Sufficient conditions are established for the forced oscillation of fractional partial differential equations with damping term of the form (∂/∂t)(D+,tαu(x,t))+p(t)D+,tαu(x,t)=a(t)Δu(x,t)-q(x,t)u(x,t)+f(x,t), (x,t)∈Ω×R+≡G, with one of the two following boundary conditions: ∂u(x,t)/∂N=ψ(x,t), (x,t)∈∂Ω×R+ or u(x,t)=0, (x,t)∈∂Ω×R+, where Ω is a bounded domain in Rn with a piecewise smooth boundary, ∂Ω,R+=[0,∞), α∈(0,1) is a constant, D+,tαu(x,t) is the Riemann-Liouville fractional derivative of order α of u with respect to t, Δ is the Laplacian in Rn, N is the unit exterior normal vector to ∂Ω, and ψ(x,t) is a continuous function on ∂Ω×R+. The main results are illustrated by some examples.
1. Introduction
Fractional differential equations are generalizations of classical differential equations to an arbitrary (noninteger) order and have gained increasing attention because of their varied applications in various fields of applied sciences and engineering. In the past few years, the theory of fractional differential equations and their applications have been investigated extensively; for example, see the monographs [1–4].
Recently, the oscillatory behavior of solutions for fractional differential equations was discussed in [5–13] and so forth. In [13], the authors studied the oscillation of solutions to nonlinear fractional differential equations of this formED0+1+αyt+ptD0+αyt+qtfyt=gt,t>0,with initial condition I0+1-αy(0+)=b, where b is a real number, α∈(0,1) is a constant, and D0+αy is the Riemann-Liouville fractional derivative of order α of y.
However, to the best of author’s knowledge very little is known regarding the oscillatory behavior of fractional partial differential equations up to now; we refer to [14–16]. In particular, nothing is known regarding the oscillation properties of the problems (1)-(2) and (1)–(3) up to now.
In this paper we investigate the forced oscillation of fractional partial differential equations with damping term of the form(1)∂∂tD+,tαux,t+ptD+,tαux,t=atΔux,t-qx,tux,t+fx,t,x,t∈Ω×R+≡G,where Ω is a bounded domain in Rn with a piecewise smooth boundary ∂Ω, R+=[0,∞), α∈(0,1) is a constant, D+,tαu(x,t) is the Riemann-Liouville fractional derivative of order α of u with respect to t, and Δ is the Laplacian in Rn.
We assume throughout this paper that
a∈C(R+;(0,∞)), p∈C(R+;R);
q∈C(G¯;R+), and q(t)=minx∈Ω¯q(x,t);
f∈C(G¯;R).
Consider one of the two following boundary conditions:(2)∂ux,t∂N=ψx,t,x,t∈∂Ω×R+,or(3)ux,t=0,x,t∈∂Ω×R+,where N is the unit exterior normal vector to ∂Ω and ψ(x,t) is a continuous function on ∂Ω×R+.
By a solution of the problems (1)-(2) (or (1)–(3)), we mean a function u(x,t) which satisfies (1) on G¯ and the boundary condition (2) (or (3)).
A solution u(x,t) of the problems (1)-(2) (or (1)–(3)) is said to be oscillatory in G if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.
Definition 1.
The Riemann-Liouville fractional partial derivative of order 0<α<1 with respect to t of a function u(x,t) is given by(4)D+,tαux,t≔1Γ1-α∂∂t∫0tt-ξ-αux,ξdξ,provided the right hand side is pointwise defined on R+, where Γ is the gamma function.
Definition 2.
The Riemann-Liouville fractional integral of order α>0 of a function y:R+→R on the half-axis R+ is given by(5)I+αyt≔1Γα∫0tt-ξα-1yξdξfort>0,provided the right hand side is pointwise defined on R+.
Definition 3.
The Riemann-Liouville fractional derivative of order α>0 of a function y:R+→R on the half-axis R+ is given by(6)D+αyt≔dαdtαI+α-αyt=1Γα-αdαdtα∫0tt-ξα-α-1yξdξfort>0,provided the right hand side is pointwise defined on R+, where ⌈α⌉ is the ceiling function of α.
