American Institute of Mathematical Sciences

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

Small data solutions for semilinear wave equations with effective damping

Marcello D'Abbicco 1,

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 $b(t)u_t - \Delta u = f(t,u)$.
Keywords: global existence., effective damping, Semilinear waves, decay estimates, small data.
Mathematics Subject Classification: 35L7.
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.  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), 315-332, doi:10.1016/j.jmaa.2012.10.017.  Google Scholar

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M. D'Abbicco, S. Lucente, A modified test function method for damped wave equations Adv. Nonlinear Studies 13 (2013), 867-892. Google Scholar

[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.  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), 109-124.  Google Scholar

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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.  Google Scholar

[7]

R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbbR^N$, J. Math. Anal. Appl., 269 (2002), 87-97.  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), 1189-1208.  Google Scholar

[9]

R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential, Funkcial. Ekvac. 52 (2009), 411-435.  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), 4307-4320, http://www.aimsciences.org/journals/pdfs.jsp?paperID=7562&mode=full  Google Scholar

[11]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS. 12 (1976), 169-189.  Google Scholar

[12]

G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. of Differential Equations 174 (2001), 464-489.  Google Scholar

[13]

J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations 232 (2007), 74-103.  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), 109-114.  Google Scholar

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.  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), 315-332, doi:10.1016/j.jmaa.2012.10.017.  Google Scholar

[3]

M. D'Abbicco, S. Lucente, A modified test function method for damped wave equations Adv. Nonlinear Studies 13 (2013), 867-892. Google Scholar

[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.  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), 109-124.  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), 365-373.  Google Scholar

[7]

R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbbR^N$, J. Math. Anal. Appl., 269 (2002), 87-97.  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), 1189-1208.  Google Scholar

[9]

R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential, Funkcial. Ekvac. 52 (2009), 411-435.  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), 4307-4320, http://www.aimsciences.org/journals/pdfs.jsp?paperID=7562&mode=full  Google Scholar

[11]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS. 12 (1976), 169-189.  Google Scholar

[12]

G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. of Differential Equations 174 (2001), 464-489.  Google Scholar

[13]

J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations 232 (2007), 74-103.  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), 109-114.  Google Scholar

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