# 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 & Continuous Dynamical Systems - A, 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. doi: 10.1006/jfan.2001.3829. Google Scholar [2] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'', Progress in Mathematics \textbf{223}, 223 (2004). Google Scholar [3] M. Arisawa and Y. Giga, Anisotropic curvature flow in a very thin domain,, Indiana Univ. Math. J., 52 (2003), 257. doi: 10.1512/iumj.2003.52.2099. Google Scholar [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. Google Scholar [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'', Noordhoff, (1976). Google Scholar [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. doi: 10.1512/iumj.1981.30.30014. Google Scholar [7] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,'', Grundlehren der Mathematischen Wissenschaften \textbf{223}, 223 (1976). Google Scholar [8] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert,'', North-Holland Mathematics Studies \textbf{5}, 5 (1973). Google Scholar [9] H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces,, J. Functional Analysis, 9 (1972), 63. doi: 10.1016/0022-1236(72)90014-6. Google Scholar [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). doi: 10.1103/PhysRevB.70.245403. Google Scholar [11] E. DiBenedetto, "Degenerate Parabolic Equations,'', Springer-Verlag, (1993). Google Scholar [12] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'', Studies in Mathematics and its Applications \textbf{1}, 1 (1976). Google Scholar [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature III,, J. Geom. Anal., 2 (1992), 121. Google Scholar [14] M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature,, Arch. Rational Mech. Anal., 141 (1998), 117. doi: 10.1007/s002050050075. Google Scholar [15] M.-H. Giga and Y. Giga, Very singular diffusion equations - second and fourth order problems,, Japan J. Indust. Appl. Math., 27 (2010), 323. doi: 10.1007/s13160-010-0020-y. Google Scholar [16] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations,, in Adv. Stud. Pure Math., 31 (2001), 93. Google Scholar [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 \textbf{79}, 79 (2010). Google Scholar [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. Google Scholar [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. Google Scholar [20] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'', Monographs in Mathematics \textbf{80}, 80 (1984). Google Scholar [21] J. Hager and H. Spohn, Self-similar morphology and dynamics of periodic surface profiles below the roughening transition,, Surf. Sci., 324 (1995), 365. doi: 10.1016/0039-6028(94)00771-3. Google Scholar [22] Y. Kashima, A subdifferential formulation of fourth order singular diffusion equations,, Adv. Math. Sci. Appl., 14 (2004), 49. Google Scholar [23] R. Kobayashi and Y. Giga, Equations with singular diffusivity,, J. Statist. Phys., 95 (1999), 1187. doi: 10.1023/A:1004570921372. Google Scholar [24] R. V. Kohn and F. Otto, Upper bounds on coarsening rates,, Comm. Math. Phys., 229 (2002), 375. doi: 10.1007/s00220-002-0693-4. Google Scholar [25] Y. Komura, Nonlinear semi-groups in Hilbert space,, J. Math. Soc. Japan, 19 (1967), 493. doi: 10.2969/jmsj/01940493. Google Scholar [26] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem,, J. Differential Equations, 30 (1978), 340. Google Scholar [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. doi: 10.1137/06065297X. Google Scholar [28] M. V. Ramana Murty, Morphological stability of nanostructures,, Phys. Rev. B, 62 (2000), 17004. doi: 10.1103/PhysRevB.62.17004. Google Scholar [29] M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface,, Phys. Rev. B, 42 (1990), 5013. doi: 10.1103/PhysRevB.42.5013. Google Scholar [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. doi: 10.1051/jphys:01988004902025700. Google Scholar [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. doi: 10.1016/S0039-6028(03)00276-0. Google Scholar [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). doi: 10.1103/PhysRevLett.92.256101. Google Scholar [33] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, "Analysis and Geometry on Groups,'', Cambridge Tracts in Mathematics \textbf{100}, 100 (1992). Google Scholar [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 \textbf{33}, 33 (2006). Google Scholar [35] J. Watanabe, Approximation of nonlinear problems of a certain type,, in, 1 (1979), 147. Google Scholar

