116,688 research outputs found
Well-posedness for degenerate third order equations with delay and applications to inverse problems
[EN] In this paper, we study well-posedness for the following third-order in time equation with delay <disp-formula idoperators defined on a Banach space X with domains D(A) and D(B) such that t)is the state function taking values in X and u(t): (-, 0] X defined as u(t)() = u(t+) for < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue-Bochner spaces Lp(T,X), periodic Besov spaces Bp,qs(T,X) and periodic Triebel-Lizorkin spaces Fp,qs(T,X). A novel application to an inverse problem is given.The first, second and third authors have been supported by MEC, grant MTM2016-75963-P. The second author has been supported by AICO/2016/30. The fourth author has been supported by MEC, grant MTM2015-65825-P.Conejero, JA.; Lizama, C.; Murillo-Arcila, M.; Seoane Sepúlveda, JB. (2019). Well-posedness for degenerate third order equations with delay and applications to inverse problems. Israel Journal of Mathematics. 229(1):219-254. https://doi.org/10.1007/s11856-018-1796-8S2192542291K. Abbaoui and Y. Cherruault, New ideas for solving identification and optimal control problems related to biomedical systems, International Journal of Biomedical Computing 36 (1994), 181–186.M. Al Horani and A. Favini, Perturbation method for first- and complete second-order differential equations, Journal of Optimization Theory and Applications 166 (2015), 949–967.H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Mathematische Nachrichten 186 (1997), 5–56.U. A. Anufrieva, A degenerate Cauchy problem for a second-order equation. A wellposedness criterion, Differentsial’nye Uravneniya 34 (1998), 1131–1133; English translation: Differential Equations 34 (1999), 1135–1137.W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Mathematische Zeitschrift 240 (2002), 311–343.W. Arendt and S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proceedings of the Edinburgh Mathematical Society 47 (2004), 15–33.W. Arendt, C. Batty and S. Bu, Fourier multipliers for Holder continuous functions and maximal regularity, Studia Mathematica 160 (2004), 23–51.V. Barbu and A. Favini, Periodic problems for degenerate differential equations, Rendiconti dell’Istituto di Matematica dell’Università di Trieste 28 (1996), 29–57.A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, Vol. 10, A K Peters, Wellesley, MA, 2005.S. Bu, Well-posedness of second order degenerate differential equations in vector-valued function spaces, Studia Mathematica 214 (2013), 1–16.S. Bu and G. Cai, Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces, Israel Journal of Mathematics 212 (2016), 163–188.S. Bu and G. Cai, Well-posedness of second-order degenerate differential equations with finite delay in vector-valued function spaces, Pacific Journal of Mathematics 288 (2017), 27–46.S. Bu and Y. Fang, Periodic solutions of delay equations in Besov spaces and Triebel–Lizorkin spaces, Taiwanese Journal of Mathematics 13 (2009), 1063–1076.S. Bu and J. Kim, Operator-valued Fourier multipliers on periodic Triebel spaces, Acta Mathematica Sinica 21 (2005), 1049–1056.G. Cai and S. Bu, Well-posedness of second order degenerate integro-differential equations with infinite delay in vector-valued function spaces, Mathematische Nachrichten 289 (2016), 436–451.R. Chill and S. Srivastava, Lp-maximal regularity for second order Cauchy problems, Mathematische Zeitschrift 251 (2005), 751–781.R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs of the American Mathematical Society 166 (2003).O. Diekmann, S. A. van Giles, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Applied Mathematical Sciences, Vol. 110, Springer, New York, 1995.K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Vol. 194, Springer, New York, 2000.M. Fabrizio, A. Favini and G. Marinoschi, An optimal control problem for a singular system of solid liquid phase-transition, Numerical Functional Analysis and Optimization 31 (2010), 989–1022.A. Favini and G. Marinoschi, Periodic behavior for a degenerate fast diffusion equation, Journal of Mathematical Analysis and Applications 351 (2009), 509–521.A. Favini and G. Marinoschi, Identification of the time derivative coefficients in a fast diffusion degenerate equation, Journal of Optimization Theory and Applications 145 (2010), 249–269.A. Favini and A. Yagi, Degenerate differential equations in Banach spaces, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 215, Marcel Dekker, New York, 1999.X. L. Fu and M. Li, Maximal regularity of second-order evolution equations with infinite delay in Banach spaces, Studia Mathematica 224 (2014), 199–219.G. C. Gorain, Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in Rn, Journal of Mathematical Analysis and Applications 319 (2006), 635–650.P. Grisvard, Équations différentielles abstraites, Annales Scientifiques de l’école Normale Superieure 2 (1969), 311–395.J. K. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations, Journal of Mathematical Analysis and Applications 178 (1993), 344–362.Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer, Berlin, 1991.B. Kaltenbacher, I. Lasiecka and M. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Moore-Gibson-Thomson equation arising in high intensity ultrasound, Mathematical Models & Methods in Applied Sciences 22 (2012), 1250035.V. Keyantuo and C. Lizama, Fourier multipliers and integro-differential equations in Banach spaces, Journal of the London Mathematical Society 69 (2004), 737–750.V. Keyantuo and C. Lizama, Maximal regularity for a class of integro-differential equations with infinite delay in Banach spaces, Studia Mathematica 168 (2005), 25–50.V. Keyantuo, C. Lizama and V. Poblete, Periodic solutions of integro-differential euations in vector-valued function spaces, Journal of Differential Equations 246 (2009), 1007–1037.C. Lizama, Fourier multipliers and periodic solutions of delay equations in Banach spaces, Journal of Mathematical Analysis and Applications 324 (2006), 921–933.C. Lizama and V. Poblete, Maximal regularity of delay equations in Banach spaces, Studia Mathematica 175 (2006), 91–102.C. Lizama and R. Ponce, Periodic solutions of degenerate differential equations in vector valued function spaces, Studia Mathematica 202 (2011), 49–63.C. Lizama and R. Ponce, Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces, Proceedings of the Edinburgh Mathematical Society 56 (2013), 853–871.R. Marchand, T. Mcdevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in highintensity ultrasound: structural decomposition, spectral analysis, exponential stability, Mathematical Methods in the Applied Sciences 35 (2012), 1896–1929.V. Poblete, Solutions of second-order integro-differential equations on periodic Besov spaces, Proceedings of the Edinburgh Mathematical Society 50 (2007), 477–492.V. Poblete and J. C. Pozo, Periodic solutions of an abstract third-order differential equation, Studia Mathematica 215 (2013), 195–219.J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87, Birkhäuser, Heidelberg, 1993.G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev type Equations and Degenerate Semigroups of Operators, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2003.L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Mathematische Annalen 319 (2001), 735–758
Advanced Methods in Black-Hole Perturbation Theory
Black-hole perturbation theory is a useful tool to investigate issues in
astrophysics, high-energy physics, and fundamental problems in gravity. It is
often complementary to fully-fledged nonlinear evolutions and instrumental to
interpret some results of numerical simulations. Several modern applications
require advanced tools to investigate the linear dynamics of generic small
perturbations around stationary black holes. Here, we present an overview of
these applications and introduce extensions of the standard semianalytical
methods to construct and solve the linearized field equations in curved
spacetime. Current state-of-the-art techniques are pedagogically explained and
exciting open problems are presented.Comment: Lecture notes from the NRHEP spring school held at IST-Lisbon, March
2013. Extra material and notebooks available online at
http://blackholes.ist.utl.pt/nrhep2/. To be published by IJMPA (V. Cardoso,
L. Gualtieri, C. Herdeiro and U. Sperhake, Eds., 2013); v2: references
updated, published versio
Perturbations of Pulsating Strings
We discuss semiclassical quantization of circular pulsating strings in AdS3 x
S3 background with and without the Neveu-Schwarz- Neveu-Schwarz (NS-NS) flux.
We find the equations of motion corresponding to the quadratic action in
bosonic sector in terms of scalar quantities and invariants of the geometry.
The general equations for studying physical perturbations along the string in
an arbitrary curved spacetime are written down using covariant formalism. We
discuss the stability of these string configurations by studying the solutions
of the linearized perturbed equations of motion.Comment: 18 pages, to appear in EPJ
Alternative analysis to perturbation theory
We develop an alternative approach to time independent perturbation theory in
non-relativistic quantum mechanics. The method developed has the advantage to
provide in one operation the correction to the energy and to the wave function,
additionally we can analyze the time evolution of the system. To verify our
results, we apply our method to the harmonic oscillator perturbed by a
quadratic potential. An alternative form of the Dyson series, in matrix form
instead of integral form, is also obtained.Comment: 12 pages, no figure
Periodic Travelling Waves in Dimer Granular Chains
We study bifurcations of periodic travelling waves in granular dimer chains
from the anti-continuum limit, when the mass ratio between the light and heavy
beads is zero. We show that every limiting periodic wave is uniquely continued
with respect to the mass ratio parameter and the periodic waves with the
wavelength larger than a certain critical value are spectrally stable.
Numerical computations are developed to study how this solution family is
continued to the limit of equal mass ratio between the beads, where periodic
travelling waves of granular monomer chains exist
A numerical canonical transformation approach to quantum many body problems
We present a new approach for numerical solutions of ab initio quantum
chemistry systems. The main idea of the approach, which we call canonical
diagonalization, is to diagonalize directly the second quantized Hamiltonian by
a sequence of numerical canonical transformations.Comment: 10 pages, 3 encapsulated figures. Parts of the paper are
substantially revised to refer to previous similar method
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