# 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), 14 (1991), 427.   Google Scholar [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), 408 (2013), 247.   Google Scholar [3] A. Córdoba and D. Córdoba, A Maximum Principle Applied to Quasi-Geostrophic Equations,, Commun. Math. Phys. 249 (2004), 249 (2004), 511.   Google Scholar [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), 431 (2015), 569.   Google Scholar [5] M. D'Abbicco, The influence of a nonlinear memory on the damped wave equation,, Nonlinear Analysis, 95 (2014), 130.   Google Scholar [6] M. D'Abbicco, A wave equation with structural damping and nonlinear memory,, Nonlinear Differential Equations and Applications, 21 (2014), 751.   Google Scholar [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, (2015), 209.   Google Scholar [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.   Google Scholar [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.   Google Scholar [10] M. D'Abbicco and M. Reissig, Semilinear structural damped waves,, Math. Methods in Appl. Sc., 37 (2014), 1570.   Google Scholar [11] R. Ikehata and M. Natsume, Energy Decay Estimates for Wave Equations with a Fractional Damping,, Differential and Integral Equations, 25 (2012), 9.   Google Scholar [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.   Google Scholar [13] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations,, Commun. Pure Appl. Math., 33 (1980), 501.   Google Scholar [14] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. RIMS., 12 (1976), 169.   Google Scholar [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, (2013), 215.   Google Scholar [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 (2012), 331.   Google Scholar [17] K. Nishihara and Y. Wakasugi, Critical exponent for the Cauchy problem to the weakly coupled damped wave systems,, Nonlinear Analysis, 108 (2014), 249.   Google Scholar [18] G. Todorova and B. Yordanov, Critical Exponent for a Nonlinear Wave Equation with Damping,, Journal of Differential Equations, 174 (2001), 464.   Google Scholar [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.   Google Scholar

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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), 14 (1991), 427.   Google Scholar [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), 408 (2013), 247.   Google Scholar [3] A. Córdoba and D. Córdoba, A Maximum Principle Applied to Quasi-Geostrophic Equations,, Commun. Math. Phys. 249 (2004), 249 (2004), 511.   Google Scholar [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), 431 (2015), 569.   Google Scholar [5] M. D'Abbicco, The influence of a nonlinear memory on the damped wave equation,, Nonlinear Analysis, 95 (2014), 130.   Google Scholar [6] M. D'Abbicco, A wave equation with structural damping and nonlinear memory,, Nonlinear Differential Equations and Applications, 21 (2014), 751.   Google Scholar [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, (2015), 209.   Google Scholar [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.   Google Scholar [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.   Google Scholar [10] M. D'Abbicco and M. Reissig, Semilinear structural damped waves,, Math. Methods in Appl. Sc., 37 (2014), 1570.   Google Scholar [11] R. Ikehata and M. Natsume, Energy Decay Estimates for Wave Equations with a Fractional Damping,, Differential and Integral Equations, 25 (2012), 9.   Google Scholar [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.   Google Scholar [13] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations,, Commun. Pure Appl. Math., 33 (1980), 501.   Google Scholar [14] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. RIMS., 12 (1976), 169.   Google Scholar [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, (2013), 215.   Google Scholar [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 (2012), 331.   Google Scholar [17] K. Nishihara and Y. Wakasugi, Critical exponent for the Cauchy problem to the weakly coupled damped wave systems,, Nonlinear Analysis, 108 (2014), 249.   Google Scholar [18] G. Todorova and B. Yordanov, Critical Exponent for a Nonlinear Wave Equation with Damping,, Journal of Differential Equations, 174 (2001), 464.   Google Scholar [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.   Google Scholar
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