NON-EXISTENCE OF GLOBAL SOLUTIONS TO SYSTEMS OF NON-AUTONOMOUS NONLINEAR PARABOLIC EQUATIONS

TitleNON-EXISTENCE OF GLOBAL SOLUTIONS TO SYSTEMS OF NON-AUTONOMOUS NONLINEAR PARABOLIC EQUATIONS
Publication TypeJournal Article
Year of Publication2010
AuthorsKERBAL, SEBTI
Secondary TitleCommunications in Applied Analysis
Volume14
Issue2
Start Page203
Pagination212
Date Published04/2010
Type of Workscientific: mathematics
ISSN1083–2564
AMS35B33, 35K57, 35K65.
Abstract
We consider the non-autonomous system of nonlinear parabolic equations
posed in , subject to the initial data (u(0, x) = u0(x), v(0, x) = v0(x)), where p > 1 and q > 1 are positive real numbers, α, β ∈[0, 2] and ∆γ := (−∆)γ/2 is the (−∆)γ/2 fractional power of −∆ in the x variable defined via the Fourier transform and its inverse by (−∆)γ/2w(x, t) = (|ξ|γ(w)(ξ)) (x, t), where r > −1 and s > −1.
The Fujita critical exponent which separates the case of blowing-up solutions from the case of globally in time existing solutions is determined.
URLhttp://www.acadsol.eu/en/articles/14/2/6.pdf
Short TitleNON-LOCAL PARABOLIC SYSTEMS
Refereed DesignationRefereed
Full Text

REFERENCES

[1] N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Res. Notes in Math. Ser. 246, Longman Scientific and Technical and John Wiley, London, New York, 1991.
[2] B. Baeumer, M. Kovacs, and M. Meerschaert, Fractional reaction-diffusion equation for species growth and dispersal, Journal of Mathematical Biology, Manuscript online: http://www.maths.otago.ac.nz/mcubed/JMBseed.pdf
[3] A. Cordoba, D. Cordoba, A pointwise estimate for fractionary derivatives with applications to partial differential equations, PNAS vol 100 26 (2003).
[4] K. Deng and H. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000) 85–126.
[5] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Diff. Equations 89 (1991), 176–202.
[6] M. Fila, H. A. Levine, and Y. Uda. A Fujita-type global existence-global non-existence theorem for a system of reaction diffusion equations with differing diffusivities, Math. Methods Appl. Sci. 17 10 (1994), 807–835.
[7] H. Fujita, On the blowing-up o f solutions of the Cauchy problems for u t = ∆u + u 1+p , J. Fac. Sci. Univ. Tokyo sect. I A 13 (1966), 109–124.
[8] J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.
[9] M. Guedda and M. Kirane, A note on nonexistence of global solutions to a nonlinear integral equation, Bull. Belg. Math. Soc. 6 (1999), 491–497.
[10] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic diiferential equations, Proc. Japan Acad. 49 (1973), 503–505.
[11] N. Ju , The Maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Pure. Appl. Ana. (2005), 161–181.
[12] M. Kirane, Y. Laskri,and N. Tatar, Critical exponents of Fujita’s type for certain evolution equations and systems with spatio-temporal fractional derivatives, J. Math. Anal. Appl. 312 (2005) 488–501.
[13] M. Kirane, and M. Qafsaoui, Global nonexistence for the Cauchy problem of some nonlinear Reaction Diffusion systems, Journal of Mathematical Analysis and Applications 268(1) (2002), 217–243.
[14] Y. Kobayashi, T. Sirao, and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), 407–424.
[15] R. Mancinelli, D. Vergni, and A. Vulpiani, Front propagation in reactive systems with anomalous diffusion, Physica D, Nonlinear Phenomena 185 (2003), 175–195.
[16] E. Mitidieri, S. I. Pohozaev, Absence of global positive solutions of quasilinear elliptic inequalities, (Russian) Dokl. Akad. Nauk 359 4 (1998), 456–460.
[17] E. Mitidieri, S. I. Pohozaev, Existence of positive solutions for a systems of quasilinear elliptic equations and inequalities in Rn, Dokl. Math. 59 3 (1999), 351–355.
[18] M. Nagasawa and T. Sirao, Probabilistic treatment of the blowing-up of solutions for a nonlinear integral
equation, Trans. Amer. Math. Soc. 139 (1969), 301–310.
[19] S.I. Pokhozhaev and A. Tesei, Critical exponent for the absence of solutions for systems of quasilinear parabolic inequalities, Differential Equations Vol.37 4 (2001), 551–558.
[20] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math. 12 (1975), 45–51.