Critical exponents for a nonlinear reaction-diffusion system with fractional derivatives

Global Journal of Pure Applied Mathematics. ISSN 0973-768 Volume Number 6 (06 pp. 5343 535 Research India Publications http://www.ripublication.com/gjpam.htm Critical exponents f a nonlinear reaction-diffusion system with fractional derivatives Belgacem Rebiai Salim Rouar Kamel Haouam Department of Mathematics Infmatics LAMIS Labaty University of Tebessa 00 Algeria. Abstract This paper deals with critical exponents f a parabolic fractional reaction diffusion system with the non linear terms u p i v q i i = q > p q 0 are constants. This wk improve extend our result obtained in [] the conditions q 0 p q > were supposed. We also show that this result extends wks of [3] [4] done in the classical case. AMS subject classification: 6A33 35B33 35K45. Keywds: Fractional derivatives critical exponents nonlinear reaction diffusion system.. Introduction Recent studies demonstrate the role played by fractional derivatives in the mathematical modelling of various scientist situation in mechanics physics chemistry biology finance see f example [ 6]. This paper deal with the following Cauchy problem f a nonlinear fractional reaction diffusion system. D α 0 t u + ( β/ u = (tx R + R N D α 0 t v + ( β/ v = (tx R + R N u(0x= u 0 (x 0 0 x R N ( v(0x= v 0 (x 0 0 x R N p i q i α i β i i= are constants such that

5344 B. Rebiai S. Rouar K. Haouam (A > q 0 p 0 q > 0 <α i < β i (A N max { β ( β α (q }. α q are the conjugate values of q respectively. D α i 0 t denotes the derivatives of der α i in the sense of Caputo (see e.g. [0] ( β i is the fractional power of the Laplacian ( defined by ( β i/ u(t x = F ( ξ β if(u(ξ(t x F is the Fourier transfm F its inverse. In the case α i = β i = i= the problem ( was treated by many auths in several contexts see f example [ 3 4 3 4]. Escobedo Herrero [] proved that if p q > = q = 0 (γ +/(p q N/ with γ = max(p q then the only solution of the problem ( is the trivial one i.e. u v 0. Later in [3] Escobedo Levine showed that if p + q + q > 0 then the problem ( behaves like the Cauchy problem f the single equation u t u = u +q with respect to Fujita-type blowup theems (see [5]. In [3] Yamauchi considired the problem u t u = x σ v t v = x σ u(0x= u 0 (x 0 0 v(0x= v 0 (x 0 0 p i q i 0 σ i max( N i = q =. He proved a nonexistence results under some conditions concerning relation between exponents p i q i σ i initial data. In the case of real der 0 <α i < β i Kirane et al. [8] considered the following Cauchy problem D α 0 t u + ( β/ u = D α 0 t v + ( β/ v = u(0x= u 0 (x 0 v(0x= v 0 (x 0 they proved that if q > p > q q = q + q p p = p + p N max { α p + α ( α β p q + α β p p q α q + α ( α β q p + α β q q p } (

Reaction-diffusion system with fractional derivatives 5345 then the problem ( does not admit nontrivial global weak nonnegative solutions. In [] Rebiai Haouam proved a nonexistence results which is me general than the interesting result obtained in [8]. They proved that the problem ( does not admit global weak solutions under a suitable restrictions on the exponents p i q i the initial conditions u 0 v 0. Our wk improve extend our result obtained in [] the conditions 0 q > p > q 0 were supposed.. Preliminaries In this section we describe some necessary tools of the fractional derivatives required f the reminder of this wk. Definition.. Let 0 < α < φ L (0T. The left-sided right-sided Riemann-Liouville derivatives of der α f φ are defined respectively by: D0 t α φ(t = d Ɣ( α dt Dt T α φ(t = d Ɣ( α dt Ɣ denotes the gamma function of Euler. t 0 T t φ(σ (t σ α dσ φ(σ (σ t α dσ Definition.. Let 0 <α< φ L (0T. The left-sided respectively right-sided Caputo derivatives of der α f φ are defined as follows: D α 0 t φ(t = t Ɣ( α 0 φ (σ (t σ α dσ D α t T φ(t = T φ (σ Ɣ( α t (σ t α dσ Ɣ denotes as usual the gamma function of Euler. The relation between Caputo Riemann-Liouville derivatives is written as D α 0 t φ(t = Dα 0 t [φ(t φ(0]. Finally taking into account the following integration by parts fmula: T 0 T f (t(d0 t α g(tdt = (Dt T α f (tg(tdt we adopt the following definition concerning the weak fmulation f the problem (. 0

