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May  2020, 40(5): 2963-2986. doi: 10.3934/dcds.2020157

General initial data for a class of parabolic equations including the curve shortening problem

Kai-Seng Chou 1,, and  Ying-Chuen Kwong 2,

1. 

The Chinese University of Hong Kong, Hong Kong
2. 

Northern Illinois University, DeKalb, IL60115, U.S.A

* Corresponding author: Kai-Seng Chou

Received  September 2019 Revised  November 2019 Published  March 2020

The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When $ m\in(1,2] $, it is shown that there exists a smooth solution for $ t>0 $ when the initial data belongs to $ L_{\text{loc}}^p, p>1 $. When $ m>2 $, the same results holds when the initial data belongs to $ W_{\text{loc}}^{1,p}, p\geq m-1 $. An example is displayed to show that a smooth solution may not exist when the initial data is merely in $ L^p_{\text{loc}}, p>1 $. Solvability of the weak solution is also studied.

Keywords: Initial trace, curve shortening problem, porous medium equations, Widder's theorem, Cauchy problem.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76A20, 35B35, 93D20, 37B30.
Citation: Kai-Seng Chou, Ying-Chuen Kwong. General initial data for a class of parabolic equations including the curve shortening problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2963-2986. doi: 10.3934/dcds.2020157
References:
[1]

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

[2]

S. Angenent, Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. of Math., 133 (1991), 171-215.  doi: 10.2307/2944327.  Google Scholar

[3]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 25 (1971), 221-228.   Google Scholar

[4]

D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc, 280 (1983), 351-366.  doi: 10.1090/S0002-9947-1983-0712265-1.  Google Scholar

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Ph. Benilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equationin $\mathbb R^n$ under optimal conditions on initial values, Indiana Univ. Math J., 33 (1984), 51-87.  doi: 10.1512/iumj.1984.33.33003.  Google Scholar

[6]

Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.  doi: 10.4310/jdg/1214446564.  Google Scholar

[7]

K. S. Chou and Y. C. Kwong, On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth, Calc. Var. Partial Differential Equations, 12 (2001), 281-315.  doi: 10.1007/PL00009915.  Google Scholar

[8]

K. S. Chou and X. P. Zhu, The Curve Shortening Problem, Chapman and Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[9]

B. E. Dahlberg and C. E. Kenig, Non-negative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Soc., 1 (1988), 401-412.  doi: 10.1090/S0894-0347-1988-0928264-9.  Google Scholar

[10]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[11]

K. Ecker, An interior gradient bound for solutions of equations of capillary type, Arch. Rational Mech. Anal., 92 (1986), 137-151.  doi: 10.1007/BF00251254.  Google Scholar

[12]

K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991), 547-569.  doi: 10.1007/BF01232278.  Google Scholar

[13]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geom., 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.  Google Scholar

[14]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature, II, Trans. Amer. Math. Soc., 330 (1992), 321-332.  doi: 10.1090/S0002-9947-1992-1068927-8.  Google Scholar

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L. C. Evans and J. Spruck, Motion of level sets by mean curvature, III, J. Geom. Anal., 2 (1992), 121-150.  doi: 10.1007/BF02921385.  Google Scholar

[16]

A. Friedman, Partial Differential Equations, Holt, Rinehart, Winston, 1969.  Google Scholar

[17]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.  doi: 10.4310/jdg/1214439902.  Google Scholar

[18]

C. Gerhardt, Evolutionary surfaces of prescribed mean curvature, J. Differential Equations, 36 (1980), 139-172.  doi: 10.1016/0022-0396(80)90081-9.  Google Scholar

[19]

Y. Giga, Interior derivative blow-up for quailinear parabolic equations, Discrete Contin. Dynam. Systems, 1 (1995), 449-461.  doi: 10.3934/dcds.1995.1.449.  Google Scholar

[20]

M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.  doi: 10.4310/jdg/1214441371.  Google Scholar

[21]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, , vol. 23, Translations of Mathematical Monographs, American Mathematical Society, Providence, 1968.  Google Scholar

[22]

G. Leoni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, 2009. doi: 10.1090/gsm/105.  Google Scholar

[23]

