Abstract
The large time behavior of non-negative solutions to the reaction–diffusion equation \({\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}\), \({(\alpha\in(0,2], \;p > 1)}\) posed on \({\mathbb{R}^N}\) and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.
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References
Ben-Artzi M., Koch H.: Decay of mass for a semilinear parabolic equation. Comm. Partial Differ. Equ. 24, 869–881 (1999)
Biler P., Karch G., Woyczyński W.A.: Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann. I.H. Poincaré-Analyse non linéare 18, 613–637 (2001)
Birkner M., Lopez-Mimbela J.A., Wakolbinger A.: Blow-up of semilinear PDE’s at the critical dimension. Probab. Approach Proc. Am. Math. Soc. 130, 2431–2442 (2002)
Córdoba A., Córdoba D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249, 511–528 (2004)
Droniou J., Imbert C.: Fractal first-order partial differential equations. Arch. Ration. Mech. Anal. 182, 299–331 (2006)
Fujita H.: On the blowing up of solutions of the problem for u t = Δu + u 1+α. J. Fac. Sci. Univ. Tokyo 13, 109–124 (1966)
Guedda, M., Kirane, M.: Criticality for some evolution equations. Differ. Uravn. 37, 511–520, 574–575 (2001)
Guedda M., Kirane M.: A note on nonexistence of global solutions to a nonlinear integral equation. Bull. Belg. Math. Soc. Simon Stevin 6, 491–497 (1999)
Hayakawa K.: On nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Japan Acad. 49, 503–505 (1973)
Ju N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Comm. Math. Phys. 255, 161–181 (2005)
Karch G., Woyczyński W.A.: Fractal Hamilton–Jacobi–KPZ equations. Trans. Am. Math. Soc. 360, 2423–2442 (2008)
Kobayashi K., Sirao T., Tanaka H.: On the growing up problem for semilinear heat equations. J. Math. Soc. Japan 29, 407–424 (1977)
Laurençot Ph., Souplet Ph.: On the growth of mass for a viscous Hamilton–Jacobi equation. J. Anal. Math. 89, 367–383 (2003)
Mitidieri, È., Pokhozhaev, S.I.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234, 1–384 (2001); translation in Proc. Steklov Inst. Math. 234(3), 1–362 (2001)
Mitidieri E., Pokhozhaev S.I.: Nonexistence of weak solutions for some degenrate elliptic and parabolic problems on \({{\mathbb{R}}^N}\). J. Evol. Equ. 1, 189–220 (2001)
Pinsky R.G.: Decay of mass for the equation u t = Δu−a(x)u p|∇u|q. J. Differ. Equ. 165, 1–23 (2000)
Sugitani S.: On nonexistence of global solutions for some nonlinear integral equations. Osaka J. Math. 12, 45–51 (1975)
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Communicated by A. Jüngel.
The preparation of this paper was supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. A. Fino gratefully thanks the Mathematical Institute of Wrocław University for the warm hospitality. The preparation of this paper by G. Karch was also partially supported by the MNiSW grant N201 022 32/09 02.
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Fino, A., Karch, G. Decay of mass for nonlinear equation with fractional Laplacian. Monatsh Math 160, 375–384 (2010). https://doi.org/10.1007/s00605-009-0093-3
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DOI: https://doi.org/10.1007/s00605-009-0093-3