OSCILLATORY BEHAVIOR OF A FRACTIONAL PARTIAL DIFFERENTIAL EQUATION

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1 Journa of Appied Anaysis Compuaion Voume 8, Number 3, June 28, 2 Websie:hp://jaa-onine.om/ DOI:.948/28. OSCILLATORY BEHAVIOR OF A FRACTIONAL PARTIAL DIFFERENTIAL EQUATION Jiangfeng Wang,2, Fanwei Meng, Absra In his paper, a fraiona paria differenia equaion subje o he Robin boundary ondiion is onsidered. Based on he properies of Riemann-Liouvie fraiona derivaive a generaized Riai ehnique, we obained suffiien ondiions for osiaion of he souions of suh equaion. Exampes are given o iusrae he main resus. Keywords Osiaion, paria differenia equaion, fraiona derivaive. MSC(2 35B5, 35R, 34K37.. Inroduion Fraiona differenia equaions are generaizaions of assia differenia equaions of ineger order have gained inreasing aenion due o heir various appiaions in various fieds of engineering, hemia physis, eeria neworks, onro heory of dynamia sysems, indusria robois, eonomis so on. The researh of fraiona differenia equaions heir appiaions have reeived more more aenion very reeny, see he monographs 7, 4, 7. Reeny, he osiaion behavior of souions for paria differenia equaion has been deveoped rapidy some resus are esabished. However, o he bes of our knowedge very ie is known abou he osiaory behavior of fraiona paria differenia equaions up o now. Some aries abou osiaion heory of paria differenia equaions fraiona paria differenia equaions have been pubished, suh as 6, 8 3, 5, 6, 8 2. A few paper sudied he osiaion of fraiona paria differenia equaions whih invove he Riemann-Liouvie fraiona derivaive. Prakash e a. 5 onsidered he osiaion of he fraiona differenia equaion ( r(d, u(x, ( q(f ( v u(x, vdv =a( u(x,, (. he orresponding auhor. Emai address:fwmeng@mai.qfnu.edu.n(f. Meng Shoo of Mahemaia Siene, Qufu Norma Universiy, No.57 Jingxuanxi Road, Qufu 27365, Shong, China 2 Deparmen of Mahemais, Jining Universiy, No. Xingan Road, Qufu, 27355, Shong, China The auhors were suppored by Naura Siene Foundaion of China (Grans 67227, he Naura Siene Foundaion of Shong Provine (Gran ZR24AL7, ZR25PA5, ZR27BF2 he Siene Tehnoogy Proje of High Shoos of Shong Provine (Gran J4LJ9.

2 2 J. Wang & F. Meng where (x, G = R. Harikrishnan e a. 6 esabished he osiaion of he fraiona differenia equaion of he form D, ( r(d, u(x, q(x, f (u(x, = a( u(x, g(x,, (x, G. (.2 Li sudied he fored osiaion of fraiona paria differenia equaions wih he damping erm of he form ( D, u(x, p(x, D,u(x, = a( u(x, q(x, u(x, f(x,, (x, G. (.3 However, o he bes of our knowedge, very ie is known regarding he osiaory behavior of fraiona differenia equaions. To deveop he quaiaive properies of fraiona paria differenia equaion, i is of grea ineres o sudy he osiaory behavior of fraiona paria differenia equaion. In his paper, we esabish severa osiaion rieria for fraiona paria differenia equaion by appying a generaized Riai ransformaion ehnique by using he properies of he Riemann-Liouvie fraiona derivaive. These resus are onsidered esseniay new. Exampes are given o iusrae he main resus. In his paper, we onsider he osiaory properies of souions o he fraiona paria differenia equaions of he form D, ( r(d, u(x, ( p(d,u(x, q(x, f ( v u(x, vdv =a( u(x,, (x, G = R, (.4 wih he Robin boundary ondiion u(x, N g(x, u(x, =, (x, R, (.5 where (, is a onsan, D, is he Riemann-Liouvie fraiona derivaive of order of u wih respe, is a bounded domain in R n wih pieewise smooh boundary, is he Lapaian operaor N is he uni exerior norma veor o, g(x, is a nonnegaive oninuous funion on R. Throughou his paper, we assume ha: (A r( C (R, R, p( C(R, R, a( C(R, R ; (A 2 q(x, C(Ḡ, R min x q(x, = q( (A 3 f : R R is a oninuous funion suh ha f(x/x > µ for erain onsan µ > for a x. By a souion of (.4 we mean a nonrivia funion u(x, C (, suh ha ( v u(x, vdv C (Ḡ; R, D,u(x, C (Ḡ; R saisfies (.4 on Ḡ. A souion u(x, of (.4 is aed osiaory in G if i is neiher evenuay posiive nor evenuay negaive. Oherwise, i is aed non-osiaory. Equaion (.4 is said o be osiaory if a is souions are osiaory. 2. Preiminaries emmas In his seion, we rea severa definiions of fraiona inegra fraiona derivaive, whih wi be used in he foowing proof. There are severa kinds of

