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.

[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.

[3]

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

[4]

M. D'Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping,, Chinese Ann. Math. 34B (2013), 34B (2013), 345.

[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.

[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.

[7]

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

[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.

[9]

R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential,, Funkcial. Ekvac. 52 (2009), 52 (2009), 411.

[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.

[11]

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

[12]

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

[13]

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

[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.

show all references

References:
[1]

M. D'Abbicco, M.R. Ebert, Hyperbolic-like estimates for higher order equations,, J. Math. Anal. Appl. 395 (2012), 395 (2012), 747.

[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.

[3]

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

[4]

M. D'Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping,, Chinese Ann. Math. 34B (2013), 34B (2013), 345.

[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.

[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.

[7]

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

[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.

[9]

R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential,, Funkcial. Ekvac. 52 (2009), 52 (2009), 411.

[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.

[11]

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

[12]

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

[13]

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

[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.

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