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Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. (English) Zbl 0934.35095

The author defines stable and unstable sets of the initial data for the Cauchy problem to the equation \(u_{tt}-\triangle u+q^2(x)u+u_t|u_t|^{m-1}=u|u|^{p-1}\), \(t\in \mathbb{R}\), \(x\in \mathbb{R}^n\) and proves a global existence and power decay of a solution of the Cauchy problem entering into the stable set and blow-up in a finite time of the solution entering into unstable set.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B60 Continuation and prolongation of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L15 Initial value problems for second-order hyperbolic equations
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References:

[1] Brezis, H., Analyse Fonctionnelle, Théorie et Applications (1983), Masson: Masson Paris · Zbl 0511.46001
[2] Georgiev, V.; Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109, 295-308 (1994) · Zbl 0803.35092
[3] Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27, 1165-1175 (1996) · Zbl 0866.35071
[4] Levine, H. A., Instability and nonexistence of global solutions of nonlinear wave equations of the form \(Pu_{tt} \)=−\(Au}+F(u)\), Trans. Amer. Math. Soc., 192, 1-21 (1974) · Zbl 0288.35003
[5] Levine, H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5, 138-146 (1974) · Zbl 0243.35069
[6] Levine, H. A.; Park, S.; Serrin, J., Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228, 181-205 (1998) · Zbl 0922.35094
[7] Levine, H. A.; Pucci, P.; Serrin, J., Some remarks on global nonexistence for nonautonomous abstract evolution equations, Contemp. Math., 208, 253-263 (1997) · Zbl 0882.35081
[8] Levine, H. A.; Serrin, J., A global nonexistence theorem for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137, 341-361 (1997) · Zbl 0886.35096
[9] Nakao, M.; Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214, 325-342 (1993) · Zbl 0790.35072
[10] Nakao, M.; Ono, K., Global existence to the Cauchy problem of the semilinear wave equations with a nonlinear dissipation, Funkcial. Ekvac., 38, 417-431 (1995) · Zbl 0855.35081
[11] Ohta, M., Remarks on blow up of solutions for nonlinear evolution equations of second order, Adv. Math. Sci. Appl., 8, 901-910 (1998) · Zbl 0920.35025
[12] Ono, K., On global solutions and blowup solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216, 321-342 (1997) · Zbl 0893.35078
[13] Pucci, P.; Serrin, J., Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 109, 203-214 (1998) · Zbl 0915.35012
[14] Payne, L.; Sattinger, D., Saddle points and instability on nonlinear hyperbolic equations, Israel Math. J., 22, 273-303 (1981) · Zbl 0317.35059
[15] Todorova, G., Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, C. R. Acad. Sci. Paris Sér. I, 326, 191-196 (1998) · Zbl 0922.35095
[16] G. Todorova, Cauchy problem for a nonlinear wave equations with nonlinear damping and source terms, Nonlinear Anal, in press.; G. Todorova, Cauchy problem for a nonlinear wave equations with nonlinear damping and source terms, Nonlinear Anal, in press. · Zbl 0922.35095
[17] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation and applications, Arch. Rational Mech. Anal, in press.; E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation and applications, Arch. Rational Mech. Anal, in press. · Zbl 0934.35101
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