On asymptotic behavior of solution to a nonlinear wave equation with Space-time speed of propagation and damping terms
Keywords:Space-time speed of propagation, Space-time dependent damping, Asymptotic behavior, Weighted energy method
In this paper, we consider the asymptotic behavior of solution to the nonlinear damped wave equation
utt – div(a(t, x)∇u) + b(t, x)ut = −|u|p−1u t ∈ [0, ∞), x ∈ Rn
u(0, x) = u0(x), ut(0, x) = u1(x) x ∈ Rn
with space-time speed of propagation and damping potential. We obtained L2 decay estimates via the weighted energy method and under certain suitable assumptions on the functions a(t, x) and b(t, x). The technique follows that of Lin et al. with modification to the region of consideration in Rn. These decay result extends the results in the literature.
T.B. N. Bui, "Wave models with time-dependent speed and dissipation", Ph. D. thesis, Technical University Bergakademie Freiberg, 2013.
T. B. N. Bui and M. Reissig, “The interplay between time-dependent speed of propagation and dissipation in wave models,” in Fourier analysis, M. Ruzhansky and V. Turunen, Eds. Cham: Birkhäuser, 2014, pp. 9–45.
M. D’Abbicco and M. R. Ebert, ”A class of dissipative wave equations with time-dependent speed and damping”, Journal of mathematical analysis and applications, vol. 399, no. 1, pp. 315-332, 2013.
M. R. Ebert and W. N. Nascimento, “A classiﬁcation for wave models with time-dependent mass and speed of propagation”, Advances in differential equations, vol. 23, no. 11, pp. 847-888, 2018.
T. Hosono and T. Ogawa, ”Large time behavior and Lp − Lq estimate of solutions of 2-dimensional nonlinear damped wave equations”, Journal of differential equations, vol. 203, no. 1, 82-118, 2004.
R. Ikehata, Y. Inoue, ”Total energy decay for semilinear wave equations with a critical potential type of damping”, Nonlinear analysis, vol. 69, no.1, pp. 1396-1401, 2008.
R. Ikehata, G. Todorova, and B. Yordanov, ”Wave equations with strong damping in Hilbert spaces”, Journal of differential equations, vol. 254, no. 8, pp. 3352-3368, 2013.
J. Lin, K. Nishihara, and J. Zhai, ”L2-estimates of solutions for damped wave equations with space-time dependent damping term”, Journal of differential equations, vol. 248, no. 2, pp. 403-422, 2010.
P. Marcati and K. Nishihara, ”The Lp − Lq estimates of solutions to one- dimensional damped wave equations and their application to the compressible ﬂow through porous media”, Journal of differential equations, vol. 191, no. 2, pp. 445-469, 2003.
T. Narazaki, ”Lp − Lq estimates for damped wave equations and their applications to semi-linear problem”, Journal of the Mathematical Society of Japan, vol. 56, no. 2, pp. 585-626, 2004.
K. Nishihara, ”Lp − Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application”, Mathematische zeitschrift, vol. 244, pp. 631-649, 2003.
K. Nishihara, ”Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term”, Communications in partial differential equations, vol. 35, no. 8, pp. 1402-1418, 2010.
K. Nishihara and J. Zhai, ”Asymptotic behaviors of solutions for time dependent damped wave equations”, Journal of mathematical analysis and applications, vol. 360, no. 2, pp. 412-421, 2009.
P. Radu, G. Todorova, and B. Yordanov, ”Decay estimates for wave equations with variable coeﬃcients”, Transactions of the American Mathematical Society, vol. 362, no. 5, pp. 2279-2299, 2009.
P. Radu, G. Todorova, and B. Yordanov, ”Diﬀusion phenomenon in Hilbert spaces and applications”, Journal of differential equations, vol. 250, no. 11, pp. 4200-4218, 2011.
M. Roberts, ”Decay estimates for nonlinear wave equations with variable coeﬃcients”, Electronic journal of diﬀerential equations, vol. 2019, Art. ID. 91, 2019.
B. Said-Houari, ”Asymptotic behaviors of solutions for viscoelastic wave equation with space-time dependent damping term”, Journal of mathematical analysis and applications, vol. 387, no. 2, pp. 1088-1105, 2012.
Y. Todorova, B. Yordanov, ”Critical exponent for a nonlinear wave equation with damping”, Journal of differential equations, vol. 174, no. 2 pp. 464- 489, 2001.
G. Todorova, B. Yordanov, ”Nonlinear dissipative wave equations with potential”, in Control methods in PDE-dynamical systems, F. Ancona, I. Lasiecka, W. Littman and R. Triggiani, Eds. Providence, RI: American Mathematical Society, 2007, pp. 317-337.
K. Yagdjian, ”Parametric resonance and nonexistence of the global solution to nonlinear wave equations”, Journal of mathematical analysis and applications, vol. 260, no. 1, pp. 251-268, 2001.
How to Cite
Copyright (c) 2021 Paul Ogbiyele, Peter Arawomo
This work is licensed under a Creative Commons Attribution 4.0 International License.
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.