The Cauchy problem for the Finsler heat equation
-
and
Abstract
Let H be a norm of
where
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 16H03946
Award Identifier / Grant number: 15H02058
Funding statement: Goro Akagi was partially supported by the Grant-in-Aid for Scientific Research (B) (No. 16H03946) from Japan Society for the Promotion of Science. Kazuhiro Ishige was partially supported by the Grant-in-Aid for Scientific Research (A) (No. 15H02058) from Japan Society for the Promotion of Science. Ryuichi Sato was supported in part by Research Fellow of Japan Society for the Promotion of Science.
Acknowledgements
The authors of this paper would like to thank the referees for their useful advices. They proved Theorem 2.1 by following the suggestions by the referees. Kazuhiro Ishige would like to thank Professor Paolo Salani for his valuable suggestions. Furthermore, he also would like to thank Professor Juha Kinnunen for informing him of the papers [28] and [29].
References
[1] D. Andreucci and E. DiBenedetto, A new approach to initial traces in nonlinear filtration, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 4, 305–334. 10.1016/s0294-1449(16)30294-3Search in Google Scholar
[2] D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 18 (1991), no. 3, 363–441. Search in Google Scholar
[3] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sc. Norm. Sup. Pisa (3) 22 (1968), 607–694. Search in Google Scholar
[4] D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Grad. Texts in Math. 200, Springer, New York, 2000. 10.1007/978-1-4612-1268-3Search in Google Scholar
[5] P. Baras and M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 185–212. 10.1016/s0294-1449(16)30402-4Search in Google Scholar
[6] G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J. 25 (1996), no. 3, 537–566. 10.14492/hokmj/1351516749Search in Google Scholar
[7]
P. Bénilan, M. G. Crandall and M. Pierre,
Solutions of the porous medium equation in
[8] C. Bianchini, G. Ciraolo and P. Salani, An overdetermined problem for the anisotropic capacity, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Article ID 84. 10.1007/s00526-016-1011-xSearch in Google Scholar
[9] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland Publishing, Amsterdam, 1973. Search in Google Scholar
[10] A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann. 345 (2009), no. 4, 859–881. 10.1007/s00208-009-0386-9Search in Google Scholar
[11] M. Cozzi, A. Farina and E. Valdinoci, Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations, Comm. Math. Phys. 331 (2014), no. 1, 189–214. 10.1007/s00220-014-2107-9Search in Google Scholar
[12] F. Della Pietra and G. di Blasio, Blow-up solutions for some nonlinear elliptic equations involving a Finsler–Laplacian, Publ. Mat. 61 (2017), no. 1, 213–238. 10.5565/PUBLMAT_61117_08Search in Google Scholar
[13] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, 1993. 10.1007/978-1-4612-0895-2Search in Google Scholar
[14] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22; addendum, J. Reine Angew. Math. 363 (1985), 217–220. Search in Google Scholar
[15] E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc. 314 (1989), no. 1, 187–224. 10.1090/S0002-9947-1989-0962278-5Search in Google Scholar
[16]
E. DiBenedetto and M. A. Herrero,
Nonnegative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when
[17] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Search in Google Scholar
[18] V. Ferone and B. Kawohl, Remarks on a Finsler–Laplacian, Proc. Amer. Math. Soc. 137 (2009), no. 1, 247–253. 10.1090/S0002-9939-08-09554-3Search in Google Scholar
[19] Y. Giga and N. Umeda, On instant blow-up for semilinear heat equations with growing initial data, Methods Appl. Anal. 15 (2008), no. 2, 185–195. 10.4310/MAA.2008.v15.n2.a5Search in Google Scholar
[20]
M. A. Herrero and M. Pierre,
The Cauchy problem for
[21] K. Hisa and K. Ishige, Solvability of the heat equation with a nonlinear boundary condition, preprint (2017), https://arxiv.org/abs/1704.07992. 10.1137/17M1131416Search in Google Scholar
[22] K. Ishige, On the existence of solutions of the Cauchy problem for porous medium equations with Radon measure as initial data, Discrete Contin. Dyn. Syst. 1 (1995), no. 4, 521–546. 10.3934/dcds.1995.1.521Search in Google Scholar
[23] K. Ishige, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation, SIAM J. Math. Anal. 27 (1996), no. 5, 1235–1260. 10.1137/S0036141094270370Search in Google Scholar
[24] K. Ishige, On the existence of solutions of the Cauchy problem for a quasi-linear parabolic equation with unbounded initial data, Adv. Math. Sci. Appl. 9 (1999), no. 1, 263–289. Search in Google Scholar
[25] K. Ishige and J. Kinnunen, Initial trace for a doubly nonlinear parabolic equation, J. Evol. Equ. 11 (2011), no. 4, 943–957. 10.1007/s00028-011-0119-xSearch in Google Scholar
[26]
K. Ishige and R. Sato,
Heat equation with a nonlinear boundary condition and uniformly local
[27] K. Ishige and R. Sato, Heat equation with a nonlinear boundary condition and growing initial data, Differential Integral Equations 30 (2017), no. 7–8, 481–504. 10.57262/die/1493863391Search in Google Scholar
[28] T. Kuusi and G. Mingione, Gradient regularity for nonlinear parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 4, 755–822. 10.2422/2036-2145.201103_006Search in Google Scholar
[29] T. Kuusi and G. Mingione, The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 4, 835–892. 10.4171/JEMS/449Search in Google Scholar
[30] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. 10.1090/mmono/023Search in Google Scholar
[31] S.-I. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math. 62 (2009), no. 10, 1386–1433. 10.1002/cpa.20273Search in Google Scholar
[32] S.-I. Ohta and K.-T. Sturm, Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds, Adv. Math. 252 (2014), 429–448. 10.1016/j.aim.2013.10.018Search in Google Scholar
[33] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, 1993. 10.1017/CBO9780511526282Search in Google Scholar
[34] A. Tachikawa, A partial regularity result for harmonic maps into a Finsler manifold, Calc. Var. Partial Differential Equations 16 (2003), no. 2, 217–224. 10.1007/s005260100129Search in Google Scholar
[35] D. V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc. 55 (1944), 85–95. 10.1090/S0002-9947-1944-0009795-2Search in Google Scholar
[36] Q. Xia, On the first eigencone for the Finsler Laplacian, Bull. Aust. Math. Soc. 94 (2016), no. 2, 316–325. 10.1017/S0004972716000034Search in Google Scholar
[37] K. Yamada, On viscous conservation laws with growing initial data, Differential Integral Equations 18 (2005), no. 8, 841–854. 10.57262/die/1356060148Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
