BLOW UP RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS AND SYSTEMS. Ali Hakem and Mohamed Berbiche

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série tome 93 (107 ( DOI: /PIM H BLOW UP RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS AND SYSTEMS Ali Hakem and Mohamed Berbiche Communicated by Stevan Piliović Abstract The aim of this research aer is to establish sufficient conditions for the nonexistence of global solutions for the following nonlinear fractional differential euation D α 0 t u + ( /2 u m 1 u + a(x u 1 u = h(x t u (t x Q u(0 x = u 0 (x x R N where ( /2 0 < < 2 is the fractional ower of and D α (0 < α < 1 0 t denotes the time-derivative of arbitrary α (0; 1 in the sense of Cauto The results are shown by the use of test function theory and extended to systems of the same tye 1 Introduction In his article [3] Fujita considered the Cauchy roblem (11 u t = u + u 1+ in Q = R n R + u(0 x = a(x in R n where > 0 If c = 2 n he roved that: 1 If 0 c and a(x 0 > 0 for some x 0 then any solution to (11 blows-u in a finite time 2 If > c then there exist a solution on Q The critical case = c was decided later by Hayakawa [6] for N = 1 2 and by Kobayashi Sirao and Tanaka [9] for n Mathematics Subject Classification: 58J45 26A33 35B44 Key words and hrases: blow-u; fractional derivatives; critical exonent This research was artly suorted by MESRS-ALGERIA (CNEPRU B

2 174 HAKEM AND BERBICHE In a more recent article Guedda and Kirane [5] extended the revious results to the euations u t = ( /2 u + h(x tu 1+ in Q = R N R + u(0 x = a(x in R N where h(x t = O(t σ x ρ for x large Finally Kirane and Qafsaoui [8] treated the more general euation u t + ( /2 (u m + a(x t u = h(x tu 1+ in Q The techniue we use has been introduced by Mitidieri and Pohozaev [10] [11] Pohozaev and Tesei [12] Pohozaev and Veron [14] and used by Hakem and Berbiche [1] Let us consider the following nonlinear fractional differential euation (12 D α 0 t u + ( 2 ( u m 1 u + a(x ( u 1 u = h(x t u u(0 x = u 0 (x x R N where D α 0 t denotes the time-derivative of an arbitrary order α (0 α in the sense of Cauto [14] ( /2 [1 2] is the ( 2 -fractional ower of the Lalacian x in the x variable; a(x := (a 1 (x a N (x and h(x t are given functions a(x ( u 1 u is the scalar roduct of a(x and ( u 1 u and the exonents > 1 1 and m 1 are ositive constants The nonlocal oerator ( 2 is defined by ( 2 v(x = F 1( ξ F(v(ξ (x for every v D(( 2 = H (R N where H (R N is the homogeneous Sobolev sace of order defined by H (R N = { u S ; ( 2 u L 2 (R N } if / N H (R N = { u L 2 (R N ; ( 2 u L 2 (R N } if N where S is the sace of Schwartz distributions; F denotes the Fourier transform and F 1 its inverse The fractional Lalacian ( 2 is related to Lévy flights in hysics Many observations and exeriments related to Lévy flights (suerdiffusion eg collective sli diffusion on solid surfaces uantum otics or Richardson turbulent diffusion have been recently erformed The symmetric -stable rocesses ( (0 2 are the basic characteristics for a class of juming Lévy s rocesses Comared with the continuous Brownian motion ( = 2 symmetric -stable rocesses have infinite jums in an arbitrary time interval The large jums of these rocesses make their variances and exectations infinite according to (0 2 or (0 1] resectively It is worth mentioning that when = 3 2 the symmetric -stable rocesses aear in the study of stellar dynamics The time fractional derivative has been found to be very effective means to describe the anomalous attenuation behaviors We here recall some definitions of fractional derivative