5346 B. Rebiai S. Rouar K. Haouam Definition.3. Let Q T = (0T R N 0 < T < +. We say that (u v (L loc (Q T is a local weak solution to problem ( on Q T ifu p i v q i L loc (Q T i = it is such that u 0 (xd α t T ϕ (t xdtdx + ϕ (t xdtdx Q T Q T = Q T ud α t T ϕ (t xdtdx + Q T u( β ϕ (t xdtdx (3 v 0 (xd α t T ϕ (t xdtdx + ϕ (t xdtdx Q T Q T = vd α t T ϕ (t xdtdx + v( β ϕ (t xdtdx. (4 Q T Q T f all test functions ϕ i C tx (Q T such that ϕ i (T x = 0i =. 3. Main results We now state our main result as follows. Theem 3.. Let u 0 v 0 in L (R N suth that u 0 v 0 0 u 0 v 0 0. Assume that the assumptions (A (A hold then any solution to problem ( blows up in a finite time. Proof. The proof is by contradiction. Suppose that (u v is a global weak solution to problem (. Since u 0 v 0 0 u 0 v 0 0 then u(t v(t > 0 f all t (0T T denotes the eventual blow up time. { Let T θ be two real numbers such that 0 <T <T α θ = min α }. β β Let C0 (R + a smooth nonnegative nonincreasing function such that { if 0 r (r = 0 if r 0 (r f all r 0 { ( s γ if 0 s (s = 0 if s γ is any positive real number if max ( α p α q 0

Reaction-diffusion system with fractional derivatives 5347 γ>max ( α p α q if min ( α p α q > 0 q are respectively the conjugate exponents of q. We choose ϕ i (t x = φ l (xψ(t i =. ( ( x t with φ(x = T θ ψ(t = l max { T } q. We note that D α i t T ϕ i (t x = Ɣ( + ( γt α i Ɣ( + γ α i φl (x t γ αi. (5 T Using Young s inequality to the right h side of the fmulation (3 on = (0T { x R N : x T θ} we obtain ud α t T ϕ ε ϕ + C(ε D α t T ϕ p p q u( β ϕ ε ϕ + C(ε ( β ϕ p p q F ε enough small we obtain. ϕ C.A (6 A = D α t T ϕ p p q + ( β ϕ p p q. Similarly we obtain via the fmulation (4 the next estimate ϕ C.A (7 A = D α t T ϕ q u pq q ϕ q q + ( β ϕ q u p q q ϕ q q.

5348 B. Rebiai S. Rouar K. Haouam Therefe as u > 0 v > 0 then using (5 Ju s inequality ( β i φ l lφ l ( β α i i φ (see [7] introducing the change of variables t = Tτ x = T β i ξ in A i we obtain ϕ CT γ (8 ϕ CT γ (9 γ = α p α β N γ = α q α β N. Now if we choose N<N pass to the limit in (8 (9 as T goes to infinity we get R + R N ϕ = 0 R + R N ϕ = 0. Using the dominated convergence theem the continuity in time space of u v we infer that = 0 R + R N = 0. R + R N This implies that u 0v 0 which is a contradiction. In the case N = N we modify the previous function φ by introducing a new number R 0 <R<T such that x φ(x = ( (T /R θ we set R = (0T { x R N : x (T /R θ} R = (0T { x R N : (T /R θ x (T /R θ}. Since from (8 (9 we find that R + R N ϕ C

