Microlocal analysis of fractional wave equations
aa r X i v : . [ m a t h - ph ] D ec Microlocal analysis of fractional wave equations
Günther Hörmann ∗ , Ljubica Oparnica † , Dušan Zorica ‡ February 25, 2018
Abstract
We determine the wave front sets of solutions to two special cases of the Cauchy problemfor the space-time fractional Zener wave equation, one being fractional in space, the otherbeing fractional in time. For the case of the space fractional wave equation, we show thatno spatial propagation of singularities occurs. For the time fractional Zener wave equation,we show an analogue of non-characteristic regularity.
Key words : wave front set, space-time fractional wave equation, Cauchy problem,fractional Zener model, fractional strain measure
This paper is devoted to the microlocal analysis of the solution to the generalized Cauchy problemfor the space-time fractional Zener wave equation Zu ( x, t ) = ∂ t u ( x, t ) − L αt ∂ x E βx u ( x, t ) = u ( x ) ⊗ δ ′ ( t ) + v ( x ) ⊗ δ ( t ) (1)considered as an equation on all R with supp( u ) ⊆ { ( x, t ) ∈ R | t ≥ } and u , v ∈ E ′ ( R ). Thegeneralized Cauchy problem (1) is derived and analyzed in [1], where existence and uniquenessof distributional solutions has been shown. In the present paper we study the wave front set forspecial cases of (1) when α = 0, or β = 1.The operators L αt and E βx are of convolution type, with respect to one variable only, denotedby ∗ t and ∗ x , respectively, and act on a distribution w = w ( x, t ) in the following way L αt w = F − τ → t (cid:20) b e i απ ( τ − i0) α a e i απ ( τ − i0) α (cid:21) ∗ t w, α ∈ [0 , , < a < b, (2) E βx w = F − ξ → x (cid:20) i sin( βπ ξ ) | ξ | β (cid:21) ∗ x w, β ∈ (0 , . (3)Note that E x w = ∂∂x w and, in case 0 < β <
1, we have E βx w = 12Γ (1 − β ) 1 | x | β ∗ x ∂∂x w. ∗ Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria, [email protected] † Faculty of Education in Sombor, University of Novi Sad, Podgorička 4, 25000 Sombor, Serbia, [email protected] ‡ Mathematical Institute, Serbian Academy of Arts and Sciences, Kneza Mihaila 36, 11000 Belgrade, Serbia,[email protected] and Department of Physics, Faculty of Sciences, University of Novi Sad, Trg D.Obradovića 4, 21000 Novi Sad, Serbia L αt , considered as a convolution operator in one variable, is linear and bounded L p ( R ) → L p ( R ), 1 < p < ∞ , by Hörmander’s multiplier theorem (cf. [3, Corollary 8.11] or [5,Theorem 7.9.5]), since l α , defined by l α ( τ ) = 1 + b e i απ ( τ − i0) α a e i απ ( τ − i0) α (4)is in L ∞ ( R ) ∩ C ( R \ { } ) with derivative bounded by a constant times | τ | − . Note that in [1],the operator L αt is employed in its Laplace transform variant L αt w = L − s → t h bs α as α i ∗ t w .The operator E βx , acting by convolution in one variable with h ( x ) := | x | − β (apart from a con-stant), following a differentiation, is a bounded linear operator W ,p ( R ) → L q ( R ), 1 < p < q < ∞ and p + β = q + 1, since by the Hardy-Littlewood-Sobolev-inequality [5, Theorem 4.5.3] the map w h ∗ w is continuous L p ( R ) → L q ( R ) in this setting.The space-time fractional Zener wave equation Zu ( x, t ) := ∂ t u ( x, t ) − L αt ∂ x E βx u ( x, t ) = 0 , x ∈ R , t > , (5)subject to initial conditions u ( x,
0) = u ( x ) , ∂∂t u ( x,
0) = v ( x ) , x ∈ R , (6)is derived in [1] from the system of three equations: The equation of motion of a (one-dimensional)deformable body, the constitutive equation and the non-local strain measure. In dimensionlessform, the system of equations reads ∂∂x σ ( x, t ) = ∂ ∂t u ( x, t ) , (7) σ ( x, t ) + a D αt σ ( x, t ) = ε ( x, t ) + b D αt ε ( x, t ) , (8) ε ( x, t ) = E βx u ( x, t ) , (9)where α ∈ [0 , β ∈ (0 , D αt is the fractional differential operator, defined as follows. Let t, γ ∈ R and H denote the Heaviside function. Then one defines f γ ( t ) = t γ − Γ( γ ) H ( t ) , γ > , d N d t N f γ + N ( t ) , γ ≤ , γ + N > , N ∈ N and for g ∈ S ′ , with support in the region t > D γt g = f − γ ∗ t g = dd t f − γ ∗ t g. Upon Fourier transform we may solve (8) with respect to σ by σ ( x, t ) = F − τ → t (cid:20) b e i απ ( τ − i0) α a e i απ ( τ − i0) α (cid:21) ∗ t ε ( x, t ) . (10)Indeed, by [5, Example 7.1.17], [ f − α = e − i (1 − α ) π ( τ − i0) α − , implying also d f − α = F ( ddt f − α ) =i τ e − i (1 − α ) π ( τ − i0) α − = e i απ ( τ − i0) α , hence we find that (10) solves (8). Finally, inserting this2nto (7) and observing (9), we arrive at Equation (5). As in [1], we study the Cauchy problem(5-6) with distributional initial values in the form (1).In Section 2 we analyze the microlocal properties of the space-fractional wave equation andin Section 3 we address an analogue of the non-characteristic regularity of solutions to the time-fractional Zener wave equation. We consider the solution for the special case of problem (1) with u , v ∈ E ′ ( R ) when α = 0 and0 < β <