2. Main Results
In this section, we establish the oscillation of the problems (1)-(2) and (1)–(3). We firstly introduce the following lemmas which are very useful in the proof of our main results.
Lemma 4 (see [17]).
The smallest eigenvalue β0 of the Dirichlet problem, (7)Δωx+βωx=0,inΩ,ωx=0,on∂Ω,is positive and the corresponding eigenfunction φ(x) is positive in Ω.
Lemma 5 (see [2, page 74]).
Let α≥0, m∈N, and D=d/dx. If the fractional derivatives D+αy(t) and D+α+my(t) exist, then(8)DmD+αyt=D+α+myt.
Lemma 6 (see [2, page 75]).
Let α∈(0,1) and I+1-αy(t) be the fractional integral (5) of order 1-α, then(9)I+αD+αyt=yt-I+1-αy0Γαtα-1.
For convenience, one uses the following notations in this paper: (10)Ut=∫Ωux,tdx,Ψt=∫∂Ωψx,tdS,Ft=∫Ωfx,tdx,Ht=Ft+atΨt,Wt=∫Ωux,tφxdx,Yt=∫Ωfx,tφxdx,t≥0,where dS is the surface element on ∂Ω.
Theorem 7.
Assume that(11)limt→0I+1-αUt=C0,where C0 is a constant. If(12)liminft→∞∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ<0,limsupt→∞∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ>0,then every solution u(x,t) of the problems (1)-(2) is oscillatory in G, where V(t)=exp∫t0tp(ξ)dξ and C is a constant.
Proof.
Suppose to the contrary that there is a nonoscillatory solution u(x,t) of the problems (1)-(2) which has no zero in Ω×[t0,∞) for some t0≥0. Then u(x,t)>0 or u(x,t)<0 for t≥t0.
Case 1 (u(x,t)>0, t≥t0). Integrating (1) with respect to x over the domain Ω, we have(13)ddt∫ΩD+,tαux,tdx+pt∫ΩD+,tαux,tdx=at∫ΩΔux,tdx-∫Ωqx,tux,tdx+∫Ωfx,tdx,t≥t0.Green’s formula and (2) yield (14)∫ΩΔux,tdx=∫∂Ω∂ux,t∂NdS=∫∂Ωψx,tdS=Ψt,t≥t0.From (A2), it is easy to see that(15)∫Ωqx,tux,tdx≥qt∫Ωux,tdx,t≥t0.By Lemma 5, it follows from (13)–(15) that(16)D+1+αUt+ptD+αUt≤-qtUt+Ht,t≥t0.From (16), we easily see that (17)D+αUtVt′=D+1+αUtVt+D+αUtptVt≤-qtUtVt+HtVt≤HtVt,t≥t0.Integrating both sides of the above inequality from t0 to t, we obtain(18)D+αUtVt≤D+αUt0Vt0+∫t0tHsVsds=C+∫t0tHsVsds,where C=(D+αU(t0))V(t0). Using Lemma 6, it follows from (18) that(19)Ut≤I+1-αU0Γαtα-1+I+αCVt+1Vt∫t0tHsVsds=C0Γαtα-1+1Γα∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ.Taking t→∞ in (19), we have (20)liminft→∞Ut≤limsupt→∞C0Γαtα-1+liminft→∞1Γα·∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ<0,which contradicts the fact that U(t)>0.
Case 2 (u(x,t)<0, t≥t0). Using the procedure of the proof of Case 1, we conclude that (13) and (14) are satisfied. From (A2), we have(21)∫Ωqx,tux,tdx≤qt∫Ωux,tdx,t≥t0.Combining (13), (14), and (21), we have(22)D+1+αUt+ptD+αUt≥-qtUt+Ht,t≥t0.It follows from (22) that (23)D+αUtVt′=D+1+αUtVt+D+αUtptVt≥-qtUtVt+HtVt≥HtVt,t≥t0.Integrating both sides of the above inequality from t0 to t, we obtain(24)D+αUtVt≥D+αUt0Vt0+∫t0tHsVsds=C+∫t0tHsVsds.Using Lemma 6, from (24) we obtain(25)Ut≥I+1-αU0Γαtα-1+I+αCVt+1Vt∫t0tHsVsds=C0Γαtα-1+1Γα∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ.Taking t→∞ in (25), we have(26)limsupt→∞Ut≥liminft→∞C0Γαtα-1+limsupt→∞1Γα·∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ>0,which contradicts the fact that U(t)<0. This completes the proof of Theorem 7.
