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March  2021, 20(3): 1213-1227. doi: 10.3934/cpaa.2021017

Existence of solution and asymptotic behavior for a class of parabolic equations

Anderson L. A. de Araujo 1, and  Marcelo Montenegro 2,,

1. 

Universidade Federal de Viçosa, Departamento de Matemática, Avenida Peter Henry Rolfs, s/n, CEP 36570-900, Viçosa, MG, Brasil
2. 

Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brasil

* Corresponding author

Received  July 2020 Revised  December 2020 Published  March 2021 Early access  February 2021

Fund Project: The authors have been supported by FAPESP and CNPq

We prove existence and uniqueness of a positive solution for a class of quasilinear parabolic equations. We also show some maximum principles on the derivatives of the solution and study the asymptotic behavior of the solution near the maximal time of existence.

Keywords: Parabolic equation, existence of solution, uniqueness, maximum principle, asymptotic behavior.
Mathematics Subject Classification: Primary: 35K10, 35B50; Secondary: 35K90, 35K67.
Citation: Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
References:
[1]

S. Altschuler, S. B. Angenent and Y. Giga, Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal., 5 (1995), 293-358.  doi: 10.1007/BF02921800.

[2]

S. Angenent, Parabolic equations for curves on surfaces: part I. Curves with $p$–integrable curvature, Ann. Math., 132 (1990), 451-483.  doi: 10.2307/1971426.

[3]

M. Athanassenas, Behaviour of singularities of the rotationally symmetric, volume–preserving mean curvature flow, Calc. Var. PDE, 17 (2003), 1-16.  doi: 10.1007/s00526-002-0098-4.

[4] K. A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, 2015. 
[5]

J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. AMS, 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3.

[6]

J. Escher and B. V. Matioc, Neck pinching for periodic mean curvature flows, Analysis, 30 (2010), 253-260.  doi: 10.1524/anly.2010.1039.

[7]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differ. Geom., 33 (1991), 635-681. 

[8]

M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96. 

[9]

Z. Gang and I. M. Sigal, Neck pinching dynamics under mean curvature flow, J. Geom. Anal., 19 (2009), 36-80.  doi: 10.1007/s12220-008-9050-y.

[10]

Y. Giga, Y. Seki and N. Umeda, Mean curvature flow, closes open ends of noncompact surfaces of rotation, Comm. Part. Diff. Eq., 34 (2009), 1508-1529.  doi: 10.1080/03605300903296926.

[11]

M. A. Grayson, The shape of afigure eight under the curve shortening flow, Invent. Math., 96 (1989), 177-180.  doi: 10.1007/BF01393973.

[12]

G. Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differ. Equ., 77 (1989), 369-378.  doi: 10.1016/0022-0396(89)90149-6.

[13]

G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Differ. Geom., 31 (1990), 285-299. 

[14]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946.

[15]

I. Kim and D. Kwon, On mean curvature flow with forcing, Commun. Partial Differ. Equ., 45 (2020), 414-455.  doi: 10.1080/03605302.2019.1695262.

[16]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 2005. doi: 10.1142/3302.

[17]

B. V. Matioc, value problems for rotationally symmetric mean curvature flows, Arch. Math., 89 (2007), 365-372.  doi: 10.1007/s00013-007-2141-3.

[18]

J. A. McCoy, F. Y. Y. Mofarreh and G. H. Williams, Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions, Ann. Mat. Pura Appl., 193 (2014), 1443-1455.  doi: 10.1007/s10231-013-0337-7.

[19]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, 1992.

[20]

K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Manuscr. Math., 95 (1998), 225-236.  doi: 10.1007/s002290050025.

[21]

H. M. Soner and P. E. Souganidis, Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Commun. Partial Differ. Equ., 18 (1993), 859-894.  doi: 10.1080/03605309308820954.

show all references

References:
[1]

S. Altschuler, S. B. Angenent and Y. Giga, Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal., 5 (1995), 293-358.  doi: 10.1007/BF02921800.

[2]

S. Angenent, Parabolic equations for curves on surfaces: part I. Curves with $p$–integrable curvature, Ann. Math., 132 (1990), 451-483.  doi: 10.2307/1971426.

[3]

M. Athanassenas, Behaviour of singularities of the rotationally symmetric, volume–preserving mean curvature flow, Calc. Var. PDE, 17 (2003), 1-16.  doi: 10.1007/s00526-002-0098-4.

[4] K. A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, 2015. 
[5]

J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. AMS, 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3.

[6]

J. Escher and B. V. Matioc, Neck pinching for periodic mean curvature flows, Analysis, 30 (2010), 253-260.  doi: 10.1524/anly.2010.1039.

[7]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differ. Geom., 33 (1991), 635-681. 

[8]

M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96. 

[9]

Z. Gang and I. M. Sigal, Neck pinching dynamics under mean curvature flow, J. Geom. Anal., 19 (2009), 36-80.  doi: 10.1007/s12220-008-9050-y.

[10]

Y. Giga, Y. Seki and N. Umeda, Mean curvature flow, closes open ends of noncompact surfaces of rotation, Comm. Part. Diff. Eq., 34 (2009), 1508-1529.  doi: 10.1080/03605300903296926.

[11]

M. A. Grayson, The shape of afigure eight under the curve shortening flow, Invent. Math., 96 (1989), 177-180.  doi: 10.1007/BF01393973.

[12]

G. Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differ. Equ., 77 (1989), 369-378.  doi: 10.1016/0022-0396(89)90149-6.

[13]

G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Differ. Geom., 31 (1990), 285-299. 

[14]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946.

[15]

I. Kim and D. Kwon, On mean curvature flow with forcing, Commun. Partial Differ. Equ., 45 (2020), 414-455.  doi: 10.1080/03605302.2019.1695262.

[16]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 2005. doi: 10.1142/3302.

[17]

B. V. Matioc, value problems for rotationally symmetric mean curvature flows, Arch. Math., 89 (2007), 365-372.  doi: 10.1007/s00013-007-2141-3.

[18]

J. A. McCoy, F. Y. Y. Mofarreh and G. H. Williams, Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions, Ann. Mat. Pura Appl., 193 (2014), 1443-1455.  doi: 10.1007/s10231-013-0337-7.

[19]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, 1992.

[20]

K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Manuscr. Math., 95 (1998), 225-236.  doi: 10.1007/s002290050025.

[21]

H. M. Soner and P. E. Souganidis, Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Commun. Partial Differ. Equ., 18 (1993), 859-894.  doi: 10.1080/03605309308820954.

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