3 BLOW UP RESULTS 175 The left-handed derivative and the right-handed derivative in the Riemann Liouville sense for Ψ L 1 (0 T 0 < α < 1 are defined as follows: (D0 t α Ψ(t = 1 d Γ(1 α dt t 0 Ψ(σ (t σ α dσ where the symbol Γ stands for the usual Euler gamma function and (Dt T α Ψ(t = 1 d Γ(1 α dt resectively The Cauto derivative reuires Ψ L 1 (0 T Clearly we have (13 (D α 0 t Ψ(t = 1 Γ(1 α [ (D0 t α g(t = 1 g(0 Γ(1 α t α + (D α t T f(t = 1 Γ(1 α t 0 T t t 0 [ f(t (T t α Ψ(σ (σ t α dσ Ψ (σ (t σ α dσ g ] (σ (t σ α dσ T t f (σ (σ t α dσ Therefore the Cauto derivative is related to the Riemann Liouville derivative by D α 0 t Ψ(t = Dα 0 t [Ψ(t Ψ(0] We will use the formula of integration by arts [13 46] T 0 f(t(d α 0 t g(t dt = T 0 g(t(dt T α f(t dt Solutions to roblem (12 are meant in the following sense Definition 11 A function u L loc (Q T where Q T := R N (0 T is a local weak solution to (12 defined on Q T if uh 1/ L 1 loc (Q T dx dt such that (14 h(x tξ u dx dt + u 0 Dt T α ξ dx dt Q T Q T = udt T α ξ dx dt + u m 1 u( 2 ξ dx dt Q T Q T N u 1 uξ a i dx dt u 1 ua ξ dx dt Q T x i Q T for any test function ξ C 21 xt (Q T such that ξ(x T = 0 The integrals in the definition are suosed to be convergent If in the above definition T = + the solution is called global To begin we set some hyotheses For the function h we reuire the condition (H h h(yr τt /α C h R σ T ρ/α C h > 0 ]

4 176 HAKEM AND BERBICHE for some σ ρ > 0 to be determined later R T large and τ 0 y in a bounded domain It can easily be seen that there is no conditions imosed on σ The vector a(x = (a 1 (x a N (x is reuired to satisfy (H a a i (x c x δi for x large and δ i > 2 For later use we define δ = max(δ i 11 The Results Now we may state our first result Theorem 11 Let N 1 > max(m 1 The exonent ρ satisfies { (ρ + 1 > max m (1 α } Assume that (H h and (H a are satisfied and u 0 (x satisfies u 0 (x 0 If ( α(σ + + ρ ((αn + + (ασ + ρ min 1 + αn + (1 α ((δ 1α + (Nα + then roblem (12 admits no global weak solutions other than the trivial one Proof The roof roceeds by contradiction Suose that u is a nontrivial solution which exists globally in time That is exists in (0 T for any arbitrary T > 0 Let T and R be two ositive real numbers such that 0 < T R /α < T For later use let Φ be a smooth nonincreasing function such that { 1 if z 1 Φ(z = 0 if z 2 and 0 Φ 1 The test function ξ is chosen so that (hξ m/( m ( /2 ξ /( m dx dt< (hξ 1/( 1 Dt T α ξ /( 1 dx dt< Q T Q T N /( h /( ξ a Q T x i dx dt < (hξ /( a ξ /( dx dt < Q T To estimate the right-hand side of (12 on we write u m 1 u( /2 ξdx dt = u m 1 u(hξ m/ (hξ m/ ( /2 ξdx dt Using the ε-young ineuality we have the estimate u m 1 u( 2 ξdx dt Similarly XY εx + C(εY + = X Y 0 ε hξ u dx dt + C(ε (hξ m m ( 2 ξ m dx dt