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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. doi: 10.1006/jfan.2001.3829. Google Scholar [2] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'', Progress in Mathematics \textbf{223}, 223 (2004). Google Scholar [3] M. Arisawa and Y. Giga, Anisotropic curvature flow in a very thin domain,, Indiana Univ. Math. J., 52 (2003), 257. doi: 10.1512/iumj.2003.52.2099. Google Scholar [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. Google Scholar [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'', Noordhoff, (1976). Google Scholar [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. doi: 10.1512/iumj.1981.30.30014. Google Scholar [7] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,'', Grundlehren der Mathematischen Wissenschaften \textbf{223}, 223 (1976). Google Scholar [8] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert,'', North-Holland Mathematics Studies \textbf{5}, 5 (1973). Google Scholar [9] H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces,, J. Functional Analysis, 9 (1972), 63. doi: 10.1016/0022-1236(72)90014-6. Google Scholar [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). doi: 10.1103/PhysRevB.70.245403. Google Scholar [11] E. DiBenedetto, "Degenerate Parabolic Equations,'', Springer-Verlag, (1993). Google Scholar [12] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'', Studies in Mathematics and its Applications \textbf{1}, 1 (1976). Google Scholar [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature III,, J. Geom. Anal., 2 (1992), 121. Google Scholar [14] M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature,, Arch. Rational Mech. Anal., 141 (1998), 117. doi: 10.1007/s002050050075. Google Scholar [15] M.-H. Giga and Y. Giga, Very singular diffusion equations - second and fourth order problems,, Japan J. Indust. Appl. Math., 27 (2010), 323. doi: 10.1007/s13160-010-0020-y. Google Scholar [16] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations,, in Adv. Stud. Pure Math., 31 (2001), 93. Google Scholar [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 \textbf{79}, 79 (2010). Google Scholar [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. Google Scholar [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. Google Scholar [20] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'', Monographs in Mathematics \textbf{80}, 80 (1984). Google Scholar [21] J. Hager and H. Spohn, Self-similar morphology and dynamics of periodic surface profiles below the roughening transition,, Surf. Sci., 324 (1995), 365. doi: 10.1016/0039-6028(94)00771-3. Google Scholar [22] Y. Kashima, A subdifferential formulation of fourth order singular diffusion equations,, Adv. Math. Sci. Appl., 14 (2004), 49. Google Scholar [23] R. Kobayashi and Y. Giga, Equations with singular diffusivity,, J. Statist. Phys., 95 (1999), 1187. doi: 10.1023/A:1004570921372. Google Scholar [24] R. V. Kohn and F. Otto, Upper bounds on coarsening rates,, Comm. Math. Phys., 229 (2002), 375. doi: 10.1007/s00220-002-0693-4. Google Scholar [25] Y. Komura, Nonlinear semi-groups in Hilbert space,, J. Math. Soc. Japan, 19 (1967), 493. doi: 10.2969/jmsj/01940493. Google Scholar [26] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem,, J. Differential Equations, 30 (1978), 340. Google Scholar [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. doi: 10.1137/06065297X. Google Scholar [28] M. V. Ramana Murty, Morphological stability of nanostructures,, Phys. Rev. B, 62 (2000), 17004. doi: 10.1103/PhysRevB.62.17004. Google Scholar [29] M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface,, Phys. Rev. B, 42 (1990), 5013. doi: 10.1103/PhysRevB.42.5013. Google Scholar [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. doi: 10.1051/jphys:01988004902025700. Google Scholar [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. doi: 10.1016/S0039-6028(03)00276-0. Google Scholar [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). doi: 10.1103/PhysRevLett.92.256101. Google Scholar [33] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, "Analysis and Geometry on Groups,'', Cambridge Tracts in Mathematics \textbf{100}, 100 (1992). Google Scholar [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 \textbf{33}, 33 (2006). Google Scholar [35] J. Watanabe, Approximation of nonlinear problems of a certain type,, in, 1 (1979), 147. Google Scholar
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