Nonexistence of global solutions to system of semi-linear fractional evolution equations

Year 2018, , 171 - 177, 30.09.2018

Medjahed Djilali , Ali Hakem

https://doi.org/10.32323/ujma.396407

Cited By: 1

Abstract

In this research we are interested to Cauchy problem for system of semi-linear fractional evolution equations. Some authors were concerned with studying of global existence of solutions for the hyperbolic nonlinear equations with a damping term. Our goal is to extend some results obtained by the authors, by studying the system of semi-linear hyperbolic equations with fractional damping term and fractional Laplacian .Thanks to the test functions method, we prove the nonexistence of nontrivial global weak solutions to the problem. 

Keywords

Derivatives in the sense of Caputo, Fractional Laplacian, Fujita's critical exponent, Test function, Weak solution

References

• [1] M. BERBICHE, A. HAKEM, Necessary conditions for the existence and sufficient conditions for the nonexistence of solutions to a certain fractional telegraph equation. Memoirs on Differential Equations and Mathematical physics. vol 56, 2012, 37-55.
• [2] M. ESCOBEDO & H. A. LEVINE, Critical blow up and global existence numbers of a weakly coupled system of reaction-diffusion equation. Arch. Rational. Mech. Anal. 129 (1995),47-100.
• [3] A. Z. FINO, Critical exponent for damped wave equations with nonlinear memory. Nonlinear Analysis 74 (2011) 5495-5505.
• [4] A. Z. FINO, H. IBRAHIM & A. WEHBE, A blow-up result for a nonlinear damped wave equation in exterior domain: The critical case, Computers & Mathematics with Applications, Volume 73, Issue 11, (2017), pp. 2415-2420.
• [5] H. FUJITA, On the blowing up of solutions of the problem for ut = Du+u1+a , J. Fac. Sci.Univ. Tokyo 13 (1966), 109 - 124.
• [6] M. GUEDDA & M. KIRANE, Local and global nonexistence of solutions to semilinear evolution equations. Electronic Journal of Differential Equations, Conference 09 (2002), pp. 149-160.
• [7] B. GUO, X. PU & F. HUANG, Fractional Partial Differential Equations and Their Numerical Solutions. World Scientific Publishing Co. Pte. Ltd. Beijing, China (2011).
• [8] A. HAKEM, Nonexistence of weak solutions for evolution problems on RN , Bull. Belg. Math. Soc. 12 (2005), 73-82.
• [9] A. HAKEM & M. BERBICHE, On the blow-up of solutions to semi-linear wave models with fractional damping . IAENG International Journal of Applied Mathematics, (2011) 41:3, IJAM-41-3-05.
• [10] M. KIRANE, Y. LASKRI & N.-E.TATAR, Critical exponents of fujita type for certain evolution equations and systems with spation-temporal fractional derivatives.J. Math. Anal. Appl. 312 (2005) 488-501.
• [11] W. MINGXIN, Global existence and finite time blow up for a reaction-diffusion system. Z. Angew. Math. Phys. 51 (2000) 160-167.
• [12] T. OGAWA & H. TAKIDA, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, J. Nonliniear analysis 70 (2009), 3696-3701.
• [13] I. PODLUBNY, Fractional differential equations. Mathematics in Science and Engineering, vol 198, Academic Press, New York, 1999.
• [14] S.I. POHOZAEV & A. TESEI, Nonexistence of Local Solutions to Semilinear Partial Differential Inequalities, Nota Scientifica 01/28, Dip. Mat. Universit´a ”La Sapienza”, Roma (2001).
• [15] S. POHOZAEV & L. VERON, Blow up results for nonlinear hyperbolic enequalities, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) Vol. XXIX (2000), pp. 393-420.
• [16] C. POZRIKIDIS, The fractional Laplacian, Taylor & Francis Group, LLC /CRC Press, Boca Raton (USA), (2016).
• [17] S. G. SAMKO, A. A. KILBAS & O. I. MARICHEV, Fractional integrals and derivatives: Theory and applications. Gordan and Breach Sci. Publishers, Yverdon, 1993.
• [18] G.TODOROVA & B.YORDANOV, Critical Exponent for a Nonlinear Wave Equation with Damping. Journal of Differential Equations 174, 464-489 (2001).
• [19] Y. YAMAUCHI, Blow-up results for a reaction-deffusion system , Methods Appl. Anal. 13 (2006), 337 - 350.
• [20] Q. S. ZHANG, A blow up result for a nonlinear wave equation with damping: the critical case, C. R. Acad.Sci. paris, Volume 333, no.2, (2001), 109-114.
• [21] S-MU. ZHENG, Nonlinear evolution equations, Chapman & Hall/CRC Press, Florida (USA), (2004).