# American Institute of Mathematical Sciences

September  2019, 39(9): 5431-5463. doi: 10.3934/dcds.2019222

## $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping

 1 School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1 Dai Co Viet road, Hanoi, Vietnam 2 Faculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Prüferstr. 9, 09596, Freiberg, Germany

* Corresponding author: Tuan Anh Dao

Received  November 2018 Revised  March 2019 Published  May 2019

Fund Project: The first author is supported by Vietnamese Government's Scholarship (Grant number: 2015/911).

The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped
 $\sigma$
-evolution models:
 $\begin{equation*} u_{tt}+ (-\Delta)^\sigma u+ \mu (-\Delta)^\delta u_t = f(u, u_t), \, \, \, u(0, x) = u_0(x), \, \, \, u_t(0, x) = u_1(x) \end{equation*}$
with
 $\sigma \ge 1$
,
 $\mu>0$
and
 $\delta \in (\frac{\sigma}{2}, \sigma]$
. Our aim is to study two main models including
 $\sigma$
-evolution models with structural damping
 $\delta \in (\frac{\sigma}{2}, \sigma)$
and those with visco-elastic damping
 $\delta = \sigma$
. Here the function
 $f(u, u_t)$
stands for power nonlinearities
 $|u|^{p}$
and
 $|u_t|^{p}$
with a given number
 $p>1$
. We are interested in investigating the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on
 $L^q$
 $L^{m}$
regularity for the initial data, with
 $q\in (1, \infty)$
and
 $m\in [1, q)$
.
Citation: Tuan Anh Dao, Michael Reissig. $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222
##### References:
 [1] M. D'Abbicco and M. R. Ebert, An application of $L^{p}-L^{q}$ decay estimates to the semilinear wave equation with parabolic-like structural damping, Nonlinear Analysis, 99 (2014), 16-34.  doi: 10.1016/j.na.2013.12.021. [2] M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Analysis, 149 (2017), 1-40.  doi: 10.1016/j.na.2016.10.010. [3] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913. [4] T. A. Dao and M. Reissig, An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 476 (2019), 426-463.  doi: 10.1016/j.jmaa.2019.03.048. [5] M. R. Ebert and M. Reissig, Methods for Partial Differential Equations, Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser, 2018. doi: 10.1007/978-3-319-66456-9. [6] Cav. Francesco Faà di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math., 1 (1857), 359-360. [7] V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-up for higher-order prabolic, hyperbolic, dispersion and Schrödinger equations, in Monogr. Res. Notes Math., Chapman and Hall/CRC, 2014. [8] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2004. [9] H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kokyuroku Bessatsu, B26, Res.Inst.Math.Sci. (RIMS), Kyoto, (2011), 159–175. [10] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations, 257 (2014), 2159-2177.  doi: 10.1016/j.jde.2014.05.031. [11] R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differential Equations, 254 (2013), 3352-3368.  doi: 10.1016/j.jde.2013.01.023. [12] M. Kainane, Structural Damped $\sigma$-evolution Operators, PhD thesis, TU Bergakademie Freiberg, Germany, 2014. [13] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math., 8 (1939), 78-91.  doi: 10.4064/sm-8-1-78-91. [14] A. Miyachi, On some Fourier multipliers for $H^p(\mathbb{R}^n)$, J. Fac. Sci. Univ. Tokyo IA, 27 (1980), 157-179. [15] E. Mitidieri and S. I. Pohozaev, Non-existence of weak solutions for some degenerate elliptic and parabolic problems on $\mathbb{R}^n$, J. Evol. Equ., 1 (2001), 189-220.  doi: 10.1007/PL00001368. [16] T. Narazaki and M. Reissig, $L^1$ estimates for oscillating integrals related to structural damped wave models, in, Progr. Nonlinear Differential Equations Appl., Studies in Phase Space Analysis with Applications to PDEs (eds. M. Cicognani, F. Colombini, D. Del Santo), Birkhäuser, 84 (2013), 215–258. doi: 10.1007/978-1-4614-6348-1_11. [17] A. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, Ⅱ, Math. Nachr., 291 (2018), 1859-1892.  doi: 10.1002/mana.201700144. [18] D. T. Pham, M. Kainane Mezadek and M. Reissig, Global existence for semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 431 (2015), 569-596.  doi: 10.1016/j.jmaa.2015.06.001. [19] F. Pizichillo, Linear and Non-Linear Damped Wave Equations, Master thesis, University of Bari, 2014. [20] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411. [21] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226.  doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. [22] C. G. Simander, On Dirichlet Boundary Value Problem, An $L^p$-Theory Based on a Generalization of Gårding's Inequality, Lecture Notes in Mathematics, 268, Springer, Berlin, 1972. [23] E. Stein and G. Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030. [24] F. Weisz, Marcinkiewicz multiplier theorem and the Sunouchi operator for Ciesielski-Fourier series, Journal of Approximation Theory, 133 (2005), 195-220.  doi: 10.1016/j.jat.2004.12.017.

