A note on quasilinear parabolic equations on manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 4, pp. 857-874.

We prove short time existence, uniqueness and continuous dependence on the initial data of smooth solutions of quasilinear locally parabolic equations of arbitrary even order on closed manifolds.

Publié le :
Classification : 35K59,  35K41,  35K52
@article{ASNSP_2012_5_11_4_857_0,
     author = {Mantegazza, Carlo and Martinazzi, Luca},
     title = {A note on quasilinear parabolic equations on manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {857--874},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {4},
     year = {2012},
     zbl = {1272.35123},
     mrnumber = {3060703},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_4_857_0/}
}
Mantegazza, Carlo; Martinazzi, Luca. A note on quasilinear parabolic equations on manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 4, pp. 857-874. http://www.numdam.org/item/ASNSP_2012_5_11_4_857_0/

[1] R. Adams, “Sobolev Spaces”, Academic Press, New York, 1975. | MR 450957 | Zbl 1098.46001

[2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 623–727. | MR 125307 | Zbl 0093.10401

[3] T. Aubin, “Some Nonlinear Problems in Riemannian Geometry”, Springer-Verlag, 1998. | MR 1636569 | Zbl 0896.53003

[4] S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom. 69 (2005), 217–278. | MR 2168505 | Zbl 1085.53028

[5] M. Giaquinta and G. Modica, Local existence for quasilinear parabolic systems under nonlinear boundary conditions, Ann. Mat. Pura Appl. 149 (1987), 41–59. | MR 932775 | Zbl 0655.35049

[6] G. Huisken and A. Polden, Geometric Evolution Equations for Hypersurfaces, In: “Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996)”, Springer–Verlag, Berlin, 1999, 45–84. | MR 1731639 | Zbl 0942.35047

[7] E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom. 10 (2002), 307–339. | MR 1900754 | Zbl 1029.53082

[8] J. L. Lions and E. Magenes, “Non-homogeneous Boundary Value Problems and Applications”, Vol. I, Springer-Verlag, New York, 1972. | MR 350177 | Zbl 0223.35039

[9] A. Malchiodi and M. Struwe, Q-curvature flow on 𝕊 4 , J. Differential Geom. 73 (2006), 1–44. | MR 2217518 | Zbl 1099.53034

[10] A. Polden, Curves and Surfaces of Least Total Curvature and Fourth–Order Flows, P.h.D. thesis, Mathematisches Institut, Univ. Tübingen, 1996, Arbeitsbereich Analysis Preprint Server – Univ. Tübingen, http://poincare.mathematik.uni-tuebingen.de/mozilla/home.e.html.

[11] H. Schwetlick and M. Struwe, Convergence of the Yamabe flow for “large” energies, J. Reine Angew. Math. 562 (2003), 59–100. | MR 2011332 | Zbl 1079.53100

[12] J. J. Sharples, Linear and quasilinear parabolic equations in Sobolev space, J. Differential Equations 202 (2004), 111–142. | MR 2060534 | Zbl 1056.35091

[13] R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), 35–50. | MR 1258912 | Zbl 0846.53027