36 ( ) Ω λk(k= + )-Δ <γ < (4) L (Ω) φ k λk : (-Δ) /φ γ / k=λγ k φ k { <λ λ λk (k ) D((-Δ) γ / )= {u L (Ω)stu Ω = ; (-Δ) γ / u L (Ω) = k=+ λ γ / k u φ

5 3 ( ) Vol5 No3 5 JournalofXiamenUniversity (NaturalScience) May ( 365) : D +α u-δu+(-δ) γ/ D β u= u >-<α< <γ <β< : ; ; : O755 : A : 438-479 ()3-35-6 ρ ut=u +λu +g (xt) (6) xx xtx D +α u-δu+(-δ) γ / D β u= u () ρ λ g(xt) u(x)= u t (x)=u () (xy) Q =R n ()< + -<α< <β<δ= + + ()α=β= Kiane x x n [5] (-Δ) γ / (<γ ) :(-Δ) γ / v=f - ( ξ γ F(v)( ξ )) F F - D +α atar [] D +α u-δu+d β u=h(xt) u (5) ()γ= :-- : (9766) Email:yqx458@6com Carvalho [4] u t=δu-(-δ) γ / u t+u (7) u t-δu+(-δ) γ / D β +u=u (8) D β (Cauto) D β [6] + Riemann-Liouvile [5] lim infu = + x D β f(t)= Γ(n-β t ) (t-τ) n- β- f(n)(τ)dτn = <β (8) [5] [ β ]+ β > (3) -<α<<β< Seredynkska [] Mitidieri [7] Zhang [8] atar Kirane [5] Baras [9] D u+γd + η u+f(u)= (4) () D + η <η< +η Lalacian [5] η [5] () γ Greenberg [3] () 烄 (-Δ) /φ γ k=v φ x Ω; k 烅烆 φ k= x R n \Ω (9)

36 ( ) Ω λk(k= + )-Δ <γ < (4) L (Ω) φ k λk : (-Δ) /φ γ / k=λγ k φ k { <λ λ λk (k ) D((-Δ) γ / )= {u L (Ω)stu Ω = ; (-Δ) γ / u L (Ω) = k=+ λ γ / k u φ k <+ } () k= u D((-Δ) γ / ) k=+ (-Δ) γ / u = λ γ / k u φ k φ k k= (D t ω)()=(d t ω)()=c -α (7) uv D((-Δ) γ / ) C ( = -α+η ) Γ( η +) Γ(-α+η ) u(-δ) γ / vdx = Ω v(-δ) γ / udx () Ω Cauto D γ f(t)= Γ(-γ) t (t-τ) -γ f (τ)dτ <γ < () Γ(n-γ) t D γ f(t)= ( ) <α [] β < u L loc(q ) φ C (Q ) φ (t-τ) n-γ- f (n) (τ)dτ n = [γ]+γ > (3) Cauto Riemann-Liouvile D tf(t)= γ Γ(n-γ) (d dt t )n (t-τ) n-γ- f(τ)dτ n = [γ]+ [6] n- D tf(t)= γ f (k) ()t k-γ Γ(+k-γ) + k= Γ(n-γ) (d dt t )n (t-τ) n-γ- f (n) (τ)dτ= n- k= f (k) ()t k-γ Γ(+k-γ) +Dγ f(t) > Riemann-Liouvile Dt f(t)= ( -) n γ Γ(n-γ) (d dt )n (τ-t) n-γ- f(τ)dt t n = [γ]+γ > [9] f (t)(d γ tg)(t)dt= g (t)(d γ t f)(t)dt ω(t)= (- )η + > η α () Dt ω(t)= ( -α+η ) Γ( α η +) Γ(-α+η ) (- )η-α -α + (5) D t ω(t)= ( -α+η )( η-α)γ( η +) Γ(-α+η ) -( ) (- )η -α- + (6) Q () R n L loc(q ) v:r n R + R + R n R + K u L loc(q ) u φdxdt= u(xt)d t φ dxdt- u D α t φ dxdt- D α t φ (x)dx+ [u(xt)- ](-Δ) γ/ D β t φ (xt)dx- u(xt)δφdxdt (8) u () φ φ (x )=D t α φ (x)= <α<<β< -<α<<β<-<α<<β<3 [6] D +α tf=d D α tfd +α t f=-d D t f α (9) D n+α f(t)=d α D n f(t)<α<n= () <α β < u <α β < γ (] < * =+ β +βn-β