5 ud α t T R ξdx dt = 2/θ BLOW UP RESULTS 177 ε hξ u dx dt + C(ε u(hξ 1 1 (hξ D α t T R ξdx dt 2/θ Integrating by arts we get a ( u 1 uξdx dt = u 1 ua ξdx dt Now writing N u 1 uξ a dx dt = x i and using the ε-young ineuality we get u 1 uξ N a dx dt x i ε hξ u dx dt + C(ε Similarly we have u 1 u(a ξ dx dt 2/θ ε hξ u dx dt + C(ε (hξ 1 1 D α t T R ξ 2/θ 1 dx dt N u 1 u(hξ ( h ξ h ξ N u 1 uξ a x i dx dt N a x i dx dt a x i dx dt (hξ a ξ dx dt Combining the above estimates with (14 and taking ε small enough we infer that (15 ( C(ε u 0 D α t T R ξ dx dt + 2/θ + u ξh dx dt (hξ 1 1 D α t T R ξ 2/θ 1 dx dt + (hξ a ξ dx dt + h ξ N (hξ m m At this stage we set ( x 2 + t θ ξ(x t := Φ R 2 where R and θ are ositive real numbers to be determined latter a x i dx dt m ( 2 ξ m dx dt

6 178 HAKEM AND BERBICHE We note that ξ(x T R 2/θ = 0 for T θ 2 then by (13 we have u 0 D α t T R ξ dx dt 0 2/θ Let us erform the change of variables τ = t/r 2/θ y = x/r and set Ω := { (y τ R N R + y 2 + τ θ < 2 } µ(y τ := y 2 + τ θ We have the estimates D α t T R ξ /( 1 (hξ 1/( 1 dx dt 2/θ and R 2 θ α/( 1+N+ 2 θ 1 1 (σ+ 2ρ θ ( 2 ξ m (hξ m m dx dt R m +N+ 2 θ m m (σ+ 2ρ θ h N a x i ξdx dt R (σ+ 2 θ ρ+ (δ 2+N+ 2 θ (hξ /( a ξ /( dx dt R (σ+ 2 θ ρ+(δ 1 +N+ 2 θ Now we choose θ such that θ + ( mθn + 2( m m(θσ + 2ρ Ω Ω D α τ T Φoµ 1 (Ch y σ τ ρ Φoµ 1 1 dy dτ ( 2 Φoµ m (Ch y σ τ ρ Φoµ m m dy dτ Ω (C h y σ τ ρ N Ω a i Φoµdy dτ y i (C h y σ τ ρ Φoµ a ξ dτ dy 2α + ( 1θN + 2( 1 (θσ + 2ρ then it is sufficient to take θ = 2α We then have the estimate (16 h u ξ dx dt C(ε(R s1 + R s2 + R s3 where /α ( 1θs 1 = 2α + ( 1θN + 2( 1 (θσ + 2ρ ( mθs 2 = θ + ( mθn + 2( m m(θσ + 2ρ ( θs 3 = (δ 1θ + (Nθ + 2( (θσ + 2ρ

7 BLOW UP RESULTS 179 and C(ε is a generic ositive constant deending on ε Now if we choose max(s 1 s 2 s 3 < 0 that is ( α(σ + + ρ ((αn + + (ασ + ρ < min 1 + αn + (1 α ((δ 1α + (Nα + and let R in (16; we obtain R N R + h u dx dt 0 This imlies that u = 0 ae which is a contradiction If = c (ie max(s 1 s 2 s 3 = 0 the critical case we have from (16 (17 h u dx dt C R n R + We modify the test function ξ by introducing a new fixed constant S (0 < S < R such that ( x 2 ξ(x t := Φ R 2 + tθ (SR 2 We set C RS = {(x t R n R + : R 2 x 2 + tθ S 2 2R2} See that because of the convergence of the integral in (17 then (18 lim h u ξ dx dt = 0 R C RS By using the Hölder ineuality we get u 1 ua ξ dx dt = u 1 ua ξ dx dt C RS Q T (SR 2/θ ( ( u hξ dx dt a ξ dx dt C RS CRS(hξ where we have used that the suort of a ξ is C RS Taking into account of the scaled variables: t = (RS 2 θ τ x = Ry ξ(x t = ξ(ry (RS 2 θ τ = χ(y τ and the fact that = c then instead of estimate (15 we get (19 (1 3ε h u ξ dx dt where Q T (SR 2/θ ( ( u hξ dx dt a ξ dx dt C RS CRS(hξ + C(ε ( L 1 S 1 1 ( 2ρ θ 2 θ α θ + L2 S m m ( 2ρ θ + 2 θ + L3 S ( 2 θ ρ+ 2 θ L 1 := χ 1 1 Ω D α t T χ 1 dy dτ