Theorem 8.
Assume that(27)limt→0I+1-αWt=C1,where C1 is a constant. If(28)liminft→∞∫0tt-ξα-1VξC+∫t0ξYsVsdsdξ<0,limsupt→∞∫0tt-ξα-1VξC+∫t0ξYsVsdsdξ>0,then every solution u(x,t) of the problems (1)–(3) is oscillatory in G, where V(t)=exp∫t0tp(ξ)dξ and C is a constant.
Proof.
Suppose to the contrary that there is a nonoscillatory solution u(x,t) of the problems (1)–(3) which has no zero in Ω×[t0,∞) for some t0≥0. Then u(x,t)>0 or u(x,t)<0 for t≥t0.
Case 1 (u(x,t)>0, t≥t0). Multiplying both sides of (1) by φ(x) and integrating with respect to x over the domain Ω, we have(29)ddt∫ΩD+,tαux,tφxdx+pt∫ΩD+,tαux,tφxdx=at∫ΩΔux,tφxdx-∫Ωqx,tux,tφxdx+∫Ωfx,tφxdx,t≥t0.Green’s formula and (3) yield (30)∫ΩΔux,tφxdx=∫Ωux,tΔφxdx=-β0∫Ωux,tφxdx,t≥t0.From (A2), it is easy to see that(31)∫Ωqx,tux,tφxdx≥qt∫Ωux,tφxdx,t≥t0.Combining (29)–(31), we easily see that(32)D+1+αWt+ptD+αWt≤-β0at+qtWt+Yt,t≥t0.
The remainder of the proof is similar to that of Case 1 of Theorem 7 and we can obtain a contradiction to W(t)>0.
Case 2 (u(x,t)<0, t≥t0). Using the procedure of the proof of Case 1, we conclude that (29) and (30) are satisfied. From (A2), we have(33)∫Ωqx,tux,tφxdx≤qt∫Ωux,tφxdx,t≥t0.Combining (29), (30), and (33), we obtain(34)D+1+αWt+ptD+αWt≥-β0at+qtWt+Yt,t≥t0.
Using a similar way in the proof of Case 2 of Theorem 7, we can obtain a contradiction to W(t)<0. The proof of Theorem 8 is complete.
3. ExamplesExample 1.
Consider the following fractional partial differential equation:(35)∂∂tD+,t1/2ux,t-D+,t1/2ux,t=1πΔux,t-x2+t2ux,t+etsintsinx,x,t∈0,π×R+,with the boundary condition(36)-∂u0,t∂x=∂uπ,t∂x=-etsint,t≥0.
Here α=1/2, Ω=(0,π), n=1, p(t)=-1, a(t)=1/π, q(x,t)=x2+t2, and f(x,t)=etsintsinx. It is obvious that V(t)=et0-t, q(t)=t2, Ψ(t)=-πetsint, and (37)Ft=∫Ωfx,tdx=∫0πetsintsinxdx=2etsint,Ht=Ft+atΨt=etsint.Hence (38)∫t0ξHsVsds=∫t0ξessinset0-sds=et0-cosξ+cost0.Letting t0=π/2, we obtain(39)∫0tt-ξ-1/2VξC+∫t0ξHsVsdsdξ=∫0tt-ξ-1/2eξ-π/2C+∫π/2ξHsVsdsdξ=∫0tt-ξ-1/2eξ-π/2C-eπ/2cosξdξ.Setting t-ξ=s2, above integral (39) can be written as the following form:(40)∫t01sCet-s2-π/2-et-s2cost-s2-2sds=2Cet-π/2∫0te-s2ds-2et∫0te-s2cost-s2ds=2Cet-π/2∫0te-s2ds-2etcost∫0te-s2coss2ds-2etsint∫0te-s2sins2ds=2etCe-π/2∫0te-s2ds-cost∫0te-s2coss2ds-sint∫0te-s2sins2ds.Noting that(41)e-s2coss2≤e-s2,e-s2sins2≤e-s2,limt→∞∫0te-s2ds=π2,we obtain that(42)limt→∞∫0te-s2coss2ds,limt→∞∫0te-s2sins2dsare convergent.