3 Osiaory behavior of FPDE 3 definiions of fraiona inegra fraiona derivaives 7. In his arie, we use Riemann-Liouvie definiion. For onveniene, hroughou he res of his arie, we denoe v( = u(x, dx, = dx. (2. Definiion 2. ( 7. The Riemann-Liouvie fraiona inegra I y of order R is defined by (I y( = Γ( ( v y(vdv, >. (2.2 Here Γ( is he gamma funion defined by Γ( = s e s ds for >. This inegra is aed ef-sided fraiona inegra. Definiion 2.2 ( 7. The Riemann-Liouvie fraiona paria derivaive of order < < of a funion u(x, is defined by (D,u(x, = Γ( ( v u(x, vdv, >, (2.3 provided he righ h side is poinwise defined on R, where Γ is he gamma funion. Definiion 2.3 ( 7. The Riemann-Liouvie fraiona derivaive of order > of a funion y : R R on he haf-axis R is given by (Dy( := d (I dx y( = ( v y(vdv, >, Γ( dx (2.4 provided he righ h side is poinwise defined on R, where is he eiing funion of. d Lemma 2. (Lemma 2.4, 2. Le y be a souion of (. F ( := ( v y(vdv for (, >. (2.5 Then F ( = Γ( (D y(. ( Main resu We define he foowing funions ha wi be used in he proof of our resus, suppose ha here exiss a funion φ C,, (,, e ξ( = r(φ ( p(φ(,η(= r(φ(. Aso we rea a ass funion defined on D ={(, s : s }. A funion H C (D, R is said o beong o he ass if (i H(, = for H(, s > when s; (ii H(, s has paria derivaives on D suh ha H (, s = h (, s H(, s, H s (, s = h 2(, s H(, s for some h, h 2 (D, R. L o

4 4 J. Wang & F. Meng Theorem 3.. Le ondiions (A -(A 3 hod, suppose ha here exiss a funion φ C,, (,. If for every T, here exiss an inerva (a, b T, here exiss (a, b, H,suh ha where H(, a H(b, a b µh(s, aφ(sq(s µh(b, sφ(sq(s 4Γ( η( Φ2 (s, a ds 4Γ( η( Φ2 2(b, s Φ (s, a = h (s, a ξ(sη(s H(s, a, ds > (3. Φ 2 (b, s = h 2 (b, s ξ(sη(s H(b, s. (3.2 Then every souion u(x, of (.4 is osiaory in G. Proof. Suppose o he onrary ha u is a non-osiaory souion of (.4. Wihou oss of generaiy, we an assume ha here exiss u(x, > in G, for some >. Inegraing (.4 wih respe x over he domain, we obain d r(( (D d,u(x, dx p(d,u(x, dx q(x, f( ( v u(x, vdvdx =a( u(x, dx. (3.3 Using Green s formua, i is obvious ha u(x, u(x, dx = N ds = g(x, u(x, ds,. (3.4 where ds is surfae eemen on. By using Jensen s inequaiy (A 2, we an obain ( q(x, f ( µ u(x, µdµ dx q(f ( ( µ u(x, µdµ dx ( q(f ( µ q( dxf u(x, µdx dµ ( ( µ u(x, µdµ dx( dx =q( dxf ( µ ( u(x, mudx( dx dµ =q( f (G(. (3.5 Combining (3.3-(3.5 using definiions, we ge d r(d d v( p(dv( q(f(g(, (3.6