8 180 HAKEM AND BERBICHE L 2 := χ m m Ω Ω m ( 2 χ m dy dτ ( n L 3 := χ a y i dy dτ Using (19 we obtain via (18 after assing to the limit as R (110 h u dx dt C (S 1 1 ( 2ρ θ 2 θ α θ + S m m ( 2ρ θ + 2 θ + S ( 2 θ ρ+ 2 θ R n R + Finally we realize that the left-hand side of (110 is indeendent of S then by assing to the limit when S goes to infinity we obtain u = 0 which is contradiction and this comletes the roof Remark 11 When the vector a = 0 and = m = 1 we recover the case studied by Kirane-Tatar [6] When a = 0 = m = 1 σ = ρ = 0 α = 1 and = 2 the critical exonent coincides with the well known Fujita exonent [2] (FDS 2 System of Fractional Differential Euations This section is devoted to the following system of reaction-diffusion euations D α 0 t (u u 0 + ( 2 ( u m 1 u = h(t x v + g(t x u r D δ 0 t (v v 0 + ( 2 ( v m 1 v = k(t x u + l(t x v s in Q in Q subject to the initial conditions u(x 0 = u 0 (x 0 v(x 0 = v 0 (x 0 x R N where 0 < α δ < 1 and 0 2 The functions h g k l are assumed to satisfy the conditions h(t x C 1 t ω1 x d1 g(t x t ω2 x d2 when x large k(t x C 3 t ω3 x d3 l(t x t ω4 x d4 when x large for t > 0 x 1 ω 1 0 ω 2 0 ω 3 0 ω 4 0 d 1 0 d 2 0 d 3 0 and d 4 0 We set λ i = ω i + α d i and η i = ω i + δ d i for i = 1 4 For the system (FDS we have If Theorem 21 Let > 1 m r and s 1 and assume that (1 > max(m 2 sm sr mr (2 (η3+1 (η (λ r > 1 (λ > (λ s > 1 (λ > ms > 1 (21 N max(θ 1 θ 2 where θ 1 = min 1 j 7 θ 1j θ 2 = min 1 j 7 θ 2j θ 11 = η λ 3 + (δ ( (α (1 1 ( ( 1δ + ( 12 α θ 12 = mη λ 3 + (δ (1 m + 2 (α (1 1 ( ( mδ + ( 12 α

9 BLOW UP RESULTS 181 θ 13 = s(λ (λ λ (α (1 1 (( s + ( 1 2 α θ 14 = mη mλ 3 + m(δ ( (α (1 m ( δ( 1m + α2 ( m θ 15 = sλ 3 + s 2 (λ (λ s(α (1 1 (( 1s + ( s 2 α θ 16 = rη 1 2 (η r 2 (η r(δ (( 1r + ( r 2 δ θ 17 = rs(λ r(λ r(λ (λ (( sr + ( r 2 α θ 21 = 2 η 1 + (α (1 1 + λ (δ (1 1 [ ] ( 1 α + ( 12 δ θ 22 = (η 3 + 1r (η η (δ (( r + ( 1 2 δ θ 23 = mλ 3 + m 2 η 1 + m(α ( (δ (1 m ( m( 1α + ( m2 δ θ 24 = rm(η m(η m 2 η (δ (1 m (( rm + ( m 2 δ θ 25 = sm(λ m(λ mλ (α (1 m (( sm + 2 ( m α θ 26 = ms(λ s 2 λ 1 2 (λ s(α (1 (( ms + ( s 2 α θ 27 = rs(λ s(λ s 2 (λ (λ (( rs + ( s 2 α then the system (FDS (with the initial data does not admit nontrivial global weak solutions Proof Here again the roof roceeds by contradiction Let ( t 2 + x 2µj ξ j (x t = Φ R 2 j = 1 2 where R > 0 µ 1 = /α and µ 2 = /δ The weak formulation of solutions to (FDS reads h v ξ 1 + u 0 Dt T α ξ 1