1, which leads to the so-called space-fractional wave equation Zu ( x, t ) = ∂ t u ( x, t ) − ∂ x E βx u ( x, t ) = u ( x ) ⊗ δ ′ ( t ) + v ( x ) ⊗ δ ( t ) . (11)With b β := q sin βπ we have the solution u with supp( u ) ⊆ { t ≥ } in the form (cf. [1]) u = u ∗ x F − ξ → x h cos (cid:16) b β | ξ | β t (cid:17) H ( t ) i| {z } E + v ∗ x F − ξ → x sin (cid:16) b β | ξ | β t (cid:17) b β | ξ | β H ( t ) | {z } E . (12) Remark 2.1. (i) We observe that E is a fundamental solution of Z , i.e., ZE = δ . Furthermore,the equations E = ∂ t E and ZE ( x, t ) = δ ( x ) ⊗ δ ′ ( t ) hold on R .Furthermore, both E and E are weakly smooth with respect to t when t = 0. We notethat t E ( t ) is continuous R → S ′ ( R ) with E (0) = 0, whereas lim t → E ( t ) = δ = 0 =lim t → − E ( t ); E is weakly measurable with respect to t ∈ R .(ii) It is apparent from (12) and the assumption u , v ∈ E ′ ( R ) that the partial x -Fourier trans-form F x → ξ ( u ) of u is a continuous function with respect to ξ and of moderate growth. Hence themultiplication b g γ · F x → ξ ( u ) with b g γ ( ξ ) := | ξ | γ ( − < γ <
1) gives a locally integrable function ofmoderate growth with respect to ξ , and G γ u := F − ξ → x ( b g γ · F x → ξ ( u )) is defined in S ′ ( R ). Thesame is true, if in place of u we consider ˜ u = F − ξ → x (exp( ib β | ξ | β t )) ∗ x u and other similarlyconstructed distributions, e.g., w = G σ ˜ u . We will repeatedly make use of this fact within thecurrent section in course of the following proofs.(iii) For fixed t >
0, the linear operators of convolution with E ( t ) and E ( t ) are bounded L p ( R ) → L p ( R ), if 1 < p < ∞ , by Hörmander’s multiplier theorem, since their Fourier transformsare in L ∞ ( R ) ∩ C ( R \ { } ) with derivative bounded by a constant times | ξ | − (cf. [3, Corollary8.11] or [5, Theorem 7.9.5]). Lemma 2.2.
For j = 0 , , let E + j denote the restriction of E j to the open half-space { t > } .Then the wave front sets are given by WF( E +0 ) = WF( E +1 ) = { (0 , t ; ξ, | t > , ξ = 0 } =: W . Proof.
From ∂ t E +1 = E +0 we immediately deduceWF( E +0 ) ⊆ WF( E +1 ) ⊆ WF( E +0 ) ∪ { ( x, t ; ξ, | t > , ξ = 0 } , ( ∗ )where the right-most set corresponds to the characteristic set of ∂ t when considered as partialdifferential operator on R × ]0 , ∞ [. 3 laim 1: Both, WF( E +0 ) and WF( E +1 ), are contained in W .Let t >
0, put e +0 ( ξ, t ) := cos( b β | ξ | β t ) and e +1 ( ξ, t ) := sin( b β | ξ | β t ) / ( b β | ξ | β ), and choose ρ ∈ C ∞ ( R ) such that ρ ( ξ ) = 0 for | ξ | ≤ / ρ ( ξ ) = 1 for | ξ | ≥
1. Then at fixed arbitrary t > j ∈ { , } we have E + j ( t ) = F − ξ → x (cid:0) e + j ( ξ, t ) (cid:1) = F − ξ → x (cid:0) e + j ( ξ, t ) ρ ( ξ ) (cid:1) + F − ξ → x (cid:0) e + j ( ξ, t )(1 − ρ ( ξ )) (cid:1) =: F j, ( t ) + F j, ( t ) . We observe that ( ξ, t ) e + j ( ξ, t )(1 − ρ ( ξ )) is continuous, has compact ξ -support, and is smoothwith respect to t , more precisely, e + j (1 − ρ ) ∈ C ∞ (]0 , ∞ [ , C c ( R )), hence by linearity, commuta-tivity with ddt , and continuity of the inverse Fourier transform with respect to ξ , we have F j, ∈ C ∞ (cid:0) ]0 , ∞ [ , F − ( C c ( R )) (cid:1) ⊆ C ∞ (]0 , ∞ [ , C ∞ ( R )) ⊆ C ∞ (]0 , ∞ [ × R ), thus, WF( E + j ) = WF( F j, ).Note that a j ( x, t, ξ ) := e + j ( ξ, t ) ρ ( ξ ) define functions in C ∞ ( R × R ) ( j = 0 ,
1) and that a ( x, t, ξ ) = R t a ( x, s, ξ ) ds . Furthermore, a is a symbol of class S − β , β ( R × R ), since e +0 isthe real part of a function of a special case of the type discussed in [4, Example 1.1.5.] (withappropriate choices of parameters and variable names); here, the condition 0 ≤ β < | ξ | > x -derivatives of a vanish, any ξ -derivative of a and a —as well as a and a themselves—have essentially the same bounds when ( x, t ) varyin a compact subset and | ξ | >
