Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-12T23:58:08.822Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  19 May 2023

Arun Kumar Bhardwaj ,
Vishvesh Kumar  and
Shyam Swarup Mondal
Arun Kumar Bhardwaj
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, India (arunkrbhardwaj@gmail.com)
Vishvesh Kumar
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, B 9000 Ghent, Belgium (vishveshmishra@gmail.com)
Shyam Swarup Mondal
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Delhi 110 016, India (mondalshyam055@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More precisely, we investigate some $L^2$-estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$ utilizing the group Fourier transform on $G$. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\,\cdot )\|_{L^2(G)}$ by further assuming the $L^1(G)$-regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$

Keywords

nonlinear wave equationviscoelastic dampingL2-L2-estimatelocal well-posednesscompact Lie groups35L05

MSC classification

Primary: 35L15: Initial value problems for second-order hyperbolic equations 35L05: Wave equation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 20
Check for updates
Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chen, W.. Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms. Nonlinear Anal. 202 (2021), 112160.CrossRefGoogle Scholar
Chill, R. and Haraux, A.. An optimal estimate for the difference of solutions of two abstract evolution equations. J. Differ. Eq. 193 (2003), 385–395.CrossRefGoogle Scholar
da Luz, C. R., Ikehata, R. and Charao, R. C.. Asymptotic behavior for abstract evolution differential equations of second order. J. Differ. Eq. 259 (2015), 5017–5039.CrossRefGoogle Scholar
D'Abbicco, M. and Ebert, M. R.. Diffusion phenomena for the wave equation with structural damping in the $l^p-l^q$ framework. J. Differ. Eq. 256 (2014), 2307–2336.CrossRefGoogle Scholar
Dasgupta, A., Kumar, V. and Mondal, S. S.. Nonlinear fractional wave equation on compact Lie groups. preprint arXiv:2207.04422 (2022).Google Scholar
Fischer, V. and Ruzhansky, M.. Quantization on nilpotent Lie groups. Progress in Mathematics, vol. 314. (Birkhäuser/Springer [Cham], Springer Nature, 2016).CrossRefGoogle Scholar
Garetto, C. and Ruzhansky, M.. Wave equation for sums of squares on compact Lie groups. J. Differ. Eq. 258 (2015), 4324–4347.CrossRefGoogle Scholar
Hosono, T.. Asymptotic behavior of solutions for nonlinear partial differential equations with dissipation. PhD thesis, Doctoral Thesis, Kyushu University, 2006.Google Scholar
Ikehata, R.. New decay estimates for linear damped wave equations and its application to nonlinear problem. Math. Methods Appl. Sci. 27 (2004), 865–889.CrossRefGoogle Scholar
Ikehata, R.. Asymptotic profiles for wave equations with strong damping. J. Differ. Eq. 257 (2014), 2159–2177.CrossRefGoogle Scholar
Ikehata, R.. Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-d half space. J. Math. Anal. Appl. 421 (2015), 905–916.CrossRefGoogle Scholar
Ikehata, R., Miyaoka, Y. and Nakatake, T.. Decay estimates of solutions for dissipative wave equations in $\mathbb {R}^n$ with lower power nonlinearities. J. Math. Soc. Japan 56 (2004), 365–373.CrossRefGoogle Scholar
Ikehata, R. and Sawada, A.. Asymptotic profile of solutions for wave equations with frictional and viscoelastic damping terms. Asymptot. Anal. 98 (2016), 59–77.Google Scholar
Ikehata, R. and Takeda, H.. Critical exponent for nonlinear wave equations with frictional and viscoelastic damping terms. Nonlinear Anal. 148 (2017), 228–253.CrossRefGoogle Scholar
Ikehata, R., Todorova, G. and Yordanov, B.. Wave equations with strong damping in Hilbert spaces. J. Differ. Eq. 254 (2013), 3352–3368.CrossRefGoogle Scholar
Liu, Y., Li, Y. and Shi, J.. Estimates for the linear viscoelastic damped wave equation on the Heisenberg group. J. Differ. Eq. 285 (2021), 663–685.CrossRefGoogle Scholar
Lu, X. and Reissig, M.. Rates of decay for structural damped models with decreasing in time coefficients. Int. J. Dyn. Syst. Differ. Equ. 2 (2009), 21–55.Google Scholar
Matsumura, A.. On the asymptotic behavior of solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12 (1976), 169–189.CrossRefGoogle Scholar
Nachman, A. I.. The wave equation on the Heisenberg group. Comm. Partial Differ. Eq. 7 (1982), 675–714.CrossRefGoogle Scholar
Narazaki, T.. $l^p-l^q$ estimates for damped wave equations and their applications to semi-linear problem. J. Math. Soc. Japan 56 (2004), 585–626.CrossRefGoogle Scholar
Palmieri, A.. On the blow–up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups. J. Differ. Eq. 281 (2021), 85–104.CrossRefGoogle Scholar
Palmieri, A.. Semilinear wave equation on compact Lie groups. J. Pseudo-Differ. Oper. Appl. 12 (2021), 43.CrossRefGoogle Scholar
Palmieri, A.. A global existence result for a semilinear wave equation with lower order terms on compact Lie groups. J. Fourier Anal. Appl. 28 (2022), 21.CrossRefGoogle Scholar
Ponce, G.. Global existence of small solutions to a class of nonlinear evolution equations. Nonlinear Anal. 9 (1985), 399–418.CrossRefGoogle Scholar
Ruzhansky, M. and Taranto, C. A.. Time-dependent wave equations on graded groups. Acta Appl. Math. 171 (2021), 21.CrossRefGoogle Scholar
Ruzhansky, M. and Tokmagambetov, N.. Nonlinear damped wave equations for the sub-laplacian on the Heisenberg group and for Rockland operators on graded Lie groups. J. Differ. Eq. 265 (2018), 5212–5236.CrossRefGoogle Scholar
Ruzhansky, M. and Turunen, V.. Pseudo-differential operators and symmetries: background analysis and advanced topics. Pseudo-Differential Operators. Theory and Applications, vol. 2. (Birkhäuser Verlag, Springer Science & Business Media, 2009).CrossRefGoogle Scholar
Ruzhansky, M. and Turunen, V.. Global quantization of pseudo-differential operators on compact Lie groups, su (2), 3-sphere, and homogeneous spaces. Int. Math. Res. Not. IMRN 2013 (2013), 2439–2496.CrossRefGoogle Scholar
Ruzhansky, M. and Yessirkegenov, N.. Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups. preprint arXiv:1810.08845 (2018).Google Scholar
Ruzhansky, M. and Yessirkegenov, N.. Very weak solutions to hypoelliptic wave equations. J. Differ. Eq. 268 (2020), 2063–2088.CrossRefGoogle Scholar
Shibata, Y.. On the rate of decay of solutions to linear viscoelastic equation. Math. Methods Appl. Sci. 23 (2000), 203–226.3.0.CO;2-M>CrossRefGoogle Scholar
Taranto, C. A.. Wave equations on graded groups and hypoelliptic Gevrey spaces. preprint arXiv:1804.03544 (2018).Google Scholar
Todorova, G. and Yordanov, B.. Critical exponent for a nonlinear wave equation with damping. J. Differ. Eq. 174 (2001), 464–489.CrossRefGoogle Scholar