Non-existence of Global Solutions to a Wave Equation with Fractional Damping

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1 IAEG Inernaional Journal of Alied Mahemaics, 4:, IJAM_4 6 on-exisence of Global Soluions o a Wave Equaion wih Fracional Daming Mohamed Berbiche, Ali Haem Absrac We consider he nonlinear fracional waveequaion u u + β/ D u = u, osed in :=, + I, where D, < <, is a ime fracional derivaive, β/, < β < is fracional ower of, wih given iniial osiion velociy ux, = u x u x, = u x We find Fujia s exonen which searaes in erms of,, β,, he case of global exisence from he one of nonexisence of global soluions hen we esablish necessary condiions on u x, u x assuring non-exisence of local soluions Index erms nonlinear wave equaion, fracional ower derivaive, criical exonen I IODUCIO I his aer, we discuss he nonexisence of wea soluions o he nonlinear fracional wave equaion osed in = I,, < +, subjec o he iniial condiions: u u + β D+u = u ux, = u x, u x, = u x, x I Where = + + is he usual Lalacian in he sace variable x, u is he ime derivaive of u, β/ is β/ fracional ower of he Lalacian < β which ss for roagaion in media wih imuriies is defined by β/ vx = F ξ β F v ξ x, where F denoes he Fourier ransorm F is inverse; D+, is he fracional derivaive of order, < <, such ha D+ = D+ u where D+ is he Cauo fracional derivaives of order, u x u x are given iniial daa Before we sae our resuls, le us dwell on exising lieraure concerning equaions close o iniial value roblem he ime fracional derivaive has long been found o be very effecive means o describe he anomalous aenuaion behaviors For examle, Hanyga Seredynsa[3, considered he differenial equaion D u + γd η+ u + F u =, where D η+, < η <, reresens he η + -order fracional derivaive in he sense of Cauo which models he anomalous aenuaion, γ is he hermoviscous coefficien ecenly, Chen Holm [4, sudied he equaion v = c v + c γ y/ v, 3 Manuscri received Seember 5, ; revised January, A Haem, Comuer Dearmen, Sidi Bel Abbes Universiy, ALGEIA haemali@yahoocom M Berbiche, Comuer Dearmen, Khenchela Universiy, 4 AL- GEIA berbichemed@yahoofr which governs he roagaion of sound hrough a viscous fluid, c is he inviscid hase velociy, is he collecive hermoviscous coefficien In [5, hey exended heir sudy o he wave equaion model for frequency deenden lossy media P = c P + γ η η s/ P, 4 s, < η 3, η, where γ is he viscous consan, s η can be arbirary real numbers heir range of secificaion Equaions 3 4 can be seen as generalizaion of he earlier imoran wor of Greenberg, MacCamy Mizel [6 who considered he equaion ρ u = u xx + λu xx + gx, wih x I, >, ρ, λ are some consans ha characerize he medum; gx, is a given funcion reresening an exernal force As equaions 3 4 my be viewed as aroximaions of nonlinear equaions, Eq conains a nonlinear erm ha is a rooye of nonlineariies ha may occur in racice Le s noe also ha in [3, Cholewa Carvalho deal wih he equaion u = u + θ u + u which is a secial case of roblem If β = = in, hen we obain he wave equaion wih he linear daming u here are many auhors reaed his case, see for insance odorova Yordanov [4, i S Zhang [5, Miidieri Pohozaev[9 In [9, he auhors showed ha he Fujia exonenis equal o + / he very ineresing aricle of odorova Yordanov is in fac a comlee sudy of Eq when β = = However, when β =, < <, Kirane aar [8 + showed ha he blow-u resuls