>α, hζ ) as ζ +, and hα) = ) B / ) = E, where α is given in.). Therefore, since E) <E, there exists β>αsuch that hβ) = E).Ifwesetα = u then, by.), we have hα ) E) = hβ). Therefore, α >β. To establish.8), we suose by contradiction that t ) / gs)ds u ) + g u)t <β, for soe t >. By the continuity of t ) gs)ds u + g u)t), we can choose t such that t ) / gs)ds u ) + g u)t >α. Again, the use of.) leads to t ) / ) Et ) h gs)ds u + g u)t ) >hβ)= E). This is iossible since Et) E), for all t,t). Hence.8) is established.

7 98 S.A. Messaoudi / J. Math. Anal. Al. 3 6) 9 95 To rove.9), we exloit.). We have t gs)ds ) u + g u)t) E) + u. Consequently, we obtain u t β E) gs)ds ) u + g u)t) E) β hβ) = B β..) The roof is colete. Lea.3. Suose that.) holds. Then there exists a ositive constant C> such that u s C u ) + u.) for any u H ) and s. Proof. If u then u s u C u by Sobolev ebedding. If u > then u s u. Therefore,.) follows. This coletes the roof. We set Ht)= E Et).3) and use, throughout this aer, C to denote a generic ositive constant deending on and l only. As a result of.),.), and.3), we have Lea.4. Let u be solution of.). Assue that.) holds. Then we have u s C Ht) u t g u)t) ) + u, t,t),.4) for any s. Proof. Using.3) and.), we note that l) u t ) gs)ds u Et) u t g u)t) + u E Ht) u t g u)t) + u..5)

8 S.A. Messaoudi / J. Math. Anal. Al. 3 6) Exloiting.) and.9), sile calculations yield E u. Finally, a cobination of.5) and.6) gives the desired result. Proof of Theore.. Using.),.3) and.3), we obtain.6) <H) Ht) E u t + and, fro.8), we obtain Hence, E u t + t t gs)ds gs)ds ) u + g u)t) + u ) u + g u)t) <E β = β <, t..7) <H) Ht) u, t..8) We define Lt) := H σ t) uu t dx,.9) for sall ε to be chosen later and for { } ) ) <σ in,..) ) Taking a derivative of.9) and using Eq..), we obtain { L t) = σ)h σ t) u t g u)t) + } gt) u u t u t dx u dx ε u t u t udx gt τ) σ)h σ t) u t u t u dx ut). uτ)dxdτ

9 9 S.A. Messaoudi / J. Math. Anal. Al. 3 6) 9 95 t u dx ε gt τ) t u t u t udx Using the Schwarz inequality,.) takes on the for L t) σ)h σ t) u t u t u dx ε t t u dx ε gt τ) ut) dτ ut). uτ) ut) dxdτ..) u t u t udx gt τ) ut) uτ) ut) dτ gt τ) ut) dτ..) We now exloit Young s inequality to estiate the fifth ter on the right side of.) and use.) to substitute for ux, t) dx. Hence,.) becoes L t) σ)h σ t) u t u t t dx ε Ht) + g u)t) + u t + ε u t u t ux, t) dx ετg u)t) ε 4τ σ)h σ t) u t + ) ) τ g u)t) ε ) + ) 4τ u t u t u t udx gs)ds gs)ds t t dx H t) ) ut) gs)ds ) ) ut) gs)ds ut) ) ut),.3) for soe nuber τ with <τ</. Recalling.5), the estiate.3) reduces to

10 S.A. Messaoudi / J. Math. Anal. Al. 3 6) L t) σ)h σ t) u t + ) u t x, t) dx H t) a g u)t) a ut) ε where a = ) τ>, a = + ) gs)ds >. 4τ To estiate the last ter of.4), we again use Young s inequality XY δr r Xr + δ q q Y q, X,Y,, δ>, with r = and q = / ).Sowehave r + q = u t u dx δ u + δ / ) u t, which yields, by substitution in.4), L t) σ)h σ t) εδ / ) u t + ) u t x, t) dx a g u)t) u t u t udx,.4) a ut) H t) ε δ u, δ>..5) Of course.5) reains valid even if δ is tie-deendant since the integral is taken over the x variable. Therefore, taking δ so that δ / ) = kh σ t) for large k to be secified later and substituting in.5), we arrive at L t) σ) εk H σ t) u t + ) u t x, t) dx a g u)t) a ut) Ht) k H σ ) t) u..6) Exloiting.8) and the inequality u C u, we obtain ) σ ) H σ ) t) u C u +σ ). Hence,.6) yields

11 9 S.A. Messaoudi / J. Math. Anal. Al. 3 6) 9 95 L t) σ) ) + εk H σ t) u t u t x, t) dx a g u)t) a ut) ) Ht) k σ ) C u +σ )..7) We now use.) and Lea.4 with s = + σ ) to deduce fro.7) that L t) σ) εk H σ t) u t ) + u t x, t) dx a g u)t) a ut) Ht) C k { Ht) u t g u)t) } + u σ) εk H σ t) u t ) + + C k u t a + C k ) g u)t) a ut) + C k ) Ht) εc k u,.8) where C = /) σ ) C/. Noting that Ht) u u t u g u)t) and writing = a 3 + a 3 ), where a 3 < in{a,a,/}, the estiate.8) yields L t) σ) εk H σ t) u t ) + + C k a 3 u t a + C k ) a 3 g u)t) a a 3 ) ut) a 3 + C k ) Ht) ) a3 C k u..9) At this oint, we choose k large enough so that.9) becoes L t) σ) εk H σ t) u t γ Ht)+ u t + u + g u)t),.3) where γ>isthe iniu of the coefficients of Ht), u t, u, and g u)t) in.9). Once k is fixed hence γ also), we ick ε sall enough so that σ) εk )/