3 : 37 ()~ () = * γ () >n/(n-γ) n>γ ()~ () φ ( xt)=φ φ (t) φ = ( x / β/ ) φ (t)=(-t/) η η {x R N ; x β/ } (r)= Ω r ; { r ~(5) ε< 3 (r) C/rr D ((-Δ) r/ )(-Δ) r/ B u φ C( Ω φ- λ γ / () φ Q (7)(9)() u φ+ u D α t φ+ D α t φ (x )+ (-Δ) γ / D β t φ (xt)= u(xt)d t φ + u(-δ) γ / D β t φ (x t)+ u(xt)(-δ) φ () = dxdt R N = R Ndxdt () (5)(7) (7) (-Δ) /φ γ = -βγ/ λ γ / φ u φ +C -α u φ +C( -α + - β-βγ) φ = u(xt)d t φ + u(xt)(-δ) γ/ D β t φ (xt)+ u(-δ) Q φ ε u φ + Ω C(ε) Ω φ-q (-Δ) φ q (5) C(ε)= - ( ε) -q=/(-)ω = [] Ω φ- Ω φ- q q D = () Ωdxdt t φ q + (-Δ) γ /φ D β t φ (t) q + q (-Δ) φ φ (t) q ) (6) (6) τ= - ty= - β / x (5)(6) u φ C( + βn -q () + +β n -βq(+γ) + +β n - βq ) (7) < * (7) Q u u=aer n R + = * u C (8) Q (6) φ = φ ( x /(B - β / β/ )) B B () u(xt)(-δ) Q φ u u φ u(xt)d t φ + u(xt)(-δ) γ/ D β t φ (xt)+ u(xt)(-δ) Q φ () εyoung ud t φ = uφ φ - D t φ u ε Ω φ +C(ε) D Ω φ-q t φ q (3) u(-δ) γ/ Dt β Q φ ε u φ + Ω C(ε) Ω φ-q (-Δ) γ / D β t φ q (4) u φ C( Ω u D t φ q + Ω u (-Δ) γ/ φ D β t φ (t) q + C(B) u (-Δ) φ φ (t) ) (9) Ω = [] {x R n ; x B - β / β/ } C(B)= [] {x R n ;B - β / β/ x B - β / β/ } Ω = Ω dxdt C(B) = C(B) dxdt (9) ε Young 3 Holder u φ C( φ Q Ω φ (t) - q D t φ (t) q +

38 ( ) Ω φ -q φ (t) -q (-Δ) γ /φ D β t φ (t) q )+C( C(B) u φ )/ 3 ( C(B) φ -q φ (t) (-Δ) φ q ) / q (3) τ= - ty=(/b) - β / x u φ C( + βn -q () B -βn/ + +β n -βq(+γ) β B (-n+γq) )+C +β n -βn B β(-n+q) ( C(B) u φ )/ (3) lim + C(B) u φ =( (8) ) (3) (3) R N u dxdt CB - βn +CB β (-n+γq) (33) >n/(n-γ)b u= aer n R + γ= >n/(n-γ) () u φ C( u D t Ω φ + φ (x/r)d t β φ (t) q + φ -q (-Δ) u (-Δ) γ/ φ D C(B) t β φ (t) + φ (x/r) φ (t) q ) (38) u (-Δ) C(B) φ φ (t) ) (34) t=τ (5)~ (6) (-Δ) γ / (x/r)=r -γ λ γ / (34) ε φ (x/r) Young 3 Holder -α u φ (x/r)+ ( -α + - β R -γ ) u φ C φ Q φ - q D t Ω φ q + C( C(B) u φ )/ ( C(B) φ-q φ -q (-Δ) γ /φ D β t φ (t) q + -qφ (-Δ) C(B) φ φ q ) / q (35) :τ= - ty=(/b) - β / x > u u ()() A -α lim inf + -α lim infu x + x + A -( )q φ ( xt)=φ (x/r) φ (t) φ (t) = (-t/) η η φ D ((-Δ) r/ ) (-Δ) r/ B λ= λ γ / suφ {< x <}(su ) ()(3)~(5) u D α t φ + D α t φ (x)+ (-Δ) γ/ D β t φ (xt) C( φ -q D t φ q + φ -q (-Δ) γ / φ (x/r) C ( -( )q + - βq R -γq +R -q ) φ (x/r) (39) (39) φ (x/r) inf φ (x/r) u φ C + βn -q () B -βn/ + C( +β n -βq(+γ) β B (-n+γq) + +β n - βq B β (-n+q) ) ( C(B) u φ )/ (36) (3) (36) R N u dxdt CB - βn (37) B u=aer N R + u φ (x/r) inf u φ (x/r) φ (x/r) ( -α + - β R -γ )inf + -α inf u C( -( )q + - βq R -γq +R -q ) R