1; furthermore, any t -derivative of a brings us back to estimating a with one order less in the t -derivatives, thus a is contained in the same symbol class as a .To complete the proof of Claim 1, we observe that F j, ( j = 0 , a j ( x, t, ξ ) / (2 π ) and phase funtion φ ( x, t, ξ ) = x ξ in both cases. Thus, according to [5, Theorem8.1.9], the only contributions to the wave front sets can stem from points with stationary phase,i.e., W F (cid:0) E + j (cid:1) ⊆ { ( x, t ; ∂ x φ ( x, t, ξ ) , ∂ t φ ( x, t, ξ )) | t > , ∃ ξ = 0 : ∂ ξ φ ( x, t, ξ ) = 0 } = { (0 , t, ξ, | t > , ξ = 0 } = W . Claim 2: W ⊆ WF( E +0 ).Note that due to the symmetry of F ( E +0 ) with respect to ξ
7→ − ξ and the result of Claim1 we have (0 , t ; ξ, ∈ WF( E +0 ) ⇔ (0 , t ; − ξ, ∈ WF( E +0 ) ⇔ (0 , t ) ∈ singsupp( E +0 ). Thus, itsuffices to show that E +0 is nonsmooth along the half axis x = 0, t > e E ( t ) := F − ξ → x h exp (cid:16) ib β t | ξ | β (cid:17)i and observe that by the elementary relationscos( z ) = (exp( iz ) + exp( iz )) / F − ( v )( x ) = F − ( v )( − x ) and employing the ad-hoc notation R ∗ for the pull-back by R ( x, t ) = ( − x, t ) we may write E +0 = 12 (cid:16) e E + R ∗ e E (cid:17) . Since we are here concerned solely with the question of smoothness at the points (0 , t ) for any t > t E +0 ( t ) as well as t e E ( t ) is smooth, we may note: E +0 is non-smooth at (0 , t ) ⇔ E +0 ( t ) is non-smooth at x = 0 ⇔ e E ( t ) is non-smooth at x = 0.Let f := F − ξ → x (cid:2) e i b β | ξ | σ (cid:3) ∈ S ′ ( R ), abbreviating σ = (1 + β ) /
2, and observe that for t > ξ ξ t /σ on the Fourier transformed side, we have e E ( t ) = f (cid:0) ./t /σ (cid:1) /t /σ andtherefore may reduce the question of smoothness of ˜ E ( t ) at x = 0 further to that of smoothnessof f at zero. 4hoose η ∈ C ∞ ( R ) such that η = 0 near ξ = 0, η = 1 for | ξ | >
1, and write ˆ f = (1 − η ) ˆ f + η ˆ f . Since (1 − η ) ˆ f ∈ C c ( R ), smoothness of f is equivalent to that of F − (cid:16) η ˆ f (cid:17) .Let θ > Q θ w := F − ξ → x " η ( ξ ) | ξ | θ ∗ w and m θ ( ξ ) := F x → ξ [ Q θ f ] ( ξ ) = η ( ξ ) | ξ | θ e i b β | ξ | σ . Here, Q θ is a pseudodifferential operator with symbol q θ ( x, ξ ) = η ( ξ ) / | ξ | θ , which clearly satisfies | q θ ( ξ ) | = 1 / | ξ | θ , if | ξ | >
1, hence Q θ is elliptic (of order − θ ). Thus, smoothness of f at 0 turnsout to be equivalent to smoothness of Q θ f at 0. Note that Q θ f is smooth off 0 by essentiallythe same arguments used as with E and the symbol a in Claim 1.The non-smoothness of Q θ f at 0 is shown thanks to an asymptotic result by G. H. Hardymentioned in [10, p. 357, 5.3(ii)], with parameters a , b there to be identified with σ , θ respectively;note that 1 > σ = (1 + β ) / > / θ > (1 − σ ) / (1 − σ ) >
0; then weconclude that there is some constant c > Q θ f ) ( x ) = c e i c | x | α | x | γ + O | x | γ − α ! ( x → , where α = σ − σ and γ = θ (1 − σ ) − σ σ − > . Thus, Q θ f ( x ) cannot be bounded as x → x = 0, which completes the proof of Claim 2.From Claims 1 and 2 in conjunction with the first inclusion relation in ( ∗ ) established at thebeginning of the proof, we obtain W ⊆ WF( E +0 ) ⊆ WF( E +1 ) ⊆ W , which implies that equality holds throughout and completes the proof.Based on the results of Lemma 2.2 we will investigate the influence of the singularities in theinitial data u and v on the wave front set of the solution u to (11). We emphasize that theproof of Theorem 2.4 below uses only the inclusion relation WF( E + j ) ⊆ W in its first part andprovides an independent, more general, proof of equality in this relation—thus substituting theargument of Claim 2 above drawing on Hardy’s asymptotics by advanced microlocal techniques. Remark 2.3. If v = 0 and t > u + ( t ) = E +0 ( t ) ∗ u .Since singsupp( E +0 ( t )) = { } a smooth cut-off ρ near x = 0 implies ( E +0 ( t )(1 − ρ )) ∗ u ∈ C ∞ ( R ), hence it suffices to investigate the singularity structure of ( E +0 ( t ) ρ ) ∗ u , where nowboth convolution factors belong to E ′ ( R ). At fixed t , this enables us to employ the methods andresults from [6, Section 16.3] on singular supports of convolutions (or to extend these techniquesto wave front sets as suggested by Hörmander directly after the statement of [6, Definition16.3.2]). Though a bit technical, it is not difficult to see that one will then obtain equality of theclosed convex hulls of singsupp( u + ( t )) and singsupp( u ). However, even having information onWF( u + ( t )) for every t > u + ) in termsof the two-dimensional directions in the cotangent fiber. Theorem 2.4.
Let u , v ∈ E ′ ( R ) and denote by u + the restriction of the solution u to (11) to thehalf-space of future time R × ]0 , ∞ [ , then WF( u + ) is invariant under translations ( x, t ) ( x, t + s ) with s > and WF (cid:0) u + (cid:1) ⊆ { ( x, t ; ξ, | t > , ( x, ξ ) ∈ W F ( u ) or ( x, ξ ) ∈ W F ( v ) } . oreover, in case v is smooth we have the more precise statement WF (cid:0) u + (cid:1) = { ( x, t ; ξ, | t > , ( x, ξ ) ∈ W F ( u ) } , and similarly WF ( u + ) = { ( x, t ; ξ, | t > , ( x, ξ ) ∈ W F ( v ) } , if u is smooth. To prepare for the proof of the theorem we need a technical lemma on “symbol corrections”.
Lemma 2.5.
Let σ ∈ (0 , and y ( ξ, τ ) = − τ + b β | ξ | σ . Let Γ ⊆ R (representing the ( ξ, τ ) -plane) be the union of a closed disc around (0 , and a closed narrow cone containg the τ -axisand being symmetric with respect to both axes. Let Γ ′ be a closed set of the same shape as Γ , butwith slightly larger disc and opening angle of the cone. Let ˜ b ∈ S (cid:0) R × R (cid:1) such that ˜ b ( x, t, ξ, τ ) is real, constant with respect to ( x, t ) , homogenous of degree with respect to ( ξ, τ ) away from thedisc contained in Γ ′ , and such that ˜ b ( x, t, ξ, τ ) = 0 , if ( ξ, τ ) ∈ Γ , ˜ b ( x, t, ξ, τ ) = 1 , if ( ξ, τ ) Γ ′ .Then y ˜ b is a symbol belonging to the class S (cid:0) R × R (cid:1) .Proof. By construction of ˜ b , it suffices to check the symbol estimates when ( ξ, τ ) is outsideΓ, say | τ | < c | ξ | , and | ξ | + | τ | is large. The upper bound in order zero is clear from | y ˜ b | ≤ C ( | ξ | σ + | τ | ) ≤ C ′ (1 + | ξ | + | τ | ). A derivative ∂ α ξ ∂ α τ ( y ˜ b ) with α + α = n ≥ ∂ τ y ∂ lξ ∂ m − τ ˜ b = − ∂ lξ ∂ m − τ ˜ b with l + m = n or ∂ kξ y ∂ lξ ∂ mτ ˜ b with k + l + m = n , hence it suffices to estimate these. We have | ∂ lξ ∂ m − τ ˜ b ( ξ, τ ) | ≤ C l,m (1 + | ξ | + | τ | ) − l − ( m − = C l,m (1 + | ξ | + | τ | ) − n and (cid:12)(cid:12) ∂ kξ y ( ξ, τ ) ∂ lξ ∂ mτ ˜ b ( ξ, τ ) (cid:12)(cid:12) = C ′ | ξ | σ − k (cid:12)(cid:12) ∂ lξ ∂ mτ ˜ b ( ξ, τ ) (cid:12)(cid:12) ≤ ˜ C (1 + | ξ | + | τ | ) − l − m | ξ | k − σ ≤ ˜ C (1 + | ξ | + | τ | ) − l − m | ξ | k − + | ξ | k − ≤ ˜ C (1 + | ξ | + | τ | ) − l − m | ξ | k − + c | τ | k − ≤ C (1 + | ξ | + | τ | ) − k − l − m = C (1 + | ξ | + | τ | ) − n . Proof of the Theorem.
We consider the case when v is smooth, which we may immediatelyreduce to v = 0, since the contribution of E +1 ∗ x v to the solution is smooth. Put K = f ∗ E +0 ,where f : R × ]0 , ∞ [ × R → R × ]0 , ∞ [ =: Ω is given by f ( x, t, y ) = ( x − y, t ) , and f ∗ is thedistributional pull-back in the sense of [5, Theorem 6.1.2]. Then K ∈ D ′ (Ω × R ) and [5, Theorem8.2.4] and Lemma 2.2 implyWF ( K ) ⊆ { ( x, t, y ; ξ, , − ξ ) | x = y, ξ = 0 } = { ( x, t, x ) | x, t ∈ R } × { ( ξ, , − ξ ) | ξ = 0 } . We have u + = u | { t> } = u ∗ x E +0 , whose action on test functions ϕ ∈ D (Ω) can be written inthe form h u + , ϕ i = h K, ϕ ⊗ u i , if u ∈ D ( R ), i.e., u u + is the linear map S : D ( R ) → D ′ (Ω)with distribution kernel K . Since WF ′ ( K ) R = { ( y, η ) | ∃ ( x, t ) : ( x, t, y ; 0 , , − η ) ∈ WF( K ) } = ∅ ,[5, Theorem 8.2.13] implies that S may be extended to a map E ′ ( R ) → D ′ (Ω) and satisfiesWF ( Su ) ⊆ WF ( K ) Ω ∪ WF ′ ( K ) ◦ WF ( u ) , where WF ( K ) Ω = { ( x, t ; ξ, τ ) | ( x, t, y ; ξ, τ, ∈ WF ( K ) for some y ∈ R } = ∅ and WF ′ ( K ) = { ( x, t, y ; ξ, τ, η ) | ( x, t, y ; ξ, τ, − η ) ∈ W F ( K ) } = { ( x, t, x ; ξ, , ξ ) | t > , ξ = 0 } . Thus, we obtainWF (cid:0) u + (cid:1) ⊆ (cid:8) ( x, t ; ξ, τ ) | ∃ ( y, η ) ∈ WF ( u ) : ( x, t, y ; ξ, τ, η ) ∈ WF ′ ( K ) (cid:9) ⊆ { ( x, t ; ξ, τ ) | t > , ∃ ( y, η ) ∈ W F ( u ) : y = x, τ = 0 , η = ξ } , (cid:0) u + (cid:1) ⊆ { ( x, t ; ξ, | ( x, ξ ) ∈ WF( u ) , t > } , (13)and the remaining part of the proof is concerned with showing that equality holds in (13).As in the proof of Lemma 2.2 let ˜ E ( t ) := F − ξ → x [exp ( ib β t | ξ | σ )] with σ := (1 + β ) /
2, but thistime for any t ∈ R , and put ˜ u ( t ) := ˜ E ( t ) ∗ u . We have D t d ˜ E ( t ) = i ∂ t d ˜ E ( t ) = b β | ξ | σ e i b β | ξ | σ t = b β | ξ | σ d ˜ E ( t ), which implies Y ˜ E := − D t ˜ E + A σx ˜ E = 0 , ˜ E (0) = δ, where A σx w = F − ξ → x [ b β | ξ | σ ] ∗ x w (with w of the type as in Remark 2.1(ii)). Moreover, ˜ u solvesthe initial value problem Y ˜ u = ( − D t + A σx ) ˜ u = (cid:0) − D t ˜ E + A σx ˜ E (cid:1) ∗ x u = 0 , ˜ u (0) = u . ( ∗ )Note that, since b β | ξ | σ = sin( βπ/ | ξ | β +1 is precisely the “symbol” of −E βx ∂ x , we havea (commutative) factorization of Z by ( D t + A σx ) ( − D t + A σx ) v = − D t v + A σx A σx v = ∂ t v + F − ξ → x ( b β | ξ | σ ) ∗ x v = Zv , i.e., Z = ¯ Y · Y , where we have put ¯ Y := D t + A σx .Before entering the detailed microlocal analysis of ˜ u let us anticipate its relevance for u + : Wewill show in Equation (17) below, that in the region with t > , ˜ u provides a “lower bound” forthe wave front set of u + , in fact, we will show equality of the wave front sets at the end of theproof.In studying the propagation of singularities for problem ( ∗ ) we encounter the nuisance that y ( ξ, τ ) = − τ + b β | ξ | σ is not quite a symbol of order in ξ and τ , since y is nonsmooth atzero and, furthermore, the symbol estimates obviously fail, e.g., for | ∂ ξ y ( ξ, τ ) | = | σ ( σ − | ξ | σ − | when τ → ∞ , there would have to be a bound of decrease (1 + | τ | + | ξ | ) − for large | τ | + | ξ | . Aremedy of this second kind of “symbol failure” is discussed in [8, Theorem 18.1.35], see also acomment below [8, Theorem 23.1.4], which we essentially follow in studying the propagation ofsingularities for ˜ u considered as solution to Y B ˜ u = BY ˜ u = 0 , ˜ u (0) = u , where B = op (cid:0) ˜ b (cid:1) ∈ Ψ (cid:0) R (cid:1) is the pseudodifferential operator associated with a symbol ˜ b givenas in Lemma 2.5. Thus, Y B = op (cid:0) y ˜ b (cid:1) ∈ Ψ (cid:0) R (cid:1) has principal symbol q ( ξ, τ ) := − τ ˜ b ( ξ, τ ) , which is real and homogeneous of degree one, and, modulo a regularizing contribution, can beconsidered properly supported.By [7, Theorem 26.1.1] WF (˜ u ) is