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 $
spaces by assuming additional
$ 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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3]

M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Cav. Francesco Faà di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math., 1 (1857), 359-360.   Google Scholar

[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. Google Scholar

[8]

L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. IkehataG. 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.  Google Scholar

[12]

M. Kainane, Structural Damped $\sigma$-evolution Operators, PhD thesis, TU Bergakademie Freiberg, Germany, 2014. Google Scholar

[13]

J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math., 8 (1939), 78-91.  doi: 10.4064/sm-8-1-78-91.  Google Scholar

[14]

A. Miyachi, On some Fourier multipliers for $H^p(\mathbb{R}^n)$, J. Fac. Sci. Univ. Tokyo IA, 27 (1980), 157-179.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. T. PhamM. 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.  Google Scholar

[19]

F. Pizichillo, Linear and Non-Linear Damped Wave Equations, Master thesis, University of Bari, 2014. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Cav. Francesco Faà di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math., 1 (1857), 359-360.   Google Scholar

[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. Google Scholar

[8]

L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. IkehataG. 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.  Google Scholar

[12]

M. Kainane, Structural Damped $\sigma$-evolution Operators, PhD thesis, TU Bergakademie Freiberg, Germany, 2014. Google Scholar

[13]

J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math., 8 (1939), 78-91.  doi: 10.4064/sm-8-1-78-91.  Google Scholar

[14]

A. Miyachi, On some Fourier multipliers for $H^p(\mathbb{R}^n)$, J. Fac. Sci. Univ. Tokyo IA, 27 (1980), 157-179.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. T. PhamM. 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.  Google Scholar

[19]

F. Pizichillo, Linear and Non-Linear Damped Wave Equations, Master thesis, University of Bari, 2014. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104

[2]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[3]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[4]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304

[5]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[6]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[7]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[8]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[9]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[10]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[11]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[12]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[13]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[14]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[15]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[16]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[17]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[18]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[19]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[20]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (111)
  • HTML views (178)
  • Cited by (2)

Other articles
by authors

[Back to Top]