# American Institute of Mathematical Sciences

2013, 2013(special): 183-191. doi: 10.3934/proc.2013.2013.183

## Small data solutions for semilinear wave equations with effective damping

 1 Department of Mathematics, University of Bari, Bari, 70124, Italy

Received  September 2012 Published  November 2013

We consider the Cauchy problem for the semi-linear damped wave equation
$u_{tt} - \Delta u + b(t)u_t = f(t,u),\qquad u(0,x) = u_0(x),\qquad u_t(0,x) = u_1(x).$
We prove the global existence of small data solution in low space dimension, and we derive $(L^m\cap L^2)-L^2$ decay estimates, for $m\in[1,2)$. We assume that the time-dependent damping term $b(t)>0$ is effective, that is, the equation inherits some properties of the parabolic equation $b(t)u_t - \Delta u = f(t,u)$.
Citation: Marcello D'Abbicco. Small data solutions for semilinear wave equations with effective damping. Conference Publications, 2013, 2013 (special) : 183-191. doi: 10.3934/proc.2013.2013.183
##### References:
 [1] M. D'Abbicco, M.R. Ebert, Hyperbolic-like estimates for higher order equations,, J. Math. Anal. Appl. 395 (2012), 395 (2012), 747.   Google Scholar [2] M. D'Abbicco, M.R. Ebert, A class of dissipative wave equations with time-dependent speed and damping, J. Math. Anal. Appl. 399 (2013), 399 (2013), 315.   Google Scholar [3] M. D'Abbicco, S. Lucente, A modified test function method for damped wave equations, Adv. Nonlinear Studies 13 (2013), 13 (2013), 867.   Google Scholar [4] M. D'Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping,, Chinese Ann. Math. 34B (2013), 34B (2013), 345.   Google Scholar [5] H. Fujita, On the blowing up of solutions of the Cauchy Problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac.Sci. Univ. Tokyo 13 (1966), 13 (1966), 109.   Google Scholar [6] R. Ikehata, Y. Mayaoka, T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbbR^N$ with lower power nonlinearities,, J. Math. Soc. Japan, 56 (2004), 365.   Google Scholar [7] R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbbR^N$,, J. Math. Anal. Appl., 269 (2002), 87.   Google Scholar [8] 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), 61 (2005), 1189.   Google Scholar [9] R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential,, Funkcial. Ekvac. 52 (2009), 52 (2009), 411.   Google Scholar [10] J. Lin, K. Nishihara, J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping,, Discrete and Continuous Dynamical Systems, 32 (2012), 4307.   Google Scholar [11] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. RIMS. 12 (1976), 12 (1976), 169.   Google Scholar [12] G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. of Differential Equations 174 (2001), 174 (2001), 464.   Google Scholar [13] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation,, J. Differential Equations 232 (2007), 232 (2007), 74.   Google Scholar [14] Qi 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), 333 (2001), 109.   Google Scholar

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##### References:
 [1] M. D'Abbicco, M.R. Ebert, Hyperbolic-like estimates for higher order equations,, J. Math. Anal. Appl. 395 (2012), 395 (2012), 747.   Google Scholar [2] M. D'Abbicco, M.R. Ebert, A class of dissipative wave equations with time-dependent speed and damping, J. Math. Anal. Appl. 399 (2013), 399 (2013), 315.   Google Scholar [3] M. D'Abbicco, S. Lucente, A modified test function method for damped wave equations, Adv. Nonlinear Studies 13 (2013), 13 (2013), 867.   Google Scholar [4] M. D'Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping,, Chinese Ann. Math. 34B (2013), 34B (2013), 345.   Google Scholar [5] H. Fujita, On the blowing up of solutions of the Cauchy Problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac.Sci. Univ. Tokyo 13 (1966), 13 (1966), 109.   Google Scholar [6] R. Ikehata, Y. Mayaoka, T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbbR^N$ with lower power nonlinearities,, J. Math. Soc. Japan, 56 (2004), 365.   Google Scholar [7] R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbbR^N$,, J. Math. Anal. Appl., 269 (2002), 87.   Google Scholar [8] 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), 61 (2005), 1189.   Google Scholar [9] R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential,, Funkcial. Ekvac. 52 (2009), 52 (2009), 411.   Google Scholar [10] J. Lin, K. Nishihara, J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping,, Discrete and Continuous Dynamical Systems, 32 (2012), 4307.   Google Scholar [11] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. RIMS. 12 (1976), 12 (1976), 169.   Google Scholar [12] G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. of Differential Equations 174 (2001), 174 (2001), 464.   Google Scholar [13] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation,, J. Differential Equations 232 (2007), 232 (2007), 74.   Google Scholar [14] Qi 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), 333 (2001), 109.   Google Scholar
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