12 S.A. Messaoudi / J. Math. Anal. Al. 3 6) and L) = H σ ) Therefore,.3) takes on the for u u x) dx >. L t) εγ Ht)+ u t + u + g u)t)..3) Consequently, we have Lt) L)>, t. We now estiate uu t dx u u t C u u t, which ilies uu t dx / σ) Again, Young s inequality gives us / σ) uu t dx C u / σ) u t / σ). C u μ/ σ) + u t θ/ σ),.3) for /μ+/θ =. To obtain μ/ σ)= / σ) by.), we take θ = σ). Therefore,.3) becoes / σ) uu t x, t) dx C u s + u t, where s = / σ). Using Lea.4, we obtain / σ) uu t dx C Ht)+ u + u t + g u)t), t..33) Therefore, we have L / σ) t) = H σ t) uu t dx ) / σ) Ht)+ / σ) uu t dx / σ)) C Ht)+ u + u t + g u)t), t..34)

13 94 S.A. Messaoudi / J. Math. Anal. Al. 3 6) 9 95 Cobining.3) and.34), we arrive at L t) ΓL / σ) t), t,.35) where Γ is a ositive constant deending only on εγ and C. A sile integration of.35) over, t) then yields L σ/ σ) t) L σ/ σ) ) Γtσ/ σ)..36) Therefore,.36) shows that Lt) blowsuintie T σ ΓσL) σ/ σ)..37) This coletes the roof. Reark.. By following the stes of the roof of Theore.5 closely, one can easily see that the blow-u result holds even for = daing caused only by viscosity). A sall odification is needed in the roof. Reark.. The third inequality in.5) shows that there is a strong relation between the nonlinearity in the source and the daing caused by the viscosity. More recisely, the larger is, the closer gs)ds can be to. Reark.3. The estiate.37) shows that the larger L) is, the quicker the blow-u takes lace. Acknowledgent The author exresses his sincere thanks to KFUPM for its suort. This work has been funded by KFUPM under Project # MS/VISCO ELASTIC/7. References J. Ball, Rearks on blow-u and nonexistence theores for nonlinear evolutions equations, Quart. J. Math. Oxford 8 977) S. Berrii, S.A. Messaoudi, Exonential decay of solutions to a viscoelastic equation with nonlinear localized daing, Electron. J. Differential Equations 4 4). 3 M.M. Cavalcanti, V.N. Doingos Cavalcanti, J. Ferreira, Existence and unifor decay for a non-linear viscoelastic equation with strong daing, Math. Methods Al. Sci. 4 ) M.M. Cavalcanti, V.N. Doingos Cavalcanti, J.A. Soriano, Exonential decay for the solution of seilinear viscoelastic wave equations with localized daing, Electron. J. Differential Equations ) 4. 5 M.M. Cavalcanti, V.N. Doingos Cavalcanti, J.S. Prates Filho, J.A. Soriano, Existence and unifor decay rates for viscoelastic robles with nonlinear boundary daing, Differential Integral Equations 4 ) M.M. Cavalcanti, H.P. Oquendo, Frictional versus viscoelastic daing in a seilinear wave equation, SIAM J. Control Oti. 4 3) 3 34.

14 S.A. Messaoudi / J. Math. Anal. Al. 3 6) V. Georgiev, G. Todorova, Existence of solutions of the wave equation with nonlinear daing and source ters, J. Differential Equations 9 994) A. Haraux, E. Zuazua, Decay estiates for soe seilinear daed hyerbolic robles, Arch. Ration. Mech. Anal ) V.K. Kalantarov, O.A. Ladyzhenskaya, The occurrence of collase for quasilinear equations of arabolic and hyerbolic tye, J. Soviet Math. 978) M. Koackova, Rearks on bounded solutions of a seilinear dissiative hyerbolic equation, Coent. Math. Univ. Carolin ) H.A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the for Pu tt = Au + Fu), Trans. Aer. Math. Soc ). H.A. Levine, Soe additional rearks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal ) H.A. Levine, J. Serrin, Global nonexistence theores for quasilinear evolution equation with dissiation, Arch. Ration. Mech. Anal ) H.A. Levine, S. Ro Park, Global existence and global nonexistence of solutions of the Cauchy roble for a nonlinearly daed wave equation, J. Math. Anal. Al ) S.A. Messaoudi, Blow u in a nonlinearly daed wave equation, Math. Nachr. 3 ) 7. 6 S.A. Messaoudi, N.-E. Tatar, Global existence and asytotic behavior for a nonlinear viscoelastic roble, Math. Meth. Sci. Res. J. 7 3) S.A. Messaoudi, Blow u and global existence in a nonlinear viscoelastic equation, Math. Nachr. 6 3) S.A. Messaoudi, B. Said-Houari, Blow u of solutions of a class of wave equations with nonlinear daing and source ters, Math. Methods Al. Sci. 7 4) E. Vitillaro, Global nonexistence theores for a class of evolution equations with dissiation, Arch. Ration. Mech. Anal ) Zhijian Yang, Existence and asytotic behavior of solutions for a class of quasi-linear evolution equations with non-linear daing and source ters, Math. Methods Al. Sci. 5 )