Advanced Search
Article Contents
Article Contents

Small data solutions for semilinear wave equations with effective damping

Abstract Related Papers Cited by
  • 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 $b(t)u_t - \Delta u = f(t,u)$.
    Mathematics Subject Classification: 35L71.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


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


    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.


    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.


    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.


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


    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.


    R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential, Funkcial. Ekvac. 52 (2009), 411-435.


    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


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


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


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


    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.

  • 加载中
Open Access Under a Creative Commons license

Article Metrics

HTML views() PDF downloads(104) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint