American Institute of Mathematical Sciences

2015, 2015(special): 320-329. doi: 10.3934/proc.2015.0320

A note on a weakly coupled system of structurally damped waves

 1 Departamento de Computação e Matemática, Universidade de São Paulo (USP), FFCLRP, Av. dos Bandeirantes 3900, Ribeirão Preto, SP 14040-901

Received  September 2014 Revised  June 2015 Published  November 2015

In this note, we find the critical exponent for a system of weakly coupled structurally damped waves.
Citation: Marcello D'Abbicco. A note on a weakly coupled system of structurally damped waves. Conference Publications, 2015, 2015 (special) : 320-329. doi: 10.3934/proc.2015.0320
References:
 [1] P. Biler, Time decay of solutions of semilinear strongly damped generalized wave equations, Math. Methods Appl. Sci. 14 (1991), 6, 427-443. [2] R. C. Charão, C. R. da Luz and R. Ikehata, Sharp Decay Rates for Wave Equations with a Fractional Damping via New Method in the Fourier Space, Journal of Math. Anal. and Appl. 408 (2013), 1, 247-255. [3] A. Córdoba and D. Córdoba, A Maximum Principle Applied to Quasi-Geostrophic Equations, Commun. Math. Phys. 249 (2004), 511-528. Available from http://dx.doi.org/10.1007/s00220-004-1055-1. [4] P. T. Duong, M. Kainane and M. Reissig, Global existence for semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl. 431 (2015), 569-596. Available from http://dx.doi.org/10.1016/j.jmaa.2015.06.001. [5] M. D'Abbicco, The influence of a nonlinear memory on the damped wave equation, Nonlinear Analysis, 95 (2014), 130-145. Available from: http://dx.doi.org/10.1016/j.na.2013.09.006. [6] M. D'Abbicco, A wave equation with structural damping and nonlinear memory, Nonlinear Differential Equations and Applications, 21 5 (2014), 751-773. Available from: http://dx.doi.org/10.1007/s00030-014-0265-2. [7] M. D'Abbicco, A benefit from the $L^1$ smallness of initial data for the semilinear wave equation with structural damping, Current Trends in Analysis and its Applications, Proceedings of the $9^{th}$ ISAAC Congress, Krakow, eds V. Mityushev and M. Ruzhansky, (2015), 209-216. Available from: http://www.springer.com/br/book/9783319125763. [8] M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p-L^q$ framework, J. of Differential Equations, 256 (2014), 2307-2336. Available from: http://dx.doi.org/10.1016/j.jde.2014.01.002. [9] 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. Available from: http://dx.doi.org/10.1016/j.na.2013.12.021. [10] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods in Appl. Sc., 37 (2014), 1570-1592. Available from: http://dx.doi.org/10.1002/mma.2913. [11] R. Ikehata and M. Natsume, Energy Decay Estimates for Wave Equations with a Fractional Damping, Differential and Integral Equations, 25 (2012), no. 9-10, 939-956. [12] R. Ikehata, K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $R^N$ with noncompactly supported initial data, Nonlinear Analysis, 61 (2005), 1189-1208. [13] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Commun. Pure Appl. Math., 33 (1980), 501-505. [14] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS., 12 (1976), 169-189. [15] T. Narazaki and M. Reissig, $L^1$ estimates for oscillating integrals related to structural damped wave models, Studies in Phase Space Analysis with Applications to PDEs, eds Cicognani M, Colombini F, Del Santo D, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 2013; 215-258. [16] K. Nishihara, Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system, Osaka Journal of Mathematics, 49 2 (2012), 331-348. [17] K. Nishihara and Y. Wakasugi, Critical exponent for the Cauchy problem to the weakly coupled damped wave systems, Nonlinear Analysis, 108 (2014), 249-259. Available from: http://dx.doi.org/10.1016/j.na.2014.06.001. [18] G. Todorova and B. Yordanov, Critical Exponent for a Nonlinear Wave Equation with Damping, Journal of Differential Equations, 174 (2001), 464-489. [19] Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.

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References:
 [1] P. Biler, Time decay of solutions of semilinear strongly damped generalized wave equations, Math. Methods Appl. Sci. 14 (1991), 6, 427-443. [2] R. C. Charão, C. R. da Luz and R. Ikehata, Sharp Decay Rates for Wave Equations with a Fractional Damping via New Method in the Fourier Space, Journal of Math. Anal. and Appl. 408 (2013), 1, 247-255. [3] A. Córdoba and D. Córdoba, A Maximum Principle Applied to Quasi-Geostrophic Equations, Commun. Math. Phys. 249 (2004), 511-528. Available from http://dx.doi.org/10.1007/s00220-004-1055-1. [4] P. T. Duong, M. Kainane and M. Reissig, Global existence for semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl. 431 (2015), 569-596. Available from http://dx.doi.org/10.1016/j.jmaa.2015.06.001. [5] M. D'Abbicco, The influence of a nonlinear memory on the damped wave equation, Nonlinear Analysis, 95 (2014), 130-145. Available from: http://dx.doi.org/10.1016/j.na.2013.09.006. [6] M. D'Abbicco, A wave equation with structural damping and nonlinear memory, Nonlinear Differential Equations and Applications, 21 5 (2014), 751-773. Available from: http://dx.doi.org/10.1007/s00030-014-0265-2. [7] M. D'Abbicco, A benefit from the $L^1$ smallness of initial data for the semilinear wave equation with structural damping, Current Trends in Analysis and its Applications, Proceedings of the $9^{th}$ ISAAC Congress, Krakow, eds V. Mityushev and M. Ruzhansky, (2015), 209-216. Available from: http://www.springer.com/br/book/9783319125763. [8] M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p-L^q$ framework, J. of Differential Equations, 256 (2014), 2307-2336. Available from: http://dx.doi.org/10.1016/j.jde.2014.01.002. [9] 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. Available from: http://dx.doi.org/10.1016/j.na.2013.12.021. [10] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods in Appl. Sc., 37 (2014), 1570-1592. Available from: http://dx.doi.org/10.1002/mma.2913. [11] R. Ikehata and M. Natsume, Energy Decay Estimates for Wave Equations with a Fractional Damping, Differential and Integral Equations, 25 (2012), no. 9-10, 939-956. [12] R. Ikehata, K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $R^N$ with noncompactly supported initial data, Nonlinear Analysis, 61 (2005), 1189-1208. [13] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Commun. Pure Appl. Math., 33 (1980), 501-505. [14] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS., 12 (1976), 169-189. [15] T. Narazaki and M. Reissig, $L^1$ estimates for oscillating integrals related to structural damped wave models, Studies in Phase Space Analysis with Applications to PDEs, eds Cicognani M, Colombini F, Del Santo D, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 2013; 215-258. [16] K. Nishihara, Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system, Osaka Journal of Mathematics, 49 2 (2012), 331-348. [17] K. Nishihara and Y. Wakasugi, Critical exponent for the Cauchy problem to the weakly coupled damped wave systems, Nonlinear Analysis, 108 (2014), 249-259. Available from: http://dx.doi.org/10.1016/j.na.2014.06.001. [18] G. Todorova and B. Yordanov, Critical Exponent for a Nonlinear Wave Equation with Damping, Journal of Differential Equations, 174 (2001), 464-489. [19] Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.
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