# American Institute of Mathematical Sciences

May  2011, 30(2): 509-535. doi: 10.3934/dcds.2011.30.509

## Scale-invariant extinction time estimates for some singular diffusion equations

 1 Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914 2 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, United States

Received  July 2010 Revised  July 2010 Published  February 2011

We study three singular parabolic evolutions: the second-order total variation flow, the fourth-order total variation flow, and a fourth-order surface diffusion law. Each has the property that the solution becomes identically zero in finite time. We prove scale-invariant estimates for the extinction time, using a simple argument which combines an energy estimate with a suitable Sobolev-type inequality.
Citation: Yoshikazu Giga, Robert V. Kohn. Scale-invariant extinction time estimates for some singular diffusion equations. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 509-535. doi: 10.3934/dcds.2011.30.509
##### References:
 [1] F. Andreu, V. Caselles, J. I. Diaz and J. M. Mazón, Some qualitative properties for the total variation flow, J. Funct. Anal., 188 (2002), 516-547. doi: 10.1006/jfan.2001.3829. [2] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics 223, Birkhauser, Basel, 2004. [3] M. Arisawa and Y. Giga, Anisotropic curvature flow in a very thin domain, Indiana Univ. Math. J., 52 (2003), 257-281. doi: 10.1512/iumj.2003.52.2099. [4] H. Attouch and A. Damlamian, Application des méthodes de convexité et monotonie à l'étude de certaines équations quasi linéaires, Proc. Roy. Soc. Edinburgh Sect. A, 79 (1977), 107-129. [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leiden, 1976. [6] P. Benilan and M. G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_t-\Delta \varphi (u)=0$, Indiana Univ. Math. J., 30 (1981), 161-177. doi: 10.1512/iumj.1981.30.30014. [7] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,'' Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin and New York, 1976. [8] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert,'' North-Holland Mathematics Studies 5, Notas de Matematica 50. North-Holland, Amsterdam and London; American Elsevier, New York; 1973. [9] H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis, 9 (1972), 63-74. doi: 10.1016/0022-1236(72)90014-6. [10] W.-L. Chan, A. Ramasubramaniam, V. B. Shenoy and E. Chason, Relaxation kinetics of nano-ripples on Cu(001) surfaces, Phys. Rev. B, 70 (2004), 245403. doi: 10.1103/PhysRevB.70.245403. [11] E. DiBenedetto, "Degenerate Parabolic Equations,'' Springer-Verlag, New York, 1993. [12] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'' Studies in Mathematics and its Applications 1, North-Holland, Amsterdam and Oxford; American Elsevier, New York; 1976. [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature III, J. Geom. Anal., 2 (1992), 121-150. [14] M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal., 141 (1998), 117-198. doi: 10.1007/s002050050075. [15] M.-H. Giga and Y. Giga, Very singular diffusion equations - second and fourth order problems, Japan J. Indust. Appl. Math., 27 (2010), 323-345. doi: 10.1007/s13160-010-0020-y. [16] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, in Adv. Stud. Pure Math., 31 (2001), Taniguchi Conference on Mathematics Nara '98, 93-125. [17] M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions,'' Progress in Nonlinear Differential Equations and Their Applications 79, Birkhauser, Boston, 2010. [18] Y. Giga, M. Ohnuma and M.-H. Sato, On the strong maximum principle and the large time behavior of generalized mean curvature flow with the Neumann boundary condition, J. Differential Equations, 154 (1999), 107-131. [19] Y. Giga and K. Yama-uchi, On a lower bound for the extinction time of surfaces moved by mean curvature, Calc. Var. Partial Differential Equations, 1 (1993), 417-428. [20] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Monographs in Mathematics 80, Birkhauser Verlag, Basel, 1984. [21] J. Hager and H. Spohn, Self-similar morphology and dynamics of periodic surface profiles below the roughening transition, Surf. Sci., 324 (1995), 365-372. doi: 10.1016/0039-6028(94)00771-3. [22] Y. Kashima, A subdifferential formulation of fourth order singular diffusion equations, Adv. Math. Sci. Appl., 14 (2004), 49-74. [23] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372. [24] R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys., 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4. [25] Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. doi: 10.2969/jmsj/01940493. [26] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equations, 30 (1978), 340-364. [27] D. Margetis and R. V. Kohn, Continuum theory of interacting steps on crystal surfaces in $2+1$ dimensions, Multiscale Model. Simul., 5 (2006), 729-758. doi: 10.1137/06065297X. [28] M. V. Ramana Murty, Morphological stability of nanostructures, Phys. Rev. B, 62 (2000), 17004-17011. doi: 10.1103/PhysRevB.62.17004. [29] M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Phys. Rev. B, 42 (1990), 5013-5024. doi: 10.1103/PhysRevB.42.5013. [30] A. Rettori and J. Villain, Flattening of grooves on a crystal surface: A method of investigation of surface roughness, J. Phys. France, 49 (1988), 257-267. doi: 10.1051/jphys:01988004902025700. [31] V. B. Shenoy, A. Ramasubramaniam and L. B. Freund, A variational approach to nonlinear dynamics of nanoscale surface modulations, Surf. Sci., 529 (2003), 365-383. doi: 10.1016/S0039-6028(03)00276-0. [32] V. B. Shenoy, A. Ramasubramaniam, H. Ramanarayan, D. T. Tambe, W.-L. Chan and E. Chason, Influence of step-edge barriers on the morphological relaxation of nanoscale ripples on crystal surfaces, Phys. Rev. Lett., 92 (2004), 256101. doi: 10.1103/PhysRevLett.92.256101. [33] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, "Analysis and Geometry on Groups,'' Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992. [34] J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type,'' Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, Oxford, 2006. [35] J. Watanabe, Approximation of nonlinear problems of a certain type, in "Numerical Analysis of Evolution Equations'' (eds. H. Fujita and M. Yamaguti), Lecture Notes Numer. Appl. Anal. 1, Kinokuniya Book Store, Tokyo, 1979, 147-163.

