Global solutions and finite time blow up for damped semilinear wave equations

Gazzola, Filippo; Squassina, Marco

Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) no. 2, pp. 185-207.

[1] Ambrosetti A., Rabinowitz P.H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349-381. | MR | Zbl

[2] Ball J., Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004) 31-52. | MR | Zbl

[3] Carvalho A.N., Cholewa J.W., Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc. 66 (2002) 443-463. | MR | Zbl

[4] Cazenave T., Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal. 60 (1985) 36-55. | MR | Zbl

[5] Esquivel-Avila J., The dynamics of a nonlinear wave equation, J. Math. Anal. Appl. 279 (2003) 135-150. | MR | Zbl

[6] Esquivel-Avila J., Qualitative analysis of a nonlinear wave equation, Discrete Contin. Dyn. Syst. 10 (2004) 787-804. | MR | Zbl

[7] Gazzola F., Finite time blow-up and global solutions for some nonlinear parabolic equations, Differential Integral Equations 17 (2004) 983-1012. | MR | Zbl

[8] F. Gazzola, T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, in press. | MR

[9] Georgiev V., Todorova G., Existence of a solution of the wave equation with nonlinear damping and source term, J. Differential Equations 109 (1994) 295-308. | MR | Zbl

[10] Hale J.K., Raugel G., Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys. 43 (1992) 63-124. | MR | Zbl

[11] Haraux A., Dissipative Dynamical Systems and Applications, Res. Appl. Math., vol. 17, Masson, Paris, 1991, 132 p. | MR | Zbl

[12] Haraux A., Jendoubi M.A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations 9 (1999) 95-124. | MR | Zbl

[13] Ikehata R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 27 (1996) 1165-1175. | MR | Zbl

[14] Ikehata R., Suzuki T., Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J. 26 (1996) 475-491. | MR | Zbl

[15] Jendoubi M.A., Poláčik P., Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 1137-1153. | MR | Zbl

[16] Levine H.A., Instability and nonexistence of global solutions to nonlinear wave equations of the form $P{u}_{tt}=-Au+F\left(u\right)$, Trans. Amer. Math. Soc. 192 (1974) 1-21. | MR | Zbl

[17] Levine H.A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974) 138-146. | MR | Zbl

[18] Levine H.A., Serrin J., Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal. 137 (1997) 341-361. | MR | Zbl

[19] Levine H.A., Todorova G., Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc. 129 (2001) 793-805. | MR | Zbl

[20] Nehari Z., On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960) 101-123. | MR | Zbl

[21] Ohta M., Remarks on blowup of solutions for nonlinear evolution equations of second order, Adv. Math. Sci. Appl. 8 (1998) 901-910. | MR | Zbl

[22] Ono K., On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci. 20 (1997) 151-177. | MR | Zbl

[23] Pata V., Squassina M., On the strongly damped wave equation, Comm. Math. Phys. 253 (2004) 511-533. | MR | Zbl

[24] Payne L.E., Sattinger D.H., Saddle points and instability of nonlinear hyperbolic equations, Israel Math. J. 22 (1975) 273-303. | MR | Zbl

[25] Pucci P., Serrin J., Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations 150 (1998) 203-214. | MR | Zbl

[26] Pucci P., Serrin J., Some new results on global nonexistence for abstract evolution with positive initial energy, Topol. Methods Nonlinear Anal. 10 (1997) 241-247. | MR | Zbl

[27] Sattinger D.H., On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968) 148-172. | MR | Zbl

[28] Tsutsumi M., On solutions of semilinear differential equations in a Hilbert space, Math. Japon. 17 (1972) 173-193. | MR | Zbl

[29] Vitillaro E., Global existence theorems for a class of evolution equations with dissipation, Arch. Rational Mech. Anal. 149 (1999) 155-182. | MR | Zbl

[30] Webb G.F., Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979) 19-33. | MR | Zbl

[31] Willem M., Minimax Theorems, Progress Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Boston, MA, 1996, 162 p. | MR | Zbl