# Forced Oscillation Criteria for a Class of Fractional Partial Differential Equations with Damping Term

Authors: Wei Nian Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

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.


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## Body

## 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(A1)
a∈C(R+;(0,∞)), p∈C(R+;R);(A2)
q∈C(G¯;R+), and q(t)=min⁡x∈Ω¯q(x,t);(A3)
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 solutionu(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 order0<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 orderα>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 orderα>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 eigenvalueβ0 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]).
Letα≥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]).
Letα∈(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→0⁡I+1-αUt=C0,where C0 is a constant. If(12)liminf⁡t→∞⁡∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ<0,limsup⁡t→∞⁡∫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 solutionu(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
≥
t
0
). 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)liminf⁡t→∞⁡Ut≤limsup⁡t→∞⁡C0Γαtα-1+liminf⁡t→∞⁡1Γα·∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ<0,which contradicts the fact that U(t)>0.
Case 2 (
u
(
x
,
t
)
<
0
,
t
≥
t
0
). 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)limsup⁡t→∞⁡Ut≥liminf⁡t→∞⁡C0Γαtα-1+limsup⁡t→∞⁡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→0⁡I+1-αWt=C1,where C1 is a constant. If(28)liminf⁡t→∞⁡∫0tt-ξα-1VξC+∫t0ξYsVsdsdξ<0,limsup⁡t→∞⁡∫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 solutionu(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
≥
t
0
). 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 Theorem7 and we can obtain a contradiction to W(t)>0.
Case 2 (
u
(
x
,
t
)
<
0
,
t
≥
t
0
). 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 Theorem7, we can obtain a contradiction to W(t)<0. The proof of Theorem 8 is complete.


## 3. Examples

Example 1.
Consider the following fractional partial differential equation:(35)∂∂tD+,t1/2ux,t-D+,t1/2ux,t=1πΔux,t-x2+t2ux,t+etsin⁡tsin⁡x,x,t∈0,π×R+,with the boundary condition(36)-∂u0,t∂x=∂uπ,t∂x=-etsin⁡t,t≥0.
Hereα=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=∫0πetsin⁡tsin⁡xdx=2etsin⁡t,Ht=Ft+atΨt=etsin⁡t.Hence (38)∫t0ξHsVsds=∫t0ξessin⁡set0-sds=et0-cos⁡⁡ξ+cos⁡⁡t0.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/2cos⁡⁡ξdξ.Setting t-ξ=s2, above integral (39) can be written as the following form:(40)∫t01sCet-s2-π/2-et-s2cos⁡⁡t-s2-2sds=2Cet-π/2∫0te-s2ds-2et∫0te-s2cos⁡⁡t-s2ds=2Cet-π/2∫0te-s2ds-2etcos⁡⁡t∫0te-s2cos⁡⁡s2ds-2etsin⁡t∫0te-s2sin⁡s2ds=2etCe-π/2∫0te-s2ds-cos⁡⁡t∫0te-s2cos⁡⁡s2ds-sin⁡t∫0te-s2sin⁡s2ds.Noting that(41)e-s2cos⁡⁡s2≤e-s2,e-s2sin⁡s2≤e-s2,limt→∞⁡∫0te-s2ds=π2,we obtain that(42)limt→∞⁡∫0te-s2cos⁡⁡s2ds,limt→∞⁡∫0te-s2sin⁡s2dsare convergent.
Therefore, combining (39)–(41) and noting the fact that (42) are convergent, by careful calculation, we can get (43)liminf⁡t→∞⁡∫0tt-ξα-1VξC+∫t0ξHsVsdsdξ≤-∞<0,limsup⁡t→∞⁡∫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-πsin⁡tsin⁡x,x,t∈0,π×R+,with the boundary condition(45)u0,t=uπ,t=0,t≥0.
Hereα=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 thatV(t)=et0-t,  q(t)=t2,  β0=1, and  φ(x)=sinx. Therefore, (46)Yt=∫Ωfx,tφxdx=∫0πet-πsin⁡tsin2⁡xdx=πe-π2etsin⁡t.Hence (47)∫t0ξYsVsds=πe-π2∫t0ξessin⁡set0-sds=πet0-π2-cos⁡⁡ξ+cos⁡⁡t0.
Using a similar way in Example1, we can obtain (48)liminf⁡t→∞⁡∫0tt-ξα-1VξC+∫t0ξYsVsdsdξ≤-∞<0,limsup⁡t→∞⁡∫0tt-ξα-1VξC+∫t0ξYsVsdsdξ≥+∞>0.
Therefore, by Theorem8, it is easy to see that every solution of the problems (44)-(45) is oscillatory in (0,π)×R+.



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