American Institute of Mathematical Sciences

Advanced
  2018, 25: 72-86. doi: 10.3934/era.2018.25.008

Characterization of Log-convex decay in non-selfadjoint dynamics

Jon Johnsen

Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmark

Received  June 29, 2018 Published  December 2018

Fund Project: Supported by the Danish Research Council, Natural Sciences grant no. 4181-00042.

The short-time and global behavior are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.

Keywords: Log-convex decay, non-selfadjoint, hyponormal, positively accretive operators, short-time behavior.
Mathematics Subject Classification: Primary: 35E15; Secondary: 47D06.
Citation: Jon Johnsen. Characterization of Log-convex decay in non-selfadjoint dynamics. Electronic Research Announcements, 2018, 25: 72-86. doi: 10.3934/era.2018.25.008
References:
[1]

A.-E. Christensen and J. Johnsen, Final value problems for parabolic differential equations and their well-posedness, Axioms, 7 (2018), article no. 31, 1-36. doi: 10.3390/axioms7020031.  Google Scholar

[2]

A.-E. Christensen and J. Johnsen, On parabolic final value problems and well-posedness, C. R. Math. Acad. Sci. Paris, 356 (2018), 301-305. doi: 10.1016/j.crma.2018.01.019.  Google Scholar

[3]

G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, 252, Springer, New York, 2009.  Google Scholar

[4]

B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, 139, Cambridge University Press, Cambridge, 2013.  Google Scholar

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin, 1983, 1985. Google Scholar

[6]

J. Janas, On unbounded hyponormal operators. Ⅲ, Studia Math., 112 (1994), 75-82.  doi: 10.4064/sm-112-1-75-82.  Google Scholar

[7]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[8]

S. Lang, Differential Manifolds, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1972.  Google Scholar

[9]

C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), 3-49.  doi: 10.1137/S00361445024180.  Google Scholar

[10]

C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications. A Contemporary Approach, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, New York, 2006. doi: 10.1007/0-387-31077-0.  Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

G. K. Pedersen, Analysis Now, Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1007-8.  Google Scholar

[13]

L. Perko, Differential Equations and Dynamical Systems, Third ed., Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[14]

J. Rauch, Partial Differential Equations, Graduate Texts in Mathematics, 128, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

[15]

L. Schwartz, Théorie des Distributions, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. Ⅸ-Ⅹ. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966.  Google Scholar

[16]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  doi: 10.1016/0022-247X(74)90008-0.  Google Scholar

[17]

L. N. Trefethen and M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

show all references

References:
[1]

A.-E. Christensen and J. Johnsen, Final value problems for parabolic differential equations and their well-posedness, Axioms, 7 (2018), article no. 31, 1-36. doi: 10.3390/axioms7020031.  Google Scholar

[2]

A.-E. Christensen and J. Johnsen, On parabolic final value problems and well-posedness, C. R. Math. Acad. Sci. Paris, 356 (2018), 301-305. doi: 10.1016/j.crma.2018.01.019.  Google Scholar

[3]

G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, 252, Springer, New York, 2009.  Google Scholar

[4]

B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, 139, Cambridge University Press, Cambridge, 2013.  Google Scholar

[5]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin, 1983, 1985. Google Scholar

[6]

J. Janas, On unbounded hyponormal operators. Ⅲ, Studia Math., 112 (1994), 75-82.  doi: 10.4064/sm-112-1-75-82.  Google Scholar

[7]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[8]

S. Lang, Differential Manifolds, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1972.  Google Scholar

[9]

C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), 3-49.  doi: 10.1137/S00361445024180.  Google Scholar

[10]

C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications. A Contemporary Approach, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, New York, 2006. doi: 10.1007/0-387-31077-0.  Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

G. K. Pedersen, Analysis Now, Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1007-8.  Google Scholar

[13]

L. Perko, Differential Equations and Dynamical Systems, Third ed., Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[14]

J. Rauch, Partial Differential Equations, Graduate Texts in Mathematics, 128, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.  Google Scholar

[15]

L. Schwartz, Théorie des Distributions, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. Ⅸ-Ⅹ. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966.  Google Scholar

[16]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  doi: 10.1016/0022-247X(74)90008-0.  Google Scholar

[17]

L. N. Trefethen and M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[1]

Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867
[2]

Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787
[3]

Marcel Oliver. The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221
[4]

Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011
[5]

Gregory M. Zaverucha, Douglas R. Stinson. Short one-time signatures. Advances in Mathematics of Communications, 2011, 5 (3) : 473-488. doi: 10.3934/amc.2011.5.473
[6]

Dieter Bothe, Petra Wittbold. Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2239-2260. doi: 10.3934/cpaa.2012.11.2239
[7]

Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639
[8]

Robert M. Strain. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinetic & Related Models, 2012, 5 (3) : 583-613. doi: 10.3934/krm.2012.5.583
[9]

Alessandro Fonda, Rafael Ortega. Positively homogeneous equations in the plane. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 475-482. doi: 10.3934/dcds.2000.6.475
[10]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901
[11]

Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89
[12]

Qingshan You, Qun Wan, Yipeng Liu. A short note on strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2013, 7 (1) : 305-306. doi: 10.3934/ipi.2013.7.305
[13]

Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671
[14]

Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks & Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297
[15]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096
[16]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615
[17]

Eskil Hansen, Alexander Ostermann. Dimension splitting for time dependent operators. Conference Publications, 2009, 2009 (Special) : 322-332. doi: 10.3934/proc.2009.2009.322
[18]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134
[19]

Andrea Braides, Valeria Chiadò Piat. Non convex homogenization problems for singular structures. Networks & Heterogeneous Media, 2008, 3 (3) : 489-508. doi: 10.3934/nhm.2008.3.489
[20]

Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (82)
  • HTML views (716)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences