Decay of mass for fractional evolution equation with memory term

Fino, Ahmad; Ibrahim, Hassan; Barakeh, Bilal

doi:10.1090/S0033-569X-2012-01286-4

Authors: Ahmad Z. Fino, Hassan Ibrahim and Bilal Barakeh
Journal: Quart. Appl. Math. 71 (2013), 215-228
MSC (2010): Primary 35K55, 35B40
DOI: https://doi.org/10.1090/S0033-569X-2012-01286-4
Published electronically: August 27, 2012
MathSciNet review: 3087420
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Abstract: We investigate the decay properties of the mass $M(t)= \int _{\mathbb{R}^N} u(\cdot ,t)dx$ of the solutions of a fractional diffusion equation with nonlinear memory term. We show, using a suitable class of initial data and a restriction on the diffusion and nonlinear term, that the memory term determines the large time asymptotics; that is, tends to zero as $t\to \infty .$

1. Matania Ben-Artzi and Herbert Koch, Decay of mass for a semilinear parabolic equation, Comm. Partial Differential Equations 24 (1999), no. 5-6, 869–881. MR 1680909, https://doi.org/10.1080/03605309908821450
2. Piotr Biler, Grzegorz Karch, and Wojbor A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 5, 613–637. MR 1849690, https://doi.org/10.1016/S0294-1449(01)00080-4
3. Krzysztof Bogdan and Tomasz Byczkowski, Potential theory for the 𝛼-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1999), no. 1, 53–92. MR 1671973
4. Thierry Cazenave, Flávio Dickstein, and Fred B. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal. 68 (2008), no. 4, 862–874. MR 2382302, https://doi.org/10.1016/j.na.2006.11.042
5. Thierry Cazenave and Alain Haraux, Introduction aux problèmes d’évolution semi-linéaires, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 1, Ellipses, Paris, 1990 (French). MR 1299976
6. Jérôme Droniou and Cyril Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal. 182 (2006), no. 2, 299–331. MR 2259335, https://doi.org/10.1007/s00205-006-0429-2
7. A. Z. Fino, V. Georgiev, M. Kirane, Finite time blow-up for a wave equation with a nonlocal nonlinearity, submitted, arXiv:1008.4219.
8. Ahmad Fino and Grzegorz Karch, Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math. 160 (2010), no. 4, 375–384. MR 2661320, https://doi.org/10.1007/s00605-009-0093-3
9. A. Z. Fino, M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation, Quarterly of Applied Mathematics 70 (2012), 133-157.
10. Hiroshi Fujita, On the blowing up of solutions of the Cauchy problem for 𝑢_{𝑡}=Δ𝑢+𝑢^{1+𝛼}, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124 (1966). MR 0214914
11. M. Gidda and M. Kirane, Criticality for some evolution equations, Differ. Uravn. 37 (2001), no. 4, 511–520, 574–575 (Russian, with Russian summary); English transl., Differ. Equ. 37 (2001), no. 4, 540–550. MR 1854046, https://doi.org/10.1023/A:1019283624558
12. Ning Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys. 255 (2005), no. 1, 161–181. MR 2123380, https://doi.org/10.1007/s00220-004-1256-7
13. Grzegorz Karch and Wojbor A. Woyczyński, Fractal Hamilton-Jacobi-KPZ equations, Trans. Amer. Math. Soc. 360 (2008), no. 5, 2423–2442. MR 2373320, https://doi.org/10.1090/S0002-9947-07-04389-9
14. Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
15. Mokhtar Kirane and Mahmoud Qafsaoui, Global nonexistence for the Cauchy problem of some nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 268 (2002), no. 1, 217–243. MR 1893203, https://doi.org/10.1006/jmaa.2001.7819
16. Damien Lamberton, Équations d’évolution linéaires associées à des semi-groupes de contractions dans les espaces 𝐿^{𝑝}, J. Funct. Anal. 72 (1987), no. 2, 252–262 (French). MR 886813, https://doi.org/10.1016/0022-1236(87)90088-7
17. Philippe Laurençot and Philippe Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math. 89 (2003), 367–383. MR 1981925, https://doi.org/10.1007/BF02893088
18. Francesco Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, 2010. An introduction to mathematical models. MR 2676137
19. È. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1–384 (Russian, with English and Russian summaries); English transl., Proc. Steklov Inst. Math. 3(234) (2001), 1–362. MR 1879326
20. Masao Nagasawa and Tunekiti Sirao, Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation, Trans. Amer. Math. Soc. 139 (1969), 301–310. MR 0239379, https://doi.org/10.1090/S0002-9947-1969-0239379-X
21. Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204. MR 0295683, https://doi.org/10.1090/S0033-569X-1971-0295683-6
22. Ross G. Pinsky, Decay of mass for the equation 𝑢_{𝑡}=Δ𝑢-𝑎(𝑥)𝑢^{𝑝}|∇𝑢|^{𝑞}, J. Differential Equations 165 (2000), no. 1, 1–23. MR 1771786, https://doi.org/10.1006/jdeq.2000.3771
23. Catherine A. Roberts and W. E. Olmstead, Blow-up in a subdiffusive medium of infinite extent, Fract. Calc. Appl. Anal. 12 (2009), no. 2, 179–194. MR 2498365
24. Sadao Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math. 12 (1975), 45–51. MR 0470493
25. Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
26. Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 2, 109–114 (English, with English and French summaries). MR 1847355, https://doi.org/10.1016/S0764-4442(01)01999-1
27. Wei Zhang, Nobuyuki Shimizu, Damping properties of the viscoelastic material described by fractional Kelvin-Voigt model, JSME international journal. Series C, Mechanical systems, machine elements and manufacturing, 42 (1999), no. 1, 1-9.
28. Fred B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), no. 1-2, 29–40. MR 599472, https://doi.org/10.1007/BF02761845
29. Fred B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979), no. 3, 277–296. MR 538855, https://doi.org/10.1016/0022-1236(79)90040-5