American Institute of Mathematical Sciences

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  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

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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

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