Fast diffusion on noncompact manifolds: Well-posedness theory and connections with semilinear elliptic equations

نویسندگان

چکیده

We investigate the well-posedness of fast diffusion equation (FDE) on noncompact Riemannian manifolds. Existence and uniqueness solutions for L 1 L^1 initial data was established in Bonforte, Grillo, Vázquez [J. Evol. Equ. 8 (2008), pp. 99–128]. However, Euclidean space, it is known from Herrero Pierre [Trans. Amer. Math. Soc. 291 (1985), 145–158] that Cauchy problem associated with FDE well posed are merely Subscript normal l o c l mathvariant="normal">o mathvariant="normal">c encoding="application/x-tex">L^1_{\mathrm {loc}} . establish here such still give rise to global general If, moreover, radial Ricci curvature satisfies a suitable pointwise bound below (possibly diverging alttext="negative infinity"> − mathvariant="normal">∞ encoding="application/x-tex">-\infty at spatial infinity), we prove also holds, same type data, class strong solutions. Besides, assuming addition datum 2"> 2 encoding="application/x-tex">L^2_{\mathrm nonnegative, minimal solution shown exist, purely (nonnegative) distributional solutions, fact our knowledge not before even space. The required sharp, since model manifolds equivalent stochastic completeness, Ishige, Muratori Pures Appl. (9) 139 (2020), 63–82] fails bounded when completeness does hold. A crucial ingredient result proof nonexistence nontrivial subsolutions certain semilinear elliptic equations power nonlinearities, independent interest.

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ژورنال

عنوان ژورنال: Transactions of the American Mathematical Society

سال: 2021

ISSN: ['2330-0000']

DOI: https://doi.org/10.1090/tran/8431