Fast diffusion on noncompact manifolds: Well-posedness theory and connections with semilinear elliptic equations
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- by Gabriele Grillo, Matteo Muratori and Fabio Punzo PDF
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Abstract:
We investigate the well-posedness of the fast diffusion equation (FDE) on noncompact Riemannian manifolds. Existence and uniqueness of solutions for $L^1$ 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 $L^1_{\mathrm {loc}}$. 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 $-\infty$ 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 $L^2_{\mathrm {loc}}$ 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.References
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Additional Information
- Gabriele Grillo
- Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
- MR Author ID: 268120
- ORCID: 0000-0002-7912-5089
- Email: gabriele.grillo@polimi.it
- Matteo Muratori
- Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
- MR Author ID: 1004365
- Email: matteo.muratori@polimi.it
- Fabio Punzo
- Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
- MR Author ID: 851995
- Email: fabio.punzo@polimi.it
- Received by editor(s): May 8, 2020
- Received by editor(s) in revised form: December 3, 2020
- Published electronically: June 16, 2021
- Additional Notes: The authors were partially supported by the PRIN Project “Direct and Inverse Problems for Partial Differential Equations: Theoretical Aspects and Applications” (grant no. 201758MTR2, MIUR, Italy). The second and third authors were also supported by the GNAMPA Projects “Analytic and Geometric Problems Associated to Nonlinear Elliptic and Parabolic PDEs”, “Existence and Qualitative Properties for Solutions of Nonlinear Elliptic and Parabolic PDEs” and “Differential Equations on Riemannian Manifolds and Global Analysis”
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 6367-6396
- MSC (2020): Primary 35R01; Secondary 35K67, 35K65, 58J35, 35D30, 35J61
- DOI: https://doi.org/10.1090/tran/8431
- MathSciNet review: 4302163