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• W3137560215 abstract "We investigate the well-posedness of the fast diffusion equation (FDE) on noncompact Riemannian manifolds. Existence and uniqueness of solutions for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript 1> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>L^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> initial data was established in Bonforte, Grillo, and Vázquez [J. Evol. Equ. 8 (2008), pp. 99–128]. However, in the Euclidean space, it is known from Herrero and Pierre [Trans. Amer. Math. Soc. 291 (1985), pp. 145–158] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript normal l normal o normal c Superscript 1> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>l</mml:mi> <mml:mi mathvariant=normal>o</mml:mi> <mml:mi mathvariant=normal>c</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>L^1_{mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We establish here that such data still give rise to global solutions on general manifolds. If, moreover, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=negative normal infinity> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>-infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, assuming in addition that the initial datum is in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript normal l normal o normal c Superscript 2> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>l</mml:mi> <mml:mi mathvariant=normal>o</mml:mi> <mml:mi mathvariant=normal>c</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>L^2_{mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and nonnegative, a minimal solution is shown to exist, and we establish uniqueness of purely (nonnegative) distributional solutions, a fact that to our knowledge was not known before even in the Euclidean space. The required curvature bound is sharp, since on model manifolds it is equivalent to stochastic completeness, and it was shown in Grillo, Ishige, and Muratori [J. Math. Pures Appl. (9) 139 (2020), pp. 63–82] that uniqueness for the FDE fails even in the class of bounded solutions when stochastic completeness does not hold. A crucial ingredient of the uniqueness result is the proof of nonexistence of nonnegative, nontrivial distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest." @default.
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• W3137560215 date "2021-06-16" @default.
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• W3137560215 title "Fast diffusion on noncompact manifolds: Well-posedness theory and connections with semilinear elliptic equations" @default.
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