Frequently hypercyclic translation semigroups

Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is investigated. The results are achieved by establishing an analogy between frequent hypercyclicity for the translation semigroup and for weighted pseudo-shifts and by characterizing frequent hypercyclic weighted pseudo-shifts in spaces of vanishing sequences. Frequent hypercylic translation semigroups in weighted $L^p$-spaces are also characterized

Comportamiento caótico en compactos invariantes para operadores lineales

Se estudia el comportamiento caótico en compactos invariantes para operadores lineales. Se dan algunos resultados que permiten obtener este tipo de operadores y se proporcionan algunos ejemplos concretos. Se recopilan resultados previos para subespacios, que aquí se adaptan a compactos invariantes y absolutamente convexos.Murillo Arcila, M. (2011). Comportamiento caótico en compactos invariantes para operadores lineales. http://hdl.handle.net/10251/15597Archivo delegad

Maximal lp-regularity for discrete time Volterra equations with delay

In this paper, we investigate the existence and uniqueness of solutions belonging to the vector-valued space ℓp(Z,X) by using Blunck's theorem on the equivalence between operator-valued ℓp-multipliers and the notion of R-boundedness for the discrete time Volterra equation with delay given by u(n)=∑nj=−∞b(n−j)Au(j)+∑kj=1βju(n−τj)+f(n),n∈Z, where A is a closed linear operator with domain D(A) defined on a Banach space X, and b∈ℓ1(Z) verifies suitable conditions such as 1-regularity. We characterize maximal ℓp-regularity of solutions of such problems in terms of the data and an spectral condition, and we provide optimal estimates. Moreover, we illustrate our results providing different models that label into our general scheme such as the discrete time wave and Kuznetsov equations

Well posedness for semidiscrete fractional Cauchy problems with finite delay

We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay ∆ α u ( n ) = Tu ( n ) + β u ( n − τ ) + f ( n ) , n ∈ N , 0 < α ≤ 1 , β ∈ R , τ ∈ N 0 , (0.1) where is a bounded linear operator defined on a Banach space (typically a space of functions like ) and corresponds to the time discretization of the continuous Riemann–Liouville fractional derivative by means of the Poisson distribution. We characterize the existence and uniqueness of solutions in vector-valued Lebesgue spaces of sequences of the model (0.1) in terms of boundedness of the operator-valued symbol (( z − 1) α z 1 − α I − β z − τ − T ) − 1 , | z |= 1 , z ̸= 1 , whenever and satisfies a geometrical condition. For this purpose, we use methods from operator-valued Fourier multipliers and resolvent operator families associated to the homogeneous problem. We apply this result to show a practical and computational criterion in the context of Hilbert spaces

Maximal regularity in l(p) spaces for discrete time fractional shifted equations

In this paper, we are presenting a new method based on operator-valued Fourier multipliers to characterize the existence and uniqueness of p -solutions for discrete time fractional models in the form α u(n, x) = Au(n, x) + k j = 1 β j u(n − τ j ,x) + f (n, u(n, x)), n ∈ Z ,x ∈ ⊂ R N ,β j ∈ R and τ j ∈ Z , where A is a closed linear operator defined on a Banach space X and α denotes the Grünwald–Letnikov fractional derivative of order α> 0. If X is a UMD space, we provide this characterization only in terms of the R -boundedness of the operator-valued symbol associated to the abstract model. To illustrate our results, we derive new qualitative properties of nonlinear difference equations with shiftings, including fractional versions of the logistic and Nagumo equations

L-p-L-q-Well Posedness for the Moore-Gibson-Thompson Equation with Two Temperatures on Cylindrical Domains