10 182 HAKEM AND BERBICHE = udt T α ξ 1 + ( u m 1 u( 2 ξ1 g u r ξ 1 kξ 2 u + v 0 Dt δ T R ξ 2 = vdt T δ ξ 2 + ( v m 1 v( 2 ξ2 lξ 2 v s Using the Hölder ineuality we may write ( 1 ( udt T α ξ 1 kξ 2 u 2 QT R(kξ 1 1 D α t T ξ 1 ( ( u m 1 u( 2 ξ1 kξ 2 u ( g u r ξ 1 kξ 2 u Conseuently ( h v ξ 1 kξ 2 u where ( A = ( B = ( C = 1 A + ( 1 1 m ( m QT R ( 2 ξ1 m (kξ2 m m r ( r 2 QT R(kξ r r (gξ1 r kξ 2 u (kξ D α t T ξ 1 1 m B + ( 1 ( 2 ξ1 m (kξ2 m m (kξ 2 r r (gξ1 r r m Similarly we obtain the estimates ( 1 ( vdt T δ ξ 2 v (hξ 1 1 QT R(hξ 1 1 D δ t T ξ 2 ( m ( ( v m 1 v( 2 ξ2 v hξ 1 QT R ( lξ 2 v s v hξ 1 kξ 2 u 1 1 r C m (hξ 1 m m ( 2 ξ2 m s ( s 1 QT R(hξ s s (lξ2 s So we get ( 1 ( m ( kξ 2 u v (hξ 1 D+ v hξ 1 E+ v hξ 1 s F

11 BLOW UP RESULTS 183 with If we set then we have which yields (22 (23 ( D := ( E := ( F := (hξ D δ t T ξ 2 1 (hξ 1 m m ( 2 ξ2 m (hξ 1 s s (lξ2 s 1 s m Y := h v ξ 1 Z := kξ 2 u Y Z A + Z m B + Z r C Z Y D + Y m E + Y s F Y 3 1 (Z A + Z m B + Z r C Z 3 1 (Y D + Y m E + Y s F We have used in (22 and (23 the ineuality (a + b + c 3 1 (a + b + c 1 a b c 0 It then follows from (22 (23 that Y c( m r ( (Y D + Y m E + Y s F A +(Y m D m + Y m2 E m + Y sm F m B +(Y r D r + Y mr E r + Y sr F r C where c( m r is a ositive constant deending on m and r Using ε-young ineuality we get (24 Y c( m r ε ((DA 1 + (EA m + (FA s + (D m B m + (E m B m 2 + (F m B + (D r C r + (E r C mr sm + (F r C sr Now using the scaled variables (y τ defined by t = Rτ and x = R α y in A B F while in D E C we use the variables (y τ defined by t = Rτ and x = R δ y we obtain (25 Y c ( R l1 + R l2 + R l3 + R l4 + R l5 + R l6 + R l7 + R l8 + R l9