invariant under the flow corresponding to the Hamiltonianvector field H q ( x, t, ξ, τ ) = − ∂ ξ q − ∂ τ q∂ x q∂ t q = τ ∂ ξ ˜ b ˜ b + τ ∂ τ ˜ b and is contained in the characteristic set Char ( Y B ) = R × { ( ξ, | ξ = 0 } , i.e., WF (˜ u ) ⊆ Char ( Y B ) In fact, a refinement of the latter inclusion relation is available, since ˜ u = ˜ E ∗ x u and we may argue very similar to proof of (13), noting (as in Claim 2 of the proof of Lemma2.2) that ˜ E is microlocally equivalent to F ξ → x (cos( b β | ξ | σ t )) , and deduceWF (˜ u ) ⊆ { ( x, t ; ξ, | ( x, ξ ) ∈ WF ( u ) , t ∈ R } . (14)7olving the Hamiltonian equations ˙ x = τ ∂ ξ ˜ b ( ξ, τ ) , ˙ t = ˜ b ( ξ, τ ) + τ ∂ τ ˜ b ( ξ, τ ) , ˙ ξ = 0 , ˙ τ = 0 , with ( x (0) , t (0) , ξ (0) , τ (0)) = ( x , t , ξ , ∈ Char ( Y B ) , we obtain ∀ s ∈ R : x ( s ) = x , t ( s ) = t + s ˜ b ( ξ , , ξ ( s ) = ξ , τ ( s ) = 0 . We may supposethat ˜ b ( ξ ,
0) = 1 , since ξ = 0 and characteristic sets as well as wave front sets are conic withrespect to the cotangent fibers. Hence the bicharacteristic flow evolves along the t -direction withfixed cotangent directions ( ξ , only. Therefore we have ( x , t ; ξ , ∈ WF (˜ u ) if and only if ( x , ξ , ∈ WF (˜ u ) . We claim that the latter is in turn equivalent to ( x , ξ ) ∈ WF ( u ) , fromwhich, together with (14), we may then concludeWF (˜ u ) = { ( x, t ; ξ, | ( x, ξ ) ∈ WF ( u ) , t ∈ R } . (15)We have claimed: ( x , ξ , ∈ WF (˜ u ) ⇔ ( x , ξ ) ∈ W F ( u ) .The implication ‘ ⇒ ’ follows from (14). For the converse, note that, according to [5, Theorem8.2.4] and the fact that WF (˜ u ) ⊆ Char ( Y B ) contains no directions (0 , τ ) in the fiber, we maywrite ˜ u ( t ) = f ∗ t ˜ u for any t ∈ R , where f t ( x ) = ( x, t ) as a map R → R , and obtainWF (˜ u ( t )) ⊆ f ∗ t WF (˜ u ) = { ( x, ξ ) | ( x, t ; ξ, ∈ WF (˜ u ) } = { ( x, ξ ) | ( x, ξ, ∈ WF (˜ u ) } , where the last equality follows from the Hamiltonian invariance proven above. In particular,when t = 0 we have ˜ u (0) = u , so that WF ( u ) = WF (˜ u (0)) ⊆ { ( x, ξ ) | ( x, ξ, ∈ WF (˜ u ) } ,which proves the part ‘ ⇐ ’ of the claim and thus establishes (15).We are now ready to clarify the microlocal relation between ˜ u and u + : In the subdomainwith t > we have ¯ Y u + = (cid:0) D t E +0 + A σx E +0 (cid:1) ∗ x u = A σx ˜ E ∗ x u = A σx ˜ u, (16)since F x → ξ (cid:2) D t E +0 ( t ) + A σx E +0 ( t ) (cid:3) = − i∂ t (cos ( b β | ξ | σ t )) + b β | ξ | σ cos ( b β | ξ | σ t ) = b β | ξ | σ [ ˜ E ( t ) .Denoting by ˜ u + the restriction of ˜ u to the half-plane of positive time we claim that thefollowing two inclusions hold: ( I ) WF ( ¯ Y u + ) ⊆ WF ( u + ) , and ( II ) WF (˜ u + ) ⊆ WF ( A σx ˜ u + ) . Since ¯ Y = D t + A σx and D t clearly is a microlocal, i.e, WF ( D t w ) ⊆ WF ( w ) for any w ∈ D ′ ( R ) ,we may reduce (I) to the statement WF ( A σx u + ) ⊆ WF ( u + ) . Furthermore, A σx , acting only in the x -variable, commutes with restriction to t > , hence in intermediate steps we may consider A σx as convolution on R with F − ξ → x ( b β | ξ | σ ) ⊗ δ ( t ) and restrict to t > afterwards.Note that we have ˜ u = B σx A σx ˜ u = A σx B σx ˜ u , where B σx is x -convolution with the inverse Fouriertransform of the locally integrable function ξ / ( b β | ξ | σ ) . Thus, the statements (I) and (II)are equivalent to showing WF ( G σ u + ) ⊆ WF ( u + ) and WF ( G − σ A σx ˜ u + ) ⊆ WF ( A σx ˜ u + ) with G γ and g γ specified as in Remark 2.1(ii) with γ = σ or γ = − σ (both in the range − < γ < );clearly, G γ also commutes with restriction to t > and A σx = b β G σ , B σx = G − σ /b β .The operator G γ can be considered as convolution on R with the distribution g γ ( x ) ⊗ δ ( t ) , where b g γ ( ξ ) = | ξ | γ . The one-dimensional homogeneous distribution g γ can be determinedexplicitly via [5, Example 7.1.17], showing directly that singsupp ( g γ ) = { } , and we easily deducefrom [5, Theorem 8.1.8.] that WF ( g γ ) = { (0 , ξ ) | ξ = 0 } . Hence [5, Theorem 8.2.9] givesWF ( g γ ⊗ δ ) ⊆ { (0 , ξ, τ ) | ( ξ, τ ) = (0 , } ∪ { ( x,