[EN] We examine the Cauchy problem for a model of linear acoustics, called the Moore-Gibson-Thompson equation, describing a sound propagation in thermo-viscous elastic media with two temperatures on cylindrical domains. For an adequate combination of the parameters of the model we prove L-p-L-q-well-posedness, and we provide maximal regularity estimates which are optimal thanks to the theory of operator-valued Fourier multipliers.C.L. is partially supported by FONDECYT grant number 1180041; M.M.-A. is supported by MEC, grants MTM2016-75963-P and PID2019-105011GB-I00.Lizama, C.; Murillo Arcila, M. (2020). L-p-L-q-Well Posedness for the Moore-Gibson-Thompson Equation with Two Temperatures on Cylindrical Domains. Mathematics. 8(10):1-9. https://doi.org/10.3390/math8101748S19810Quintanilla, R. (2020). Moore-Gibson-Thompson thermoelasticity with two temperatures. Applications in Engineering Science, 1, 100006. doi:10.1016/j.apples.2020.100006Chen, P. J., & Gurtin, M. E. (1968). On a theory of heat conduction involving two temperatures. Zeitschrift für angewandte Mathematik und Physik ZAMP, 19(4), 614-627. doi:10.1007/bf01594969Chen, P. J., & Williams, W. O. (1968). A note on non-simple heat conduction. Zeitschrift für angewandte Mathematik und Physik ZAMP, 19(6), 969-970. doi:10.1007/bf01602278Chen, P. J., Gurtin, M. E., & Williams, W. O. (1969). On the thermodynamics of non-simple elastic materials with two temperatures. Zeitschrift für angewandte Mathematik und Physik ZAMP, 20(1), 107-112. doi:10.1007/bf01591120Quintanilla, R. (2004). On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mechanica, 168(1-2), 61-73. doi:10.1007/s00707-004-0073-6Youssef, H. M. (2006). Theory of two-temperature-generalized thermoelasticity. IMA Journal of Applied Mathematics, 71(3), 383-390. doi:10.1093/imamat/hxh101Magaña, A., & Quintanilla, R. (2008). Uniqueness and Growth of Solutions in Two-Temperature Generalized Thermoelastic Theories. Mathematics and Mechanics of Solids, 14(7), 622-634. doi:10.1177/1081286507087653Quintanilla, R. (2019). Moore–Gibson–Thompson thermoelasticity. Mathematics and Mechanics of Solids, 24(12), 4020-4031. doi:10.1177/1081286519862007Bazarra, N., Fernández, J. R., & Quintanilla, R. (2021). Analysis of a Moore–Gibson–Thompson thermoelastic problem. Journal of Computational and Applied Mathematics, 382, 113058. doi:10.1016/j.cam.2020.113058Pellicer, M., & Quintanilla, R. (2020). On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation. Zeitschrift für angewandte Mathematik und Physik, 71(3). doi:10.1007/s00033-020-01307-7Denk, R., & Nau, T. (2013). Discrete Fourier multipliers and cylindrical boundary-value problems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143(6), 1163-1183. doi:10.1017/s0308210511001454Nau, T. (2013). The Laplacian on Cylindrical Domains. Integral Equations and Operator Theory, 75(3), 409-431. doi:10.1007/s00020-012-2031-3Arendt, W., & Bu, S. (2002). The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Mathematische Zeitschrift, 240(2), 311-343. doi:10.1007/s002090100384Keyantuo, V., & Lizama, C. (2004). Fourier Multipliers and Integro‐Differential Equations in Banach Spaces. Journal of the London Mathematical Society, 69(3), 737-750. doi:10.1112/s0024610704005198Keyantuo, V., & Lizama, C. (2006). Periodic solutions of second order differential equations in Banach spaces. Mathematische Zeitschrift, 253(3), 489-514. doi:10.1007/s00209-005-0919-1Cai, G., & Bu, S. (2016). Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces. Israel Journal of Mathematics, 212(1), 163-188. doi:10.1007/s11856-016-1282-0Conejero, J. A., Lizama, C., Murillo-Arcila, M., & Seoane-Sepúlveda, J. B. (2018). Well-posedness for degenerate third order equations with delay and applications to inverse problems. Israel Journal of Mathematics, 229(1), 219-254. doi:10.1007/s11856-018-1796-8Guidotti, P. (2004). Elliptic and parabolic problems in unbounded domains. Mathematische Nachrichten, 272(1), 32-45. doi:10.1002/mana.200310187Desch, W., Hieber, M., & Prüss, J. (2001). $L^p$-Theory of the Stokes equation in a half space. Journal of Evolution Equations, 1(1), 115-142. doi:10.1007/pl00001362Bezerra, F. D. M., & Santos, L. A. (2020). Fractional powers approach of operators for abstract evolution equations of third order in time. Journal of Differential Equations, 269(7), 5661-5679. doi:10.1016/j.jde.2020.04.020Conti, M., Pata, V., Pellicer, M., & Quintanilla, R. (2020). On the analyticity of the MGT-viscoelastic plate with heat conduction. Journal of Differential Equations, 269(10), 7862-7880. doi:10.1016/j.jde.2020.05.043Denk, R., Hieber, M., & Prüss, J. (2003). ℛ-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Memoirs of the American Mathematical Society, 166(788), 0-0. doi:10.1090/memo/0788Keyantuo, V., & Lizama, C. (2011). A characterization of periodic solutions for time-fractional differential equations in UMD spaces and applications. Mathematische Nachrichten, 284(4), 494-506. doi:10.1002/mana.200810158Kalton, N. J., & Weis, L. (2001). The $H^{\infty}-$ calculus and sums of closed operators. Mathematische Annalen, 321(2), 319-345. doi:10.1007/s002080100231Wood, I. (2006). Maximal L p -regularity for the Laplacian on Lipschitz domains. Mathematische Zeitschrift, 255(4), 855-875. doi:10.1007/s00209-006-0055-6Norris, A. N. (2006). Dynamics of thermoelastic Thin Plates: A Comparison of Four Theories. Journal of Thermal Stresses, 29(2), 169-195. doi:10.1080/0149573050025748

L-p - L-q-Maximal regularity of the Van Wijngaarden-Eringen equation in a cylindrical domain

[EN] We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden-Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden-Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.The first author is partially supported by FONDECYT grant number 1180041 and DICYT, Universidad de Santiago de Chile, USACH. The second author is supported by MEC, grants MTM2016-75963-P and PID2019-105011GB-I00.Lizama, C.; Murillo Arcila, M. (2020). L-p - L-q-Maximal regularity of the Van Wijngaarden-Eringen equation in a cylindrical domain. Advances in Difference Equations. 2020(1):1-10. https://doi.org/10.1186/s13662-020-03054-5S11020201Anufrieva, U.A.: A degenerate Cauchy problem for a second-order equation. A well-posedness criterion. Differ. Uravn. 34(8), 1131–1133 (1998) (Russian). Translation in: Differ. Equ. 34(8), 1135–1137 (1999)Arendt, W., Bu, S.: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240(2), 311–343 (2002)Bu, S., Cai, G.: Periodic solutions of second order degenerate differential equations with delay in Banach spaces. Can. Math. Bull. 61(4), 717–737 (2018)Carroll, R.W., Showalter, R.E.: Singular and Degenerate Cauchy Problems. Academic Press, New York (1976)Chipot, M.: ℓ Goes to Plus Infinity. Birkhaüser Advanced Texts: Basler Lehrbücher. Birkhäuser Advanced Texts: Basel Textbooks. Birkhäuser, Basel (2002)Chipot, M.: Elliptic Equations: An Introductory Course. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Advanced Texts: Basel Textbooks. Birkhäuser, Basel (2009)Conejero, A., Lizama, C., Murillo, M.: On the existence of chaos for the viscous Van Wijngaarden–Eringen equation. Chaos Solitons Fractals 89, 100–104 (2016)Denk, R., Hieber, M., Prüss, J.: R-Boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166, 788 (2003)Denk, R., Nau, T.: Discrete Fourier multipliers and cylindrical boundary-value problems. Proc. R. Soc. Edinb., Sect. A 143(6), 1163–1183 (2013)Eringen, A.C.: Theory of thermo-microstretch fluids and bubbly liquids. Int. J. Eng. Sci. 28(2), 133–143 (1990)Favini, A., Yagi, A.: Degenerate Differential Equations in Banach Spaces. Chapman and Hall/CRC Pure and Applied Mathematics, New York (1998)Guidotti, P.: Elliptic and parabolic problems in unbounded domains. Math. Nachr. 272, 32–45 (2004)Hayes, M.A., Saccomandi, G.: Finite amplitude transverse waves in special incompressible viscoelastic solids. J. Elast. 