X.-J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.  doi: 10.1007/PL00004604.  Google Scholar

[24]

D. V. Widder, Positive temperatures on an infintie rod, Trans. Amer. Math. Soc., 55 (1944), 85-95.  doi: 10.1090/S0002-9947-1944-0009795-2.  Google Scholar

[25]

Z. Wu, J. Yin, H. Li and J. Zhao, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001. doi: 10.1142/4782.  Google Scholar

[26]

J. Zhao, On the Cauchy problem and initial traces for evolution p-Laplacian equation with strongly nonlinear sources, J. Differential Equations, 121 (1995), 329-383.  doi: 10.1006/jdeq.1995.1132.  Google Scholar

show all references

References:
[1]

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

[2]

S. Angenent, Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. of Math., 133 (1991), 171-215.  doi: 10.2307/2944327.  Google Scholar

[3]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 25 (1971), 221-228.   Google Scholar

[4]

D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc, 280 (1983), 351-366.  doi: 10.1090/S0002-9947-1983-0712265-1.  Google Scholar

[5]

Ph. Benilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equationin $\mathbb R^n$ under optimal conditions on initial values, Indiana Univ. Math J., 33 (1984), 51-87.  doi: 10.1512/iumj.1984.33.33003.  Google Scholar

[6]

Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.  doi: 10.4310/jdg/1214446564.  Google Scholar

[7]

K. S. Chou and Y. C. Kwong, On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth, Calc. Var. Partial Differential Equations, 12 (2001), 281-315.  doi: 10.1007/PL00009915.  Google Scholar

[8]

K. S. Chou and X. P. Zhu, The Curve Shortening Problem, Chapman and Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[9]

B. E. Dahlberg and C. E. Kenig, Non-negative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Soc., 1 (1988), 401-412.  doi: 10.1090/S0894-0347-1988-0928264-9.  Google Scholar

[10]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[11]

K. Ecker, An interior gradient bound for solutions of equations of capillary type, Arch. Rational Mech. Anal., 92 (1986), 137-151.  doi: 10.1007/BF00251254.  Google Scholar

[12]

K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991), 547-569.  doi: 10.1007/BF01232278.  Google Scholar

[13]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geom., 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.  Google Scholar

[14]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature, II, Trans. Amer. Math. Soc., 330 (1992), 321-332.  doi: 10.1090/S0002-9947-1992-1068927-8.  Google Scholar

[15]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature, III, J. Geom. Anal., 2 (1992), 121-150.  doi: 10.1007/BF02921385.  Google Scholar

[16]

A. Friedman, Partial Differential Equations, Holt, Rinehart, Winston, 1969.  Google Scholar

[17]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.  doi: 10.4310/jdg/1214439902.  Google Scholar

[18]

C. Gerhardt, Evolutionary surfaces of prescribed mean curvature, J. Differential Equations, 36 (1980), 139-172.  doi: 10.1016/0022-0396(80)90081-9.  Google Scholar

[19]

Y. Giga, Interior derivative blow-up for quailinear parabolic equations, Discrete Contin. Dynam. Systems, 1 (1995), 449-461.  doi: 10.3934/dcds.1995.1.449.  Google Scholar

[20]

M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.  doi: 10.4310/jdg/1214441371.  Google Scholar

[21]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, , vol. 23, Translations of Mathematical Monographs, American Mathematical Society, Providence, 1968.  Google Scholar

[22]

G. Leoni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, 2009. doi: 10.1090/gsm/105.  Google Scholar

[23]

X.-J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.  doi: 10.1007/PL00004604.  Google Scholar

[24]

D. V. Widder, Positive temperatures on an infintie rod, Trans. Amer. Math. Soc., 55 (1944), 85-95.  doi: 10.1090/S0002-9947-1944-0009795-2.  Google Scholar

[25]

Z. Wu, J. Yin, H. Li and J. Zhao, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001. doi: 10.1142/4782.  Google Scholar

[26]

J. Zhao, On the Cauchy problem and initial traces for evolution p-Laplacian equation with strongly nonlinear sources, J. Differential Equations, 121 (1995), 329-383.  doi: 10.1006/jdeq.1995.1132.  Google Scholar

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