0; 0 , τ ) | x ∈ R , τ = 0 } G γ w = ( g γ ⊗ δ ) ∗ w is defined in case of w = u + or w = A σx ˜ u + , we may prove by cut-off techniques the appropriate extension of [5, Equation(8.2.16)] to these cases and obtainWF ( G γ w ) ⊆ WF ( w ) ∪ { ( x, t ; 0 , τ ) | ∃ y ∈ R : ( y, t ; 0 , τ ) ∈ WF ( w ) } | {z } =:WF vert ( w ) . Equations (15) and (13) show WF vert (˜ u + ) = ∅ and WF vert ( u + ) = ∅ , respectively, hence the proofof (I) and (II) is complete.We may now put (I) and (II) to use with the outermost equalities in (16) and arrive at thefollowing: WF (˜ u + ) ⊆ WF ( A σx ˜ u + ) = WF ( ¯ Y u + ) ⊆ WF ( u + ) . (17)In summary, combining Equations (15) and (17) with (13) we obtain { ( x, t ; ξ, | t > , ( x, ξ ) ∈ WF ( u ) } = WF (˜ u + ) ⊆ WF ( u + ) ⊆ { ( x, t ; ξ, | t > , ( x, ξ ) ∈ WF ( u ) } , hence we have, in fact, equality in all places of the above chain of inclusions, thereby the proofof the theorem in case v = 0 is completed.As shown in Lemma 2.2 the microlocal structure of E +1 is equivalent to that of E +0 , hence wehave the same kind of wave front set statement with v in place of u , if u is smooth, since inthis case u + = E +1 ∗ x v plus a smooth contribution steming from u .Finally, the solution in the general case u , v ∈ E ′ ( R ) is just the sum of the two solutions forthe special cases v = 0 and u = 0 , hence its wave front set is contained in the correspondingunion. Invariance of the wave front set under positive time translations follows in this case aswell, since it was established via the operator factorization Z = ¯ Y · Y with subsequent “symbolcorrection factor” B and is valid for solutions w of Y Bw = 0 independent of initial values.
Remark 2.6.
The result on the wave front set of u + in the above theorem implies, in particular,smoothness of u + considered as a map from time into distributions on space (cf. [2, (23.65.5)]),i.e., u + ∈ C ∞ (]0 , ∞ [ , D ′ ( R )) ; in addition, we have u + ( t ) ∈ S ′ ( R ) for every t > . For the special case of (1) when β = 1 and ≤ α < we obtain the so-called time-fractionalZener wave equation Zu ( x, t ) = ∂ t u ( x, t ) − L αt ∂ x u ( x, t ) = u ( x ) ⊗ δ ′ ( t ) + v ( x ) ⊗ δ ( t ) , (18)whose unique solvability by distributions supported in a forward cone has been established in[9]. Here we show a kind of non-characteristic regularity of the solution u to problem (18).The “Fourier symbol” of Z is z ( ξ, τ ) = − τ + l α ( τ ) ξ with l α ( τ ) := 1 + b e i απ ( τ − i α a e i απ ( τ − i α = 1 + b i α sgn ( τ ) | τ | α a i α sgn ( τ ) | τ | α , to which we apply a conic cut-off to obtain a smooth symbol in both variables ( ξ, τ ) , similarlyas in Lemma 2.5 above. 9 emma 3.1. Let Γ ⊆ R (representing the ( ξ, τ ) -plane) be the union of a closed disc around (0 , and a closed narrow cone containg the ξ -axis and being symmetric with respect to bothaxes. Let Γ ′ be a closed set of the same shape as Γ , but with slightly larger disc and openingangle of the cone. Let ˜ b ∈ S (cid:0) R × R (cid:1) such that ˜ b ( x, t, ξ, τ ) is real, constant with respect to ( x, t ) , homogenous of degree with respect to ( ξ, τ ) away from the disc contained in Γ ′ , and suchthat ˜ b ( x, t, ξ, τ ) = 0 , if ( ξ, τ ) ∈ Γ , ˜ b ( x, t, ξ, τ ) = 1 , if ( ξ, τ ) Γ ′ . Then p := ˜ b z is a symbolbelonging to the class S (cid:0) R × R (cid:1) . The proof is a variation of that of Lemma 2.5.
Theorem 3.2.
For the wave front set of u + , the restriction of the solution u to (18) to forwardtime t > , we have the inclusion WF ( u + ) ⊆ { ( x, t ; ξ, τ ) | x ∈ R , t > , ξ = 0 , τ = ba ξ or τ = 0 } . Proof.