Therefore, combining (39)–(41) and noting the fact that (42) are convergent, by careful calculation, we can get (43)liminft→∞∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ≤-∞<0,limsupt→∞∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ≥+∞>0,which shows that all the conditions of Theorem 7 are fulfilled. Then every solution of the problems (35)-(36) oscillates in (0,π)×R+.
Example 2.
Consider the following fractional partial differential equation:(44)∂∂tD+,t1/2ux,t-D+,t1/2ux,t=e-tΔux,t-2x2+t2ux,t+et-πsintsinx,x,t∈0,π×R+,with the boundary condition(45)u0,t=uπ,t=0,t≥0.
Here α=1/2, Ω=0,π, n=1, p(t)=-1, at=e-t, q(x,t)=2x2+t2, and f(x,t)=et-πsintsinx.
It is obvious that V(t)=et0-t, q(t)=t2, β0=1, and φ(x)=sinx. Therefore, (46)Yt=∫Ωfx,tφxdx=∫0πet-πsintsin2xdx=πe-π2etsint.Hence (47)∫t0ξYsVsds=πe-π2∫t0ξessinset0-sds=πet0-π2-cosξ+cost0.
Using a similar way in Example 1, we can obtain (48)liminft→∞∫0tt-ξα-1VξC+∫t0ξYsVsdsdξ≤-∞<0,limsupt→∞∫0tt-ξα-1VξC+∫t0ξYsVsdsdξ≥+∞>0.
Therefore, by Theorem 8, it is easy to see that every solution of the problems (44)-(45) is oscillatory in (0,π)×R+.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (10971018). The author thanks the referee for his valuable comments and suggestions on this paper.
PodlubnyI.1999198San Diego, Calif, USAAcademic PressMR1658022KilbasA. A.SrivastavaH. M.TrujilloJ. J.2006Amsterdam, The NetherlandsElsevier Science B.VDasS.2008Berlin, GermanySpringerMR2414740ZhouY.2014SingaporeWorld ScientificÖğrekçiS.Interval oscillation criteria for functional differential equations of fractional order20152015, article 310.1186/s13662-014-0336-zBolatY.On the oscillation of fractional-order delay differential equations with constant coefficients201419113988399310.1016/j.cnsns.2014.01.005MR32127922-s2.0-84901687096ChenD.-X.Oscillation criteria of fractional differential equations20122012, article 3310.1186/1687-1847-2012-33MR29156312-s2.0-84871326781GraceS. R.AgarwalR. P.WongP. J. Y.ZaferA.On the oscillation of fractional differential equations201215222223110.2478/s13540-012-0016-1MR28977752-s2.0-84869202206ZhengB.Oscillation for a class of nonlinear fractional differential equations with damping term201361107115MR3088028HanZ.ZhaoY.SunY.ZhangC.Oscillation for a class of fractional differential equation20132013639028210.1155/2013/390282QiC.ChengJ.Interval oscillation criteria for a class of fractional differential equations with damping term20132013830108510.1155/2013/301085MR3043862ChenD.-X.QuP.-X.LanY.-H.Forced oscillation of certain fractional differential equations20132013, article 1251010.1186/1687-1847-2013-1252-s2.0-84890912170YangJ.LiuA.LiuT.Forced oscillation of nonlinear fractional differential equations with damping term20152015710.1186/s13662-014-0331-4PrakashP.HarikrishnanS.NietoJ. J.KimJ.-H.Oscillation of a time fractional partial differential equation201415110MR3183613HarikrishnanS.PrakashP.NietoJ. J.Forced oscillation of solutions of a nonlinear fractional partial differential equation2015254141910.1016/j.amc.2014.12.074MR3314430PrakashP.HarikrishnanS.BenchohraM.Oscillation of certain nonlinear fractional partial differential equation with damping term201543727910.1016/j.aml.2014.11.018MR3305632CourantR.HilbertD.19661New York, NY, USAInterscience