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##### References:
 [1] F. Andreu, V. Caselles, J. I. Diaz and J. M. Mazón, Some qualitative properties for the total variation flow, J. Funct. Anal., 188 (2002), 516-547. doi: 10.1006/jfan.2001.3829. [2] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics 223, Birkhauser, Basel, 2004. [3] M. Arisawa and Y. Giga, Anisotropic curvature flow in a very thin domain, Indiana Univ. Math. J., 52 (2003), 257-281. doi: 10.1512/iumj.2003.52.2099. [4] H. Attouch and A. Damlamian, Application des méthodes de convexité et monotonie à l'étude de certaines équations quasi linéaires, Proc. Roy. Soc. Edinburgh Sect. A, 79 (1977), 107-129. [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leiden, 1976. [6] P. Benilan and M. G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_t-\Delta \varphi (u)=0$, Indiana Univ. Math. J., 30 (1981), 161-177. doi: 10.1512/iumj.1981.30.30014. [7] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,'' Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin and New York, 1976. [8] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert,'' North-Holland Mathematics Studies 5, Notas de Matematica 50. North-Holland, Amsterdam and London; American Elsevier, New York; 1973. [9] H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis, 9 (1972), 63-74. doi: 10.1016/0022-1236(72)90014-6. [10] W.-L. Chan, A. Ramasubramaniam, V. B. Shenoy and E. Chason, Relaxation kinetics of nano-ripples on Cu(001) surfaces, Phys. Rev. B, 70 (2004), 245403. doi: 10.1103/PhysRevB.70.245403. [11] E. DiBenedetto, "Degenerate Parabolic Equations,'' Springer-Verlag, New York, 1993. [12] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'' Studies in Mathematics and its Applications 1, North-Holland, Amsterdam and Oxford; American Elsevier, New York; 1976. [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature III, J. Geom. Anal., 2 (1992), 121-150. [14] M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal., 141 (1998), 117-198. doi: 10.1007/s002050050075. [15] M.-H. Giga and Y. Giga, Very singular diffusion equations - second and fourth order problems, Japan J. Indust. Appl. Math., 27 (2010), 323-345. doi: 10.1007/s13160-010-0020-y. [16] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, in Adv. Stud. Pure Math., 31 (2001), Taniguchi Conference on Mathematics Nara '98, 93-125. [17] M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions,'' Progress in Nonlinear Differential Equations and Their Applications 79, Birkhauser, Boston, 2010. [18] Y. Giga, M. Ohnuma and M.-H. Sato, On the strong maximum principle and the large time behavior of generalized mean curvature flow with the Neumann boundary condition, J. Differential Equations, 154 (1999), 107-131. [19] Y. Giga and K. Yama-uchi, On a lower bound for the extinction time of surfaces moved by mean curvature, Calc. Var. Partial Differential Equations, 1 (1993), 417-428. [20] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Monographs in Mathematics 80, Birkhauser Verlag, Basel, 1984. [21] J. Hager and H. Spohn, Self-similar morphology and dynamics of periodic surface profiles below the roughening transition, Surf. Sci., 324 (1995), 365-372. doi: 10.1016/0039-6028(94)00771-3. [22] Y. Kashima, A subdifferential formulation of fourth order singular diffusion equations, Adv. Math. Sci. Appl., 14 (2004), 49-74. [23] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372. [24] R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys., 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4. [25] Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. doi: 10.2969/jmsj/01940493. [26] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equations, 30 (1978), 340-364. [27] D. Margetis and R. V. Kohn, Continuum theory of interacting steps on crystal surfaces in $2+1$ dimensions, Multiscale Model. Simul., 5 (2006), 729-758. doi: 10.1137/06065297X. [28] M. V. Ramana Murty, Morphological stability of nanostructures, Phys. Rev. B, 62 (2000), 17004-17011. doi: 10.1103/PhysRevB.62.17004. [29] M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Phys. Rev. B, 42 (1990), 5013-5024. doi: 10.1103/PhysRevB.42.5013. [30] A. Rettori and J. Villain, Flattening of grooves on a crystal surface: A method of investigation of surface roughness, J. Phys. France, 49 (1988), 257-267. doi: 10.1051/jphys:01988004902025700. [31] V. B. Shenoy, A. Ramasubramaniam and L. B. Freund, A variational approach to nonlinear dynamics of nanoscale surface modulations, Surf. Sci., 529 (2003), 365-383. doi: 10.1016/S0039-6028(03)00276-0. [32] V. B. Shenoy, A. Ramasubramaniam, H. Ramanarayan, D. T. Tambe, W.-L. Chan and E. Chason, Influence of step-edge barriers on the morphological relaxation of nanoscale ripples on crystal surfaces, Phys. Rev. Lett., 92 (2004), 256101. doi: 10.1103/PhysRevLett.92.256101. [33] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, "Analysis and Geometry on Groups,'' Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992. [34] J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type,'' Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, Oxford, 2006. [35] J. Watanabe, Approximation of nonlinear problems of a certain type, in "Numerical Analysis of Evolution Equations'' (eds. H. Fujita and M. Yamaguti), Lecture Notes Numer. Appl. Anal. 1, Kinokuniya Book Store, Tokyo, 1979, 147-163.
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