Let B and P be the pseudo-differential operators associated with the symbols ˜ b and p ,respectively, constructed in Lemma 3.1 according to arbitrary, but fixed, Γ and Γ ′ chosen asspecified there. We have P = BZ and therefore P u + = BZu + = 0 . By non-characteristic regularity [8, Theorem 18.1.28],WF ( u ) ⊆ Char ( P ) , where the characteristic set is Char ( P ) = (cid:0) R × (cid:0) R \ { (0 , } (cid:1)(cid:1) \ M with M being defined asthe set of all points ( x , t , ξ , τ ) such that there exist c > , R > and a conic neighborhood V of ( ξ , τ ) such that the following estimate holds: ∀ ( ξ, τ ) ∈ V, ξ + τ ≥ R : | p ( ξ, τ ) | ≥ c ( ξ + τ ) . (19)1. We have R × (Γ \ { (0 , } ) ∩ M = ∅ , since ˜ b ( ξ , τ ) = 0 whenever ( ξ , τ ) ∈ Γ . As Γ getsmore and more narrow (and smaller around the origin) only points of the form ( x , t , ξ , will remain with this property.2. We have no definite decay properties of the symbol in all of R × (Γ ′ \ Γ) , but this will notbe required as we let later shrink both Γ ′ ⊃ Γ to R × { } , causing Γ ′ \ Γ → ∅ .3. Suppose ( x , t , ξ , τ ) ∈ R × (cid:0) R \ Γ ′ (cid:1) , which will leave only points with τ = 0 upon theshrinking process of Γ ′ and Γ .(a) If τ = ba ξ , then the estimate (19) must fail in any conic neighborhood of ( ξ , τ ) ,since for λ > we have | p ( λξ , λτ ) | = | z ( λξ , λτ ) | = λ ξ (cid:12)(cid:12)(cid:12)(cid:12) − ba + b i α sgn ( τ ) | τ | α + λ − α a i α sgn ( τ ) | τ | α + λ − α (cid:12)(cid:12)(cid:12)(cid:12)| {z } =: d ( λ ) , where d ( λ ) → as λ → ∞ , which makes a lower bound of the form | p ( λξ , λτ ) | ≥ c λ to hold for all λ ≥ R/ p ξ + τ impossible. Thus, ( x , t , ξ , τ ) M .10b) If τ = ba ξ , we define a closed conic neighborhood of the point ( ξ , τ ) by V := n ( ξ, τ ) ∈ R | (cid:12)(cid:12)(cid:12) τ − ba ξ ξ + τ (cid:12)(cid:12)(cid:12) ≥ c − δ o , where c := (cid:12)(cid:12)(cid:12) τ − ba ξ ξ + τ (cid:12)(cid:12)(cid:12) and < δ < c . Let V R := V ∩ { ( ξ, τ ) | ξ + τ ≥ R } and suppose that R > is large enough and δ chosensufficiently small to ensure V R ∩ Γ ′ = ∅ as well as V R ∩ (cid:8) ( ξ, τ ) | τ = ba ξ (cid:9) = ∅ .Let ( ξ, τ ) ∈ V R , then τ ≥ ( c − δ )( ξ − τ )+ ba ξ ≥ min( c − δ, ba )( ξ + τ ) ≥ c R . Since l α ( τ ) → ba ( | τ | → ∞ ) we may thus choose R large enough to have l α ( τ ) − ba < c − δ ,if ξ + τ ≥ R and ( ξ, τ ) ∈ V . Thus, ( ξ, τ ) ∈ V R implies | p ( ξ, τ ) | = | z ( ξ, τ ) | = | τ − l α ( τ ) ξ | = | τ − ba ξ − ( l α ( τ ) + ba ) ξ |≥ | τ − ba ξ | − | l α ( τ ) + ba | ξ ≥ ( c − δ )( ξ + τ ) − c − δ ξ ≥ c − δ ξ + τ ) . Therefore we have in this case ( x , t , ξ , τ ) ∈ M .To summarize,WF ( u + ) ⊆ Char ( P ) ⊆ R × (cid:18) (Γ \ { (0 , } ) ∪ (Γ ′ \ Γ) ∪ { ( ξ , τ ) | τ = ba ξ } (cid:19) . This result holds for any Γ and Γ ′ chosen arbitrarily according to the specifications in the previouslemma. Letting Γ ′ ⊇ Γ both shrink toward the ξ -axis yields the claim of the theorem, since wemay use the intersection of all corresponding (Γ , Γ ′ ) -dependent sets in the middle and in theright part of the above chain of inclusions. Acknowledgement
This research is supported by project P25326 of the Austrian Science Fund and by projects , of the Serbian Ministry of Education and Science, as well as by project − − of the Secretariat for Science of Vojvodina. References [1] T. M. Atanackovic, M. Janev, Lj. Oparnica, S. Pilipovic, and D. Zorica. Space-time frac-tional Zener wave equation.
Proceedings of the Royal Society A: Mathematical, Physical andEngineering Sciences , 471:20140614–1–25, 2015.[2] J. Dieudonné.
Treatise on Analysis , volume VIII. Academic Press, Boston, 1993.[3] J. Duoandikoetxea.
Fourier Analysis . American Mathematical Society, Providence, 2001.[4] L. Hörmander. Fourier integral operators. I.
Acta Mathematica , 127:79–183, 1971.[5] L. Hörmander.
The Analysis of Linear Partial Differential Operators I. Distribution Theoryand Fourier Analysis . Springer-Verlag, Berlin, 1983.[6] L. Hörmander.
The Analysis of Linear Partial Differential Operators II. Differential Oper-ators with Constant Coefficients . Springer-Verlag, Berlin, 1983.[7] L. Hörmander.
The Analysis of Linear Partial Differential Operators IV. Fourier IntegralOperators . Springer-Verlag, Berlin, 1985. 118] L. Hörmander.
The Analysis of Linear Partial Differential Operators III. Pseudo-DifferentialOperators . Springer-Verlag, Berlin, 1994.[9] S. Konjik, Lj. Oparnica, and D. Zorica. Waves in fractional Zener type viscoelastic media.
Journal of Mathematical Analysis and Applications , 365:259–268, 2010.[10] E. M. Stein.