for < + he mehod of roof is raher simle consiss in a judicious choice of a es funcion his mehod develoed by Miidieri Pohozaev[9, Pohozaev esei [ for he equaion inequaliies wih olynomial nonlineariy hen in he aer of Baras Pierre [, Baras Kersner [, Kalashniov [7 i S Zhang [5 II PELIMIAIES In his secion we resen wo differen definiions of fracional derivaives, some of heir roeries We define he fracional derivaive in he Cauo sense see [ C D γ u = Γ γ τ γ uτdτ, < γ <, Advance online ublicaion: February

2 IAEG Inernaional Journal of Alied Mahemaics, 4:, IJAM_4 6 for higher ower, C D γ u = Γ m γ m = [γ + τ m γ u m, he fracional derivaive in he iemann-liouville sense L D γ d m u = Γ m γ d m τ m γ ud, m = [γ + he relaionshi beween he wo definiions is m L D γ u = C D γ γ u + Γ γ + u +, = we have also he formula inegraion by ars fd γ gd = gd γ f, < γ < see[, 46 III O-EXISECE OF GLOBAL SOLUIOS will denoe here he se :=, I, L loc, ddx will denoe he sace of all funcions v : I + I I such ha v ddx < for any comac K in I + I Definiion A funcion u L loc is a local wea soluion of he roblem defined on, < < +, if u L loc such ha: ξ u u ξ u ξ + ξu + I for any es funcion ξ C, x, K ud β ξ = β u D ξ, I I [,, such ha ξ, ξ, x = ξ, x = ξ, x = ow, we are in osiion o announce our resuls heorem 3: Assume ha if I β u >, c = + I u >, > +, hen, roblem does no admi global nonrivial soluions in ime Proof he roof is by conradicion So we assume ha he soluion is global Le Φ be a decreasing funcion C I +, Φ such ha if y, Φ y := if y We choose ξ x, := Φ λ + x 4, = τ, x = y, 4 dxd = + dydτ Where is a osiive real number λ is any real greaer han 5 he es funcion ξ is chosen so ha ξ ξ + suξ + su su ξ ξ ξ ξ D β ξ < o esimae he righ h side of 5on, we wrie by using he ε-young inequaliy ξ u ε u ξ + C ε ξ ξ, 6 su ξ su ξ Similarly, u ξ ε suξ u ξ + C ε su ξ ud β ξ ε u ξ suξ +C ε ξ D β ξ su ξ ξ, 7 Where = Summing u esimaes 6, 7 8, wih ε small enough, we infer ha β u D ξ + u ξ + u ξ I C ε suξ ξ ξ + ξ q + D β ξ 9 for some osiive consan C ε = / ε A his sage, we inroduce he scaled variables = τ, x = y se := τ, y I + I ; τ + y 4 herefore, wriing ϕ, x = ϕ τ, y := χ τ, y we have ξ ξ = + 4 ξ ξ + ξ D β ξ + +β χ χ ττ, χ χ χ D β χ ow aing ε small enough, we obain he esimae ξ u C ε + χ 8 χ + χ + D β χ In he esimae,we have o disinguish ow cases: Eiher < c : In his case, assing o he limi as in we obain lim ξ u = I I + u = Advance online ublicaion: February

3 IAEG Inernaional Journal of Alied Mahemaics, 4:, IJAM_4 6 hus u = Or = c : In his case, we obain from ha I I + u C So lim u ξ =, C where C = x, : 4 + x 4 4 we se := τ, y I + I ; τ + y 4 If insead of using ε-young inequaliy, we raher use he Hölder inequaliy, hen han he esimae 9, we find u ξ ξ u C ξ ξ + ξ D β ξ ξ ξ Passing o he limi as in aing ino accoun, we obain u ξ = hus u = ae he roof is comlee emar 3: When, he criical exonen is c = + see odorova-yordanov [4 IV ECESSAY CODIIOS FO LOCAL AD GLOBAL EXISECE heorem 4: Le u, be a local soluion o WE- where < + > hen here exis consans C, C C 3 such ha C inf β u + C inf u x C 3 q Proof Le us consider he following es funcion x q ξ x, = Φ where Φ W, I is nonnegaive wih suφ x I / < x < su ss for suor, saisfies Φ Φ β Φ Φ for some osiive consans, From 9, we have β u D ξ + u ξ I n C ξ q + ξ q + D β ξ q for some osiive consan osiive C I is clear, from our choice of ξ ha he requiremens ξ x, = ξ x, = ξ x, = are saisfied ow, we esimae he righ h side in erms of Firs, if we se = τ, we find ξ q dxd C x q Φ dx, for some C > ξ q x C q q Φ For he las erm, we comue D ξ Γ D ξ = ϕ σ dσ = σ [ q σ σ dσ [ q σ σ = 4q σ dσ = 4q σ dσ hen Γ D ξ = [ σ q σ q 4q 4q σ q σ dσ dx 3 σ q +8q q 4q σ σ q σ dσ I + J Using he Euler s change of variables we see ha y = σ σ = y y = σ, y = σ y = σ y σ = y + y herefore I = 4q 4q σ q σ dσ = 4q 4q +q since we have as y q + y + q y dy + y + = + + y y < +, for y < hen one can aly he Binomial formula for non ineger ower o I = 4q 4q +q y q + + y q y dy Advance online ublicaion: February

4 IAEG Inernaional Journal of Alied Mahemaics, 4:, IJAM_4 6 or I = 4q 4q where = Using he formula we obain C q +q+ + q y q y + dy C q q! =! q! τ u τ v dτ = Γ u Γ v Γ u + v, u, v > I = 4q 4q C q +q+ q ΓqΓ + + Γq+ + = he same hing for J J = 8qq 4q σ σ q σ dσ = 8qq 4q + y lef y q + + y q +q y dy hen J = 8q q 4q = C q = C q + y + q +q y q y + dy = 8q q 4q + [ + q +q y q y + dy + + q +q+ y q y + dy + + q +q+ y q y +3 dy Using he formula in he above J = 8q q 4q [ C q + q +q = Γq Γ + Γq q +q+ Γq Γ3 + Γq q +q+ Γq Γ4 + Γq+3 + Hence D [ q 4q 4q = Γ C q = ΓlΓ +r Γl+ +r +q+ + q + 8qq Γ 4q C q [ + q +q = Γq Γ + Γq q +q+ Γq Γ3 + Γq q +q+ Γq Γ4 + Γq+3 + if we se = τ we find D [ q 4q = Γ C q = ΓlΓ +r Γl+ +r τ +q+ + τ q + 8qq Γ C q [ τ + τ q τ +q = Γq Γ + Γq+ + +τ + τ q τ +q+ Γq Γ3 + Γq τ q τ +q+ Γq Γ4 + Γq+3 + hus D q C q, Γ 4 where C q, is consan deending of q Subsiuing exression 5 ino he following inegral D ξ = ξ Φ x q D ξ We have he esimae h β D ξ q q τ q q x τ q q dτ Γ I Φ hen we have h β D ξ q C,q Γ q q + x 5 Γ Γ 4q q + q Φ q ow we comue D q Γ D = σ σ q dσ Using he above Euler s change of variable o comue I := σ σ q σ dσ Advance online ublicaion: February

5 IAEG Inernaional Journal of Alied Mahemaics, 4:, IJAM_4 6 or I = q + y + y q + + y q y dy By using he generalized binomial formula we may wrie I = q + y + y q Cq + q y + dy = Using he formula we obain τ u τ v dτ = I = q + = y + y q dy Γ u Γ v Γ u + v, u, v > C q + q + q + Cq + q hence D = = y + y q dy, q = 4q 4q Γ q + C q + q Γ+3 Γq Γ+3 +q + q + = C q + q Γ+ Γq Γ+ +q In aricular we have D ξ = 4q + Γ = 6 C q Γ+3 Γq Γ+3 +q 7 Subsiuing he exression of D ξ, in he following erm, u D ξ = x u x Φ [ q +l Cq + q y + y q dy Γ = + q + Cq + q = y + y q dyd i easy o see ha herefore u D ξ = + Γ x u x Φ [ τ q + Cq + τ q τ = y + y q dy+ τ q + τ Cq + τ q τ = u D y + y q dydτ ξ = C,q + x u x Φ Γ We have, D β ξ q x q q βq Φ D Using 5, we find D β C 3 βq q+ I Φ ξ q x dx 8 q 9 Gahering he esimaes 9,, we obain C + Φ β u + C Φu I n I C x + C q q q + C 3 βq q+ Φ dx On he oher h we have I β u Φ inf β x x u Φ x u x Φ x inf u x Φ aing ino accoun he esimae, he inequaliy 7 x dividing by he erm Φ, imly ha C inf β u + C inf u x C 3 q + C 4 q q + C 5 βq q Passing o he limi as +, we ge C inf β u + C inf u x C 3 q 3 Corollary Assume ha roblem has a nonrivial global wea soluion hen one a leas of he following is saisfied lim inf β u =, lim inf u x = Corollary If one of he following limis is infinie Advance online ublicaion: February