Reaction-diffusion system with fractional derivatives 5349 ϕ C R + R N Then we have lim ϕ dtdx = 0 (0 T + R lim ϕ dtdx = 0. ( T + R Using Young s inequality Hölder s inequality respectively in the first second integral of the right h side of the fmulation (3 on R we obtain u D α t T ϕ ε ϕ + C(ε.B R R Consequently R u ( β ϕ ( v q ϕ.c R B = D α t T ϕ p p q R ( C = ( β ϕ p p q R p. ( ϕ C.B + v q ϕ.c. ( R R In the same way we obtain via the fmulation (4 the next estimate ( ϕ C.B + v q q ϕ.c (3 R R B = D α t T ϕ q q p u R ( C = ( β ϕ q q q u R q ϕ q q q ϕ q q q.

5350 B. Rebiai S. Rouar K. Haouam If we introduce the change of variables t = Tτ x = (T /R α i β i ξ in B i C i using ( (3 we obtain via (0 ( after passing the limit as T goes to infinity R + R N ϕ CR γ (4 R + R N ϕ CR γ (5 γ i = α i N i =. Then taking the limit when R goes to infinity we obtain β i u 0v 0 which is a contradiction completes the proof of the theem. Remark 3.. Our result can be extended to the me general system { D α 0 t u + ( β / ( u m u = h(t x + g(t x u r v s D α 0 t v + ( β / ( v m v = k(t x + l(tx u r v s under suitable conditions on h g k l. References [] K. S. Cole Electric conductance of biological systems Cold Spring Harb Symposia on Quantitative Biology (993 07 6. [] M. Escobedo M.A. Herrero Boundedness blow-up f a semilinear reaction-diffusion equation J. Diff. Equ. 89((99 76 0. [3] M. Escobedo H.A. Levine Critical blowup global existence numbers f a weakly coupled system of reaction-diffusion equations Arch. Rational. Mech. Anal. 9((995 47 00. [4] M. Fila H.A. Levine Y. Uda A Fujita-type global existence - global nonexistence theem f a system of reaction diffusion equations with differing diffusivities Math. Methods Appl. Sci. 7((994 807 835. [5] H. Fujita On the blowing-up of solutions of the Cauchy problem f u t = u + u +α J. Fac. Sci. Univ. Tokyo Sect. I(3(966 09 4. [6] R. Hilfer Applications of Fractional Calculus in Physics Wld Scientific Publishing River Edge NJ USA 000. [7] N. Ju The maximum principle the global attract f the dissipative D quasigeostrophic equations Comm. Math. Phys. 55((005 6 8. [8] M. KiraneY. Laskri N.E. Tatar Critical exponents of Fujita type f certain evolution equations systems with spatio-tempal fractional derivatives J. Math. Anal. Appl. 3((005 488 50.

Reaction-diffusion system with fractional derivatives 535 [9] M. Kirane M. Qafsaoui Global nonexistence f the Cauchy problem of some nonlinear reaction-diffusion systems J. Math. Anal. Appl. 68((00 7 43. [0] I. Podlubny Fractional Differential Equations Academic Press San Diego-Boston- New Yk-London-Tokyo-Tonto 999. [] B. Rebiai K. Haouam Nonexistence of global solutions to a nonlinear fractional reaction-diffusion system IAENG Int. J. Appl. Math. 45(4(05 59 6. [] S.G. Samko A.A. Kilbass O.I. Marichev Fractional Integrals Derivatives: They Applications Gdon Breach Sci. Publishers Yverdon 993. [3] Y. Yamauchi Blow-up results f a reaction-diffusion system Methods Appl. Anal. 3(4(006 337 350. [4] S. Zheng Global existence global non-existence of solutions to a reactiondiffusion system Nonlinear Anal. 39(3(000 37 340.