3 : 39 -α lim inf + -α C -( )q lim infu lim inf = lim infu = () 4 : inf ( x γ (q-) ) C (44) 3 () lim (44) R inf = lim infu = A = lim inf > A = lim infu > u φ (x/r) C( β -()q R γ + > < A A (+α )(q-) < A A β -βq R -γq+γ + β R -q+γ ) (x/r) )q-α (-α φ () 3 > u (46) () K K K 3 (i)lim inf( x γ (q-) ) K (i)< < +α+ β β lim inf( x γ ())(q-)-β ) K (i)lim inf( x γ ()q-α u ) K 3 (i) (39) > φ (x/r) C( -( )q + - βq R -γq + R -q ) (x/r) φ - C 4 =C 3q() β ()q[()q-β β ]β-()q C( -( )q + R -γq ) (x/r)(4) φ x γ[ β+(-q)()] φ (x/r) (4) * = (q-) ()q γ R φ (x/r) C R γ (x/r) (-q) φ (4) C =C [(q-) - + (q-) q ] (4) suφ {x R x R} C x γ (-q) φ (x/r) (4) φ inf ( x γ(q-) ) x γ (-q) φ (x/r) φ (x/r) (43) (4)(43) x γ (-q) φ (x/r) lim inf( x γ (q-) ) C (45) (i) C 3 ( β -()q R γ + β R -γq+γ ) φ (x/r) * =[ β ()q-β ]- φ (x/r) ()q R γ (46) C 4R γ[ β+(-q)()] φ (x/r) (47) inf ( x γ[(q-)()-β] φ (x/r) C4 ) x γ[ β+(-q)()] x γ[ β+(-q)()] φ (x/r) inf ( x γ[(q-)()-β] ) C 4 (48) R lim inf( x γ[(q-)()-β] (i) ) C4 u φ (x/r) C( α-( )q + φ (x/r) α- βq R -γq + α R -q ) φ (x/r)

3 ( ) C 5 ( α-( )q + α R -γq ) φ (x/r) (49) * = [ α ()q-α ]- R γ u φ (x/r) ()q C 6R γ[α-()q] φ (x/r) (5) C 6=C 5q()α - ()q[()q-α] α α-()q ()q (5) suφ {x R x R} u φ (x/r) C 6 x γ[α-()q] φ (x/r) (5) φ inf ( x γ[()q-α] u ) x γ[α-()q] φ (x/r) uφ (x/r) (5) (5)(5) x γ[α-()q] inf ( x γ[()q-α] φ (x/r) u ) C6 R : [] Seredynska MHanyga ANonlinearhamiltonianequa- tionswithfractionaldaming[j]jmathphys4: 35-56 [] atarn ENonexistenceresultsforafractionalroblem arisinginthermaldifusioninfractalmedia[j]chaos SolitonsandFractals836:5-4 [3] GreenbergJ M MacCamyRMizelVJOntheexist- enceuniquenessandstabilityofsolutionsoftheequation [J]JMath Mech967/9687:77-78 [4] Carvalho A NCholewaJ WAtractorsforstronglyd- amedwaveequationswithcriticalnonlinearities[j]pa- cificjmath7:87-3 [5] KiraneMLaskriYNonexistenceofglobalsolutionstoa hyerbolicequationwithasace-timefractionaldaming [J]Alied Mathematicsand Comutation567: 34-3 [6] SamkoS GKilbasA AMarichev OIFractionalinte- gralsandderivativestheoryandalications[m]new York:GordonandBeachSciencePublishers987 [7] MitidieriEPohozaevSIArioriestimatesandblow-u ofsolutionstononlinearartialdiferentialequationsand inequalities[j]proc SteklovInst Math34:- 383 [8] ZhangQSAblow-uresultforanonlinearwaveequa- tionwithdaming:thecriticalcase[j]cr AcadSciPar- is333:9-4 [9] BarasPKersnerRLocalandglobalsolvabilityofaclass ofsemilineararabolicequations[j]jdiferentialequa- tions98768:38-5 [] emmam RInfinite-dimensionaldynamicalsystemsin mechanicsandhysics[m]new York:Sring-Verlag 998 [] PodlubnyIFractionaldiferentialequationsmathemat- icsinscienceandengineering[m]new York/London: Sringer999 NonexistenceofWeakSolutionsforaFractionalDifferentialEquation XU Yong-qiang (SchoolofMathematicalSciencesXiamenUniversityXiamen365China) Abstract: WeconsidertheCauchyroblemforthefractionaldiferentialequationD +α u-δu+(-δ) r/ D β u= u withgiveninitial dataandwhere>-<α<<r and<β<nonexistenceresultsandnecessaryconditionsforlocalandglobalexistence areestablishedbymeansofthetest-functionmethodheseresultsimroveandextendreviousworks Keywords: nonexistence ;fractionalderivative;weaksolutions;necessaryconditions