6 IAEG Inernaional Journal of Alied Mahemaics, 4:, IJAM_4 6 lim inf β u >, lim inf u x >,hen roblem, canno have any local wea soluion If A = lim inf β u > B = lim inf u C x > hen min 6 C, 7 A q B q heorem 4: Suose he roblem has a nonrivial global wea soluion hen, here are wo osiive consans such ha lim inf β u x minq,βq q K, lim inf x minq,+β u x K, Proof In he relaion C I Φ β u + C I u x Φ x C 3 q + C 4 q q + C 5 βq q I Φ x as β < q <, we have C I Φ β u C 3 q + C 4 q q + C 5 βq q I Φ x aing in he righ h side =, we obain C I Φ β u C 3 q + C 4 q q + C 5 βq+ q Φ x hen I Φ β u C 8 minq,βq q I Φ x ow, using assumions on Φ namely, < x < Φ β u I C 8 x I minq,βq q x Φ we see ha inf β u x minq,βq q x minq,βq q Φ I C 8 x I minq,βq q x Φ 4 5 o conclude i suffices o ae he su wih resec o of 6 divide by x minq,βq q x Φ we ge lim inf β u x minq,βq q C 8 Wih similar argumen we have, Φu I C + C q q q + C 3 βq q x Φ dx We obain, x Φu C 9 minq,+β Φ dx, I hence Φu C 9 x minq,+β x Φ I hus x minq,+βq u x inf dx x minq,+βq Φ I C 9 x I minq,+βq x Φ dx Finally assing o he su, deviding boh sides of he resuling relaion by he exression x minq,+βq Φ x dx we obain lim inf x minq,+βq u x C V COCLUSIO Our analysis is cerainly robus for more general wave equaions I can surely be used, for equaions wich inerolaes hea equaion he wave equaion EFEECES [ Baras, P Kersner: Local global solvabiliy of a class of semil-inear arabolic equaions J Diff Eqs, , 38-5 [ Baras, P M Pierre: Crières d exisence de soluions osiives our des équaions semi-linéaires non monoones Ann Ins H Poincaré, Anal on linéaire, 985, 85- [3 A Carvalho J W Cholewa: Aracors for srongly damed wave equaions wih criical nonlineariies, Pacific J of Mah, 7, 87-3 [4 W Chen S Holm: Physical inerreaion of fracional diffusion- Wave equaion via lossy media obeying frequency ower law Prerin [5 W Chen S Holm: Fracional Lalacian, Levy sable disribuion ime-sace models for linear nonlinear frequency deenden lossy media Prerin [6 J M Greenberg, MacCamy V J Mizel: On he exisence, uniqueness, sabiliy of soluions of he equaion σ u x u xx +λu xx = ρ u J Mah Mech, 7 967, [7 A S Kalashniov: On a hea conducion equaion for a medium wih non uniformly disribued non-linear hea source or absorbers, Bull Univ Moscou Mah Mech3 983, -4 [8 M Kirane -e aar: onexisence of Soluions o a Hyerbolic Equaion wih a ime Fracional Daming, J Mah Anal Al, [9 E Miidieri S I Pohozaev: A riori esimaes, blow-u of soluions o nonlinear arial differenial equaions inequaliies Proc Seolov Ins Mah, v 34, -383 [ S I Pohozaev A esei: Blow-u of nonnegaive soluions o quasilinear arabolic inequaliies, Lincei CI Sci Fis Ma aur end Lincei 9 Mah Al vol, 99-9 [ I Podlubny: Fracional Differenial Equaions Mahemaics in Science Ingineering, vol98 ew Yor/London: Sringer; 999 [ S G Samo, A A Kilbas O I Marichev: Fracional inegrals derivaives, heory Alicaions, Gordon Beach Sciences Publishers, 987 [3 A Hanyga, MSeredynsa: onlinear Hamilonian equaions wih fracional daming J Mah Phys,vol4, [4 odorova, G B, Yordanov: Criical exonen for nonlinear wave equaions wih daming J Diff Eqs,vol74, [5 Zhang, S: A blow-u resul for a nonlinear wane equaion wih daming: he criical case C Mah Acad Sci Paris,vol333,, 9-4 [6 Jonahan M Blacledge: Alicaion of he fracional diffusion equaion for redicing mare behavior IAEG Inernaional Journal of Alied Mahemaics, vol4, issue 3 [7 Z Odiba, A El-ajou: Consrucion of analyical soluion o fracional differenial equaion using homooy analysis mehod IAEG Inernaional Journal of Alied Mahemaics, vol4, issue Advance online ublicaion: February