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##### References:
 [1] M. D'Abbicco and M. R. Ebert, An application of $L^{p}-L^{q}$ decay estimates to the semilinear wave equation with parabolic-like structural damping, Nonlinear Analysis, 99 (2014), 16-34.  doi: 10.1016/j.na.2013.12.021. [2] M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Analysis, 149 (2017), 1-40.  doi: 10.1016/j.na.2016.10.010. [3] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913. [4] T. A. Dao and M. Reissig, An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 476 (2019), 426-463.  doi: 10.1016/j.jmaa.2019.03.048. [5] M. R. Ebert and M. Reissig, Methods for Partial Differential Equations, Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser, 2018. doi: 10.1007/978-3-319-66456-9. [6] Cav. Francesco Faà di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math., 1 (1857), 359-360. [7] V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-up for higher-order prabolic, hyperbolic, dispersion and Schrödinger equations, in Monogr. Res. Notes Math., Chapman and Hall/CRC, 2014. [8] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2004. [9] H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kokyuroku Bessatsu, B26, Res.Inst.Math.Sci. (RIMS), Kyoto, (2011), 159–175. [10] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations, 257 (2014), 2159-2177.  doi: 10.1016/j.jde.2014.05.031. [11] R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differential Equations, 254 (2013), 3352-3368.  doi: 10.1016/j.jde.2013.01.023. [12] M. Kainane, Structural Damped $\sigma$-evolution Operators, PhD thesis, TU Bergakademie Freiberg, Germany, 2014. [13] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math., 8 (1939), 78-91.  doi: 10.4064/sm-8-1-78-91. [14] A. Miyachi, On some Fourier multipliers for $H^p(\mathbb{R}^n)$, J. Fac. Sci. Univ. Tokyo IA, 27 (1980), 157-179. [15] E. Mitidieri and S. I. Pohozaev, Non-existence of weak solutions for some degenerate elliptic and parabolic problems on $\mathbb{R}^n$, J. Evol. Equ., 1 (2001), 189-220.  doi: 10.1007/PL00001368. [16] T. Narazaki and M. Reissig, $L^1$ estimates for oscillating integrals related to structural damped wave models, in, Progr. Nonlinear Differential Equations Appl., Studies in Phase Space Analysis with Applications to PDEs (eds. M. Cicognani, F. Colombini, D. Del Santo), Birkhäuser, 84 (2013), 215–258. doi: 10.1007/978-1-4614-6348-1_11. [17] A. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, Ⅱ, Math. Nachr., 291 (2018), 1859-1892.  doi: 10.1002/mana.201700144. [18] D. T. Pham, M. Kainane Mezadek and M. Reissig, Global existence for semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 431 (2015), 569-596.  doi: 10.1016/j.jmaa.2015.06.001. [19] F. Pizichillo, Linear and Non-Linear Damped Wave Equations, Master thesis, University of Bari, 2014. [20] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411. [21] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226.  doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. [22] C. G. Simander, On Dirichlet Boundary Value Problem, An $L^p$-Theory Based on a Generalization of Gårding's Inequality, Lecture Notes in Mathematics, 268, Springer, Berlin, 1972. [23] E. Stein and G. Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030. [24] F. Weisz, Marcinkiewicz multiplier theorem and the Sunouchi operator for Ciesielski-Fourier series, Journal of Approximation Theory, 133 (2005), 195-220.  doi: 10.1016/j.jat.2004.12.017.
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