Abstract
The authors study the Cauchy problem for the semi-linear damped wave equation
in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u|p in the supercritical case of \(p > \tfrac{2} {n}\) and \(p \leqslant \tfrac{n} {{n - 2}}\) for n ≥ 3 is proved.
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D’Abbicco, M., The threshold between effective and noneffective damping for semilinear wave equations, to appear. arXiv: 1211.0731
D’Abbicco, M. and Ebert, M. R., Hyperbolic-like estimates for higher order equations, J. Math. Anal. Appl., 395, 2012, 747–765. DOI: 10.1016/j.jmaa.2012.05.070
D’Abbicco, M. and Ebert, M. R., 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
D’Abbicco, M. and Lucente, S., A modified test function method for damped wave equations, Advanced Nonlinear Studies, 2013, to appear. arXiv: 1211.0453
Friedman, A., Partial Differential Equations, Krieger, New York, 1976.
Fujita, H., On the blowing up of solutions of the Cauchy problem for u t = Δu + u 1+α, J. Fac. Sci. Univ. Tokyo, 13, 1966, 109–124.
Ikehata, R., Miyaoka, Y. and Nakatake, T., Decay estimates of solutions for dissipative wave equations in ℝN with lower power nonlinearities, J. Math. Soc. Japan, 56, 2004, 365–373.
Ikehata, R. and Ohta, M., Critical exponents for semilinear dissipative wave equations in ℝN, J. Math. Anal. Appl., 269, 2002, 87–97.
Ikehata, R. and Tanizawa, K., Global existence of solutions for semilinear damped wave equations in R N with noncompactly supported initial data, Nonlinear Analysis, 61, 2005, 1189–1208.
Ikehata, R., Todorova, G. and Yordanov, B., Critical exponent for semilinear wave equations with a subcritical potential, Funkcial. Ekvac., 52, 2009, 411–435.
Lin, J., Nishihara, K. and Zhai, J., Critical exponent for the semilinear wave equation with time-dependent damping, Disc. Contin. Dynam. Syst., 32(12), 2012, 4307–4320. DOI: 10.3934/dcds.2012.32.4307
Matsumura, A., On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS., 12, 1976, 169–189.
Nishihara, K., Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, Commun. Part. Diff. Eq., 35, 2010, 1402–1418.
Nishihara, K., Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. of Math., 34, 2011, 327–343.
Nakao, M. and Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214, 1993, 325–342.
Todorova, G. and Yordanov, B., Critical exponent for a nonlinear wave equation with damping, J. Diff. Eq., 174, 2001, 464–489.
Wirth, J., Asymptotic properties of solutions to wave equations with time-dependent dissipation, PhD Thesis, TU Bergakademie Freiberg, 2004.
Wirth, J., Wave equations with time-dependent dissipation II. Effective dissipation, J. Diff. Eq., 232, 2007, 74–103.
Zhang, Q. S., 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.
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D’Abbicco, M., Lucente, S. & Reissig, M. Semi-linear wave equations with effective damping. Chin. Ann. Math. Ser. B 34, 345–380 (2013). https://doi.org/10.1007/s11401-013-0773-0
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DOI: https://doi.org/10.1007/s11401-013-0773-0