5 Osiaory behavior of FPDE 5 where Define he funion w( by v( = G( = u(x, dx, ( ξ v(ξdξ. w( = φ( r(d,v(, for. (3.7 G( Then we have w( > for. Differeniaing (3.7 for, we have w ( = φ ( φ( w( v( φ((r(d G ( φ(r(d v( G( G 2 ( φ ( φ( p(d w( v( q(f(g( φ( G( Γ( Dv( φ(r(d v( G 2 ( = φ ( φ( p( Γ( w( w( φ(q(f(g( r( G( φ(r( w2 ( µφ(q( ( φ ( φ( p( Γ( w( r( φ(r( w2 ( µφ(q( ξ(η(w( Γ( η(w 2 (. (3.8 Muipying (3.8 by H(s, inegraing wih respe o s from o for (a,, we have µh(s, φ(sq(sds In view of (i (ii, we see ha H(s, w (sds = H(, w( Using (3. in (3.9 eads o µh(s, φ(sq(sds H(, w( Γ( η(sh(s, w 2 (s H(s, w (sds H(s, ξ(sη(sw(sds H(s, Γ( η(sw 2 (sds. (3.9 h (s, H(s, w(sds. (3. ( h (s, H(s, ξ(sη(sh(s, w(s ds

6 6 J. Wang & F. Meng = H(, w( 4Γ( η(s Φ2 (s, ds H(, w( ( 2 Γ( η(sh(s, w(s 2 Γ( η(s Φ (s, ds 4Γ( η(s Φ2 (s, ds. (3. Simiary, if (3.8 is muipied by H(, s hen inegraed from o for, b, hen we ges µh(s, φ(sq(sds H(, w( =H(, w( H(, w( Γ( η(sh(s, w 2 (s ( h 2 (, s H(s, ξ(sη(sh(s, w(s ds ( 2 Γ( η(sh(s, w(s 2 Γ( η(s Φ 2(s ds 4Γ( η(s Φ2 2(, sds a b 4Γ( η(s Φ2 2(, sds. (3.2 Leing a in (3. b in (3.2 adding he resuing inequaiies we have µh(s, aφ(sq(s H(, a 4Γ( η(s Φ2 (s, a ds H(b, µh(b, sφ(sq(s 4Γ( η(s Φ2 2(b, s whih onradis he assumpion (3.. The proof is ompee. ds (3.3 Theorem 3.2. Le ondiions (A -(A 3 hod, suppose ha here exiss a funion φ C,, (,, here exiss H suh ha imsup µh(s, φ(sq(s 4Γ( η(s Φ2 (s, ds >, (3.4 imsup µh(, sφ(sq(s 4Γ( η(s Φ2 2(, s ds >, (3.5 hod for every,, >, where Φ, Φ 2 are he same in Theorem 3., hen every souion of (.4 is osiaory. Proof. Suppose o he onrary ha u is a non-osiaory souion of (.4. Wihou oss of generaiy, we an assume ha here exiss u(x, > in G, for

