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.
@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},
mrnumber = {3060703},
zbl = {1272.35123},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2012_5_11_4_857_0/}
}
TY - JOUR AU - Mantegazza, Carlo AU - Martinazzi, Luca TI - A note on quasilinear parabolic equations on manifolds JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 857 EP - 874 VL - 11 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2012_5_11_4_857_0/ LA - en ID - ASNSP_2012_5_11_4_857_0 ER -
%0 Journal Article %A Mantegazza, Carlo %A Martinazzi, Luca %T A note on quasilinear parabolic equations on manifolds %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 857-874 %V 11 %N 4 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2012_5_11_4_857_0/ %G en %F 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://archive.numdam.org/item/ASNSP_2012_5_11_4_857_0/
[1] R. Adams, “Sobolev Spaces”, Academic Press, New York, 1975. | MR | Zbl
[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 | Zbl
[3] T. Aubin, “Some Nonlinear Problems in Riemannian Geometry”, Springer-Verlag, 1998. | MR | Zbl
[4] S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom. 69 (2005), 217–278. | MR | Zbl
[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 | Zbl
[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 | Zbl
[7] E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom. 10 (2002), 307–339. | MR | Zbl
[8] J. L. Lions and E. Magenes, “Non-homogeneous Boundary Value Problems and Applications”, Vol. I, Springer-Verlag, New York, 1972. | MR | Zbl
[9] A. Malchiodi and M. Struwe, -curvature flow on , J. Differential Geom. 73 (2006), 1–44. | MR | Zbl
[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 | Zbl
[12] J. J. Sharples, Linear and quasilinear parabolic equations in Sobolev space, J. Differential Equations 202 (2004), 111–142. | MR | Zbl
[13] R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), 35–50. | MR | Zbl