12 184 HAKEM AND BERBICHE where ( 1l 1 := N η λ 3 + (δ ( (α ( ( 1δ + ( 12 α ( ml 2 := N mη λ 3 + (δ (1 m + 2 (α (1 1 ( ( mδ + ( 12 α ( sl 3 := N s(λ (λ λ (α (1 1 (( s + ( 1 2 α ( ml 4 := N mη mλ 3 + m(δ ( (α (1 m ( δ( 1m + α2 ( m ( m 2 l 5 := N m2 η mλ 3 + m(δ (1 m + 2 (α 1 + m ( δm( m + 2 ( mα ( sml 6 := N sm(λ m(λ mλ (α (1 m (( sm + 2 ( m α ( rl 7 := N rη 1 2 (η r 2 (η r(δ (( 1r + ( r 2 δ ( mrl 8 := N mrη 1 + (η r 2 (η r(δ (1 m (( mr + ( r 2 δ ( srl 9 := N rs(λ r(λ r(λ (λ (( sr + ( r 2 α In the same way we find (26 Z c(ε ((AD 1 + (BD m + (CD r Similarly we have for Z + (A m E m + (B m E m 2 + (C m E mr + (A s F s + (B s F ms + (C s F rs (27 Z c ( R j 1 + R j 2 + R j3 + R j4 + R j5 + R j6 + R j7 + R j8 + R j9 where ( 1j 1 := N 2 η 1 + (α (1 1 + λ (δ (1 1 [ ( 1 α + ( ] 12 δ [ mλ3 + 2 η 1 + (α (1 m ( mj 2 := N + 2 (δ (1 1 ] ( ( mα + ( 12 δ ( rj 3 := N (η 3 + 1r (η η (δ (( r + ( 1 2 δ

13 BLOW UP RESULTS 185 ( mj 4 := N mλ 3 + m 2 η 1 + m(α ( (δ (1 m ( ( 1mα + ( m2 δ ( m 2 j 5 := N m2 λ 3 + m 2 η 1 + m(α (1 m + 2 (δ (1 m ( α( mm + δ( m2 ( mrj 6 := N rm(η m(η m 2 η (δ (1 m (( rm + ( m 2 δ ( sj 7 := N sλ 3 + s 2 (λ (λ s(α (1 1 (( 1s + ( s 2 α ( msj 8 := N ms(λ s 2 λ 1 2 (λ s(α (1 (( ms+( s 2 α ( rsj 9 := N rs(λ s(λ s 2 (λ (λ (( rs + ( s 2 α Condition (21 leads to either max 1 i 9 l i 0 or max 1 i 9 j i 0 In both cases the roof follows from the arguments resented above Remark 21 When α = δ = 1 = = 2 h = k = 1 and g = l = 0 we found the case studied by Escobedo and Herrero [1] however we imose the constraint > 1 > 1 while Escobedo and Herrero reuire > 1 Acknowledgments We would like to exress our gratitude to the referees and editors for their helful comments Another major debt to Dr N Guerroudj for correcting linguistic mistakes References 1 M Berbiche A Hakem Nonexistence of global solutions for a fractional wave -diffusion euation J Part Diff Es 25(1 ( M Escobedo M A Herrero Boundedness and blow-u for a semilinear reaction-diffusion euation J Differ Euations 89 ( H Fujita On the blowing u of solutions of the Cauchy roblem for u t = u + u J Fac Sci Univ Tokyo Sect I13 ( M Guedda M Kirane A note on nonexistence of global solutions to a nonlinear integral euation Bull Belg Math Soc Simon Stevin 6 ( M Guedda M Kirane Criticality for some evolution euations Differ Eu 37 ( K Hayakawa On nonexistence of global solutions of some semilinear arabolic differential euations Proc Jaan Acad Ser A 37 ( M Kirane Y Laskri N Tatar Critical exonents of Fujita tye for certain evolution euations andsystems with satio-temoral fractional derivatives J Math Anal Al 312 ( M Kirane M Qafsaoui Global nonexistence for the Cauchy roblem of some nonlinear reaction-diffusion systems J Math Anal Al 268(1 ( K Kobayashi T Sirao and H Tanaka On the growing u roblem for semilinear heat euations J Math Soc Jaan 29 (

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