59, 213–225 (2000)Jordan, P.M., Feuillade, C.: On the propagation of harmonic acoustic waves in bubbly liquids. Int. J. Eng. Sci. 42(11–12), 1119–1128 (2004)Kalton, N., Weis, L.: The $\mathcal{H}^{\infty }$-calculus and sums of closed operators. Math. Ann. 321, 319–345 (2001)Keyantuo, V., Lizama, C.: Fourier multipliers and integro-differential equations in Banach spaces. J. Lond. Math. Soc. (2) 69(3), 737–750 (2004)Keyantuo, V., Lizama, C.: Periodic solutions of second order differential equations in Banach spaces. Math. Z. 253(3), 489–514 (2006)Kostic, M.: Abstract Degenerate Volterra Integro-Differential Equations. Mathematical Institute SANU, Belgrade (2020)Nau, T.: $L^{p}$-theory of cylindrical boundary value problems. An operator-valued Fourier multiplier and functional calculus approach. Dissertation, University of Konstanz, Konstanz (2012). Springer Spektrum, Wiesbaden (2012)Nau, T.: The Laplacian on cylindrical domains. Integral Equ. Oper. Theory 75, 409–431 (2013)Nau, T., Saal, J.: $\mathcal{R}$-Sectoriality of cylindrical boundary value problems. In: Parabolic Problems. Progr. Nonlinear Differential Equations Appl., vol. 80, pp. 479–505. Birkhäuser, Basel (2011)Nau, T., Saal, J.: Jürgen $\mathcal{H}^{\infty }$-calculus for cylindrical boundary value problems. Adv. Differ. Equ. 17(7–8), 767–800 (2012)Rubin, M.B., Rosenau, P., Gottlieb, O.: Continuum model of dispersion caused by an inherent material characteristic length. J. Appl. Phys. 77, 4054–4063 (1995)Sviridyuk, G.A., Fedorov, V.E.: Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Inverse and Ill-Posed Problems, vol. 42. VSP, Utrecht (2003)Thompson, P.A.: Compressible-Fluid Mechanics. McGraw-Hill, New York (1992)Wijngaarden, L.V.: One-dimensional flow of liquids containing small gas bubbles. Annu. Rev. Fluid Mech. 4, 369–396 (1972)Wood, I.: Maximal $L_{p}$-regularity for the Laplacian on Lipschitz domains. Math. Z. 255(4), 855–875 (2007

Maximal l(p)-regularity of multiterm fractional equations with delay

[EN] We provide a characterization for the existence and uniqueness of solutions in the space of vector-valued sequences l(p) (Z, X)for the multiterm fractional delayed model in the form Delta(alpha)u(n) + lambda Delta(beta)u(n) = Lambda u(n) + u(n-tau) + f(n), n is an element of Z, alpha, beta is an element of R+, tau is an element of Z, lambda is an element of R, where X is a Banach space, A is a closed linear operator with domain D(A) defined on X, f is an element of l(p)(Z,X) and Delta(Gamma) denotes the Grunwald-Letkinov fractional derivative of order Gamma > 0. We also give some conditions to ensure the existence of solutions when adding nonlinearities. Finally, we illustrate our results with an example given by a general abstract nonlinear model that includes the fractional Fisher equation with delay.The second author was supported by MEC, MTM2016-75963-P and PID2019-105011GB-I00 and GVA/2018/110.Girona, I.; Murillo Arcila, M. (2021). Maximal l(p)-regularity of multiterm fractional equations with delay. Mathematical Methods in the Applied Sciences. 44(1):853-864. https://doi.org/10.1002/mma.6792S85386444