Abstract
Let n ≥ 3, 0 < m ≤ (n − 2)/n, p > max(1, (1 − m)n/2), and \({0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}\) satisfy \({{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}\). We prove the existence of unique global classical solution of u t = Δu m, u > 0, in \({\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}\) in \({\mathbb{R}^n}\). If in addition 0 < m < (n − 2)/n and u 0(x) ≈ A|x|−q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of \({\mathbb{R}^n}\) to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|−q as t → ∞. Note that when m = (n − 2)/(n + 2), n ≥ 3, if \({g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}\) is a metric on \({\mathbb{R}^n}\) that evolves by the Yamabe flow ∂g ij /∂t = −Rg ij with u(x, 0) = u 0(x) in \({\mathbb{R}^n}\) where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.
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References
Aronson D.G., Caffarelli L.A.: The initial trace of a solution of the porous medium equation. Trans. Am. Math. Soc. 280(1), 351–366 (1983)
Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vazquez J.L.: Asymptotics of the fast diffusion equation via entropy estimates. Arch. Ration. Mech. Anal. 191, 347–385 (2009)
Bonforte M., Vazquez J.L.: Positivity, local smoothing and Harnack inequalities for very fast diffusion equations. Adv. Math. 223, 529–578 (2010)
Bénlian, P., Crandall, M.G.: Regularizing effects of homogenous evolution equations. In: Contributions to Analysis and Geometry (suppl. to Am. J. Math.), pp. 23–39. Johns Hopkins University Press, Baltimore (1981)
Dahlberg B.E.J., Fabes E., Kenig C.E.: A Fatou thoerem for solutions of the porous medium equations. Proc. Am. Math. Soc. 91, 205–212 (1984)
Dahlberg B.E.J., Kenig C.E.: Nonnegative solutions to the generalized porous medium equation. Rev. Mat. Iberoamericana 2, 267–305 (1986)
Daskalopoulos, P., Kenig, C.E.: Degenerate Diffusion-Initial Value Problems and Local Regularity Theory, Tracts in Mathematics 1, European Mathematical Society, (2007)
Daskalopoulos P., Del Pino M.: On Nonlinear parabolic equations of very fast diffusion. Arch. Ration. Mech. Anal. 137, 363–380 (1997)
Daskalopoulos P., Sesum N.: On the extinction profile of solutions to fast diffusion. J. Reine Angew Math. 622, 95–119 (2008)
Daskalopoulos, P., Sesum, N.: The classification of locally conformally flat Yamabe solitons. http://arxiv.org/abs/1104.2242.
Del Pino M., Sáez M.: On the extinction profile for solutions of \({u_t=\Delta u^{(N-2)/(N+2)}}\). Indiana Univ. Math. J. 50(1), 611–628 (2001)
Herrero M.A., Pierre M.: The Cauchy problem for \({u_t=\Delta u^m}\) for 0 < m < 1. Trans. Am. Math. Soc. 291(1), 145–158 (1985)
Hsu S.Y.: Large time behaviour of solutions of a singular diffusion equation in \({\mathbb{R}^{n}}\). Nonlinear Anal. TMA 62(2), 195–206 (2005)
Hsu S.Y.: Singular limit and exact decay rate of a nonlinear elliptic equation. Nonlinear Anal. TMA 75(7), 3443–3455 (2012)
Hui K.M.: On some Dirichlet and Cauchy problems for a singular diffusion equation. Differ. Integral Equ. 15(7), 769–804 (2002)
Hui K.M.: Singular limit of solutions of the very fast diffusion equation. Nonlinear Anal. TMA 68, 1120–1147 (2008)
Kato T.: Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1973)
Ladyzenskaya, O.A., Solonnikov, V.A., Uraltceva, N.N.: Linear and quasilinear equations of parabolic type. Transl. Math. Mono. vol. 23, Am. Math. Soc., Providence (1968)
Peletier, L.A.: The porous medium equation. In: Amann, H., Bazley, N., Kirchgassner, K. (eds.) Applications of Nonlinear Analysis in the Physical Sciences. Pitman, Boston (1981)
Vazquez J.L.: Nonexistence of solutions for nonlinear heat equations of fast-diffusion type. J. Math. Pures Appl. 71, 503–526 (1992)
Vazquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, Oxford (2006)
Vazquez J.L.: The Porous Medium Equation—Mathematical Theory. Oxford Mathematical Monographs. Oxford University Press, Oxford (2007)
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Hsu, SY. Existence and asymptotic behaviour of solutions of the very fast diffusion equation. manuscripta math. 140, 441–460 (2013). https://doi.org/10.1007/s00229-012-0576-8
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DOI: https://doi.org/10.1007/s00229-012-0576-8