Abstract
The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L2 and Lp+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Feng, S. J. and Feng, D. X., Nonlinear internal damping of wave equations with variable coefficients, Acta Math. Sin., 20(6), (2006), 1057–1072.
Fujita, H., On the blowing up of solutions of the Cauchy problem for ut = + u1+, J. Fac. Sci. Univ. Tokyo Sec. 1A, 13, 1966, 109–124.
Georgiev, V., Lindblad, H. and Sogge, C., Weighted Strichards estimates and global existence for semilinear wave equation, Amer. J. Math., 119, 1997, 1291–1319.
Glassey, R., Existence in the large for u = F(u) in two dimensions, Math. Z., 178, 1981, 233–261.
Ikawa, M., Hyperbolic Partial Differential Equations and Wave Phenomena, American Math. Soc., Provi-dence, RI, 2000.
Ikehata, R. and Tanizawa, K., Global existence of solutions for semilinear damped wave equations in RN 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 space-dependent potential, Funkcial Ekvac., 52(3), (2009), 411–435.
John, F., Blow up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28, 1979, 235–268.
Lindblad, H. and Sogge, C., On the small-power semilinear wave equations, preprint, 1995.
Nishihara, K., Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math., 34, 2011, 327–343.
Radu, P., Todorova, G. and Yordanov, B., Decay estimates for wave equations with variable coefficients, Trans. Amer. Math. Soc., 362(5), (2010), 2279–2299.
Sideris, T., Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52, 1984, 378–406.
Strauss, W., Nonlinear scattering theory at low energy, J. Funct. Anal., 43, 1981, 281–293.
Strauss, W., Nonlinear wave equations, C.B.M.S. Lecture Notes, Vol. 73, American Math. Soc., Providence, RI, 1989.
Takamura, H. and Wakasa, K., Almost global solutions of semilinear wave equations with the critical exponent in high dimensions, Nonlinear Analysis, 109, 2004, 187–229.
Tataru, D., Stricharts estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353(2), (2001), 795–807.(electronic).
Todorova, G. and Yordanov, B., Critical exponent for a nonlinear wave equation with damping, C. R. Acad. Sci. Paris, 330, 2000, 557–562.
Todorova, G. and Yordanov, B., Critical exponent for a nonlinear wave equation with damping, J. Differ-ential Equations, 174, 2001, 464–489.
Todorova, G. and Yordanov, B., Weighted L2-estimates for dissipative wave equations with variable coef-ficients, J. Differential Equations, 246(12), (2009), 4497–4518.
Wakasugi, Y., Small data global existence for the semilinear wave equation with space-time dependent damping, J. Math. Anal. Appl., 393, 2012, 66–79.
Zhang, Q. S., A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris, 333, 2001, 109–114.
Zhou, Y., Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Partial Differential Equations, 8, 1995, 135–144.
Acknowledgement
We are grateful to the anonymous referees for a number of valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 11501395, 71572156).
Rights and permissions
About this article
Cite this article
Lei, Q., Yang, H. Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients. Chin. Ann. Math. Ser. B 39, 643–664 (2018). https://doi.org/10.1007/s11401-018-0087-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-018-0087-3