7 Osiaory behavior of FPDE 7 some 2. Se = a 2 in (3.4. Ceary, we see from (3.4 ha here exiss > a suh ha µh(s, aφ(sq(s 4Γ( η(s Φ2 (s, a ds >. (3.6 a Simiary seing = 2 in (3.5, i foows ha here exiss b > suh ha b µh(b, sφ(sq(s 4Γ( η(s Φ2 2(b, s ds >. (3.7 From (3.6 (3.7 we see ha (3. is saisfied. Therefore, in view of Theorem 3., we may onude ha every souion of (.4 is osiaory. If we hoose H(, s = ( s λ, s, where λ > is a onsan. Then, we obain he foowing usefu osiaion rierion. Coroary 3.. Le ondiions (A -(A 3 hod, suppose ha here exiss a funion φ C,, (,, suh ha he foowing wo inequaiies hod: imsup imsup λ λ (s λ {µφ(sq(s ( s λ {µφ(sq(s for eah, λ >, hen Eq.(.4 is osiaory. ( } 2 λ 4Γ( η(s (s ξ(sη(s ds > (3.8 ( } 2 λ 4Γ( η(s ( s ξ(sη(s ds > (3.9 More generay, one may onsider H(, s = R( R(s λ, where λ is onsan R( = r(s ds im R( =. If we hoose φ( =, by Theorem 3.2, we have he foowing osiaory rierion. Theorem 3.3. Le ondiions (A -(A 3 hod, p( q( for a, im R( =. Then every souion of Eq.(.3 is osiaory provided for eah, for some λ >, he foowing wo inequaiies hod: imsup R λ ( λp(s 2Γ( r(s {( µq(s R(s R( p 2 (s 4Γ( r(s λ } ds > λ R(s R( λ 2 4Γ( (λ (3.2 imsup R λ ( λp(s 2Γ( r(s {( µq(s R( R(s p 2 (s 4Γ( r(s λ } ds > λ R( R(s λ 2 4Γ( (λ. (3.2

8 8 J. Wang & F. Meng Proof. Noing ha Sine H(, s = R( R(s λ, i is easy o see ha r(sh 2 (s, ds = r(sh 2 2(s, ds = h (, s = λ R( R(s λ 2 2 h 2 (, s = λ R( R(s λ 2 2 In view of im R( =, i foows ha im im im sup 4Γ( R λ ( 4Γ( R λ ( R λ ( r(, r(s. r(sλ 2 R(s R( λ 2 r 2 (s ds = λ2 R( R(λ λ r(sλ 2 R( R(s λ 2 r 2 (s ds = λ2 λ R( R(λ. From (3.2 (3.22, we have ha { H(s, µq(s =im sup R λ ( p 2 (s 4Γ( r(s =im sup R λ ( λp(s 2Γ( r(s r(sh 2 (s, ds = r(sh 2 2(s, ds = r(s 4Γ( { R(s R( λ µq(s R(s R( {( µq(s R(s R( p 2 (s 4Γ( r(s λ } ds λ 2 4Γ( (λ (3.22 λ 2 4Γ( (λ. (3.23 h (s, p(s r(s H(s, 2 } ds R(s R( λ λp(s 2Γ( r(s λ } r(s ds imsup R λ ( 4Γ( h2 (s, ds λ R(s R( λ 2 >, (3.24 4Γ( (λ i.e., (3.4 hods. Simiary, (3.2 impies ha (3.5 hods. By Theorem 3.2, every souion of (.4 is osiaory. The proof is ompee. 4. Exampe Exampe 4.. Consider he fraiona paria differenia equaions D, ( D, u(x, ( D,u(x, ex 2 f ( ξ u(x, vdv = e u(x,, (x, (, π (, (4. 8

9 Osiaory behavior of FPDE 9 wih he Robin boundary ondiion u x (, = u x (π, =, (4.2 where (,, p( =, q( = minq(x, = min x x (,π = 2, r( =, a( = 2 e 8, f(u = u. Se µ =. Thus a he ondiions of he heorem (3. hod. Therefore every souion of (4. is osiaory. Exampe 4.2. Consider he fraiona paria differenia equaions D 2, (D 2, u(x, D 2, u(x, (x 2 ( f ( ξ 2 u(x, vdv =3e u(x,, (x, (, π (, (4.3 wih he Robin boundary ondiion u x (, = u x (π, =, (4.4 where (,, p( =, q(x, = (x 2, q( = minq(x, = min x x (,π (x2 =, r( =, a( = 3e, f(u = u. Se µ =. Thus a he ondiions of he heorem (3. hod. Therefore every souion of (4.3 is osiaory. Referenes D. Chen, Osiaory behavior of a ass of fraiona differenia equaions wih damping, U.P.B. Si. Bu., Series A, 23, 75(, D. X. Chen, P. X. Qu Y. H. Lan,Fored osiaion of erain fraiona differenia equaions, Adv. Differ. Equ., 23, 23(, Q. Feng F. Meng, Osiaion of souions o noninear fored fraiona differenia equaions, Eeron. J. Differ. Eq., 23, 23(69, S. R. Grae, R. P. Agarwa, P. J. Y. Wong, e a, On he osiaion of fraiona differenia equaions. Fra. Ca. App. Ana., 22, 5(2, Z. Han, Y. Zhao, S. Ying, e a, Osiaion Theorem for a Kind of Fraiona Differenia Equaions, Journa of Binzhou Universiy, 23, 23(9, S. Harikrishnan, P. Prakash J. J. Nieo, Fored osiaion of souions of a noninear fraiona paria differenia equaion, App. Mah. Compu., 25(254, A. A. Kibas, H. M. Srivasava J. J. Trujio, Theory Appiaions of Fraiona Differenia Equaions, Esevier, Amserdam, F. Lu F. Meng, Osiaion heorems for superinear seond-order damped differenia equaions, App. Mah. Compu., 27, 89(, W. N. Li, Osiaion of souions for erain fraiona paria differenia equaions, Advanes in Differene Equaions, (26 26: 6. W. N. Li, On he fored osiaion of erain fraiona paria differenia equaions, App. Mah. Le., 5(25, 5 9. e x

10 2 J. Wang & F. Meng W. N. Li, Fored osiaion rieria for a ass of fraiona paria differenia equaions wih damping erm, Mahemaia Probems in Engineering, vo. 25, Arie ID 494, W. N. Li W. Sheng, Osiaion properies for souions of a kind of paria fraiona differenia equaions wih damping erm, J. Noninear Si. App., 9(26, F. Meng, An osiaion heorem for seond order superinear differenia equaions, Indian J. Pure Ap. Ma., 996, 27(7. 4 I. Podubny, Fraiona Differenia Equaions, Aademi Press, San Diego, P. Prakash, S. Harikrishnan, J. J. Nieo, e a, Osiaion of a ime fraiona paria differenia equaion, Eeron. J. Qua. Theo., 24, 5(,. 6 P. Prakash, S. Harikrishnan M. Benhohra, Osiaion of erain noninear fraiona paria differenia equaion wih damping erm, App. Mah. Le., 25(43, S. G. Samko, A. A. Kibas O. I. Marihev, Fraiona Inegras Derivaives: Theory Appiaions, Esevier, Amserdam, E. Tunç H. Avi, Inerva osiaion rieria for seond order noninear differenia equaions wih noninear damping// Seminar on noninear paria differenia equaions, Springer-Verag, 984, J. Wang, F. Meng S. Liu, Inegra average mehod for osiaion of seond order paria differenia equaions wih deays, App. Mah. Compu., 27, 87(2, J. Wang, F. Meng S. Liu, Inerva osiaion rieria for seond order paria differenia equaions wih deays, J. Compu. App. Mah., 28, 22(2, B. Zheng, Osiaion for a ass of noninear fraiona differenia equaions wih damping erm, J.Adv.Mah.Sud., 23(9,