# 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), 747-765, doi:10.1016/j.jmaa.2012.05.070. [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), 315-332, doi:10.1016/j.jmaa.2012.10.017. [3] M. D'Abbicco, S. Lucente, A modified test function method for damped wave equations Adv. Nonlinear Studies 13 (2013), 867-892. [4] M. D'Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping, Chinese Ann. Math. 34B (2013), 3, 345-380, doi:10.1007/s11401-013-0773-0. [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), 109-124. [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), 365-373. [7] R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbbR^N$, J. Math. Anal. Appl., 269 (2002), 87-97. [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), 1189-1208. [9] R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential, Funkcial. Ekvac. 52 (2009), 411-435. [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), 4307-4320, http://www.aimsciences.org/journals/pdfs.jsp?paperID=7562&mode=full [11] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS. 12 (1976), 169-189. [12] G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. of Differential Equations 174 (2001), 464-489. [13] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations 232 (2007), 74-103. [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), 109-114.

show all references

##### References:
 [1] M. D'Abbicco, M.R. Ebert, Hyperbolic-like estimates for higher order equations, J. Math. Anal. Appl. 395 (2012), 747-765, doi:10.1016/j.jmaa.2012.05.070. [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), 315-332, doi:10.1016/j.jmaa.2012.10.017. [3] M. D'Abbicco, S. Lucente, A modified test function method for damped wave equations Adv. Nonlinear Studies 13 (2013), 867-892. [4] M. D'Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping, Chinese Ann. Math. 34B (2013), 3, 345-380, doi:10.1007/s11401-013-0773-0. [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), 109-124. [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), 365-373. [7] R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbbR^N$, J. Math. Anal. Appl., 269 (2002), 87-97. [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), 1189-1208. [9] R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential, Funkcial. Ekvac. 52 (2009), 411-435. [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), 4307-4320, http://www.aimsciences.org/journals/pdfs.jsp?paperID=7562&mode=full [11] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS. 12 (1976), 169-189. [12] G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. of Differential Equations 174 (2001), 464-489. [13] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations 232 (2007), 74-103. [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), 109-114.
 [1] Paschalis Karageorgis. Small-data scattering for nonlinear waves with potential and initial data of critical decay. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 87-106. doi: 10.3934/dcds.2006.16.87 [2] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [3] Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651 [4] Xianglong Duan. Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data. Kinetic and Related Models, 2022, 15 (1) : 119-146. doi: 10.3934/krm.2021049 [5] Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100 [6] Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015 [7] Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032 [8] Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 [9] Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307 [10] Marco Cappiello, Fabio Nicola. Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1869-1880. doi: 10.3934/dcds.2016.36.1869 [11] Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure and Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923 [12] Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 [13] Walter A. Strauss, Kimitoshi Tsutaya. Existence and blow up of small amplitude nonlinear waves with a negative potential. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 175-188. doi: 10.3934/dcds.1997.3.175 [14] Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 [15] Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control and Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45 [16] Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209 [17] Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025 [18] Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557 [19] Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations and Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217 [20] Andrea Cianchi, Vladimir Maz'ya. Global gradient estimates in elliptic problems under minimal data and domain regularity. Communications on Pure and Applied Analysis, 2015, 14 (1) : 285-311. doi: 10.3934/cpaa.2015.14.285

Impact Factor: