@article{AIHPC_2006__23_2_185_0, author = {Gazzola, Filippo and Squassina, Marco}, title = {Global solutions and finite time blow up for damped semilinear wave equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {23}, year = {2006}, pages = {185-207}, doi = {10.1016/j.anihpc.2005.02.007}, mrnumber = {2201151}, zbl = {1094.35082}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2006__23_2_185_0} }
Gazzola, Filippo; Squassina, Marco. Global solutions and finite time blow up for damped semilinear wave equations. Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) pp. 185-207. doi : 10.1016/j.anihpc.2005.02.007. http://gdmltest.u-ga.fr/item/AIHPC_2006__23_2_185_0/
[1] Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349-381. | MR 370183 | Zbl 0273.49063
, ,[2] Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004) 31-52. | MR 2026182 | Zbl 1056.37084
,[3] Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc. 66 (2002) 443-463. | MR 1939206 | Zbl 1020.35059
, ,[4] Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal. 60 (1985) 36-55. | MR 780103 | Zbl 0568.35068
,[5] The dynamics of a nonlinear wave equation, J. Math. Anal. Appl. 279 (2003) 135-150. | MR 1970496 | Zbl 1015.35072
,[6] Qualitative analysis of a nonlinear wave equation, Discrete Contin. Dyn. Syst. 10 (2004) 787-804. | MR 2018880 | Zbl 1047.35103
,[7] Finite time blow-up and global solutions for some nonlinear parabolic equations, Differential Integral Equations 17 (2004) 983-1012. | MR 2082457 | Zbl 1150.35336
,[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 2162421
[9] Existence of a solution of the wave equation with nonlinear damping and source term, J. Differential Equations 109 (1994) 295-308. | MR 1273304 | Zbl 0803.35092
, ,[10] Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys. 43 (1992) 63-124. | MR 1149371 | Zbl 0751.58033
, ,[11] Dissipative Dynamical Systems and Applications, Res. Appl. Math., vol. 17, Masson, Paris, 1991, 132 p. | MR 1084372 | Zbl 0726.58001
,[12] Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations 9 (1999) 95-124. | MR 1714129 | Zbl 0939.35122
, ,[13] Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 27 (1996) 1165-1175. | MR 1407454 | Zbl 0866.35071
,[14] Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J. 26 (1996) 475-491. | MR 1421221 | Zbl 0873.35010
, ,[15] Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 1137-1153. | MR 2018329 | Zbl 1046.37045
, ,[16] Instability and nonexistence of global solutions to nonlinear wave equations of the form , Trans. Amer. Math. Soc. 192 (1974) 1-21. | MR 344697 | Zbl 0288.35003
,[17] Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974) 138-146. | MR 399682 | Zbl 0243.35069
,[18] Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal. 137 (1997) 341-361. | MR 1463799 | Zbl 0886.35096
, ,[19] 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 1792187 | Zbl 0956.35087
, ,[20] On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960) 101-123. | MR 111898 | Zbl 0097.29501
,[21] Remarks on blowup of solutions for nonlinear evolution equations of second order, Adv. Math. Sci. Appl. 8 (1998) 901-910. | MR 1657188 | Zbl 0920.35025
,[22] 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 1430038 | Zbl 0878.35081
,[23] On the strongly damped wave equation, Comm. Math. Phys. 253 (2004) 511-533. | MR 2116726 | Zbl 1068.35077
, ,[24] Saddle points and instability of nonlinear hyperbolic equations, Israel Math. J. 22 (1975) 273-303. | MR 402291 | Zbl 0317.35059
, ,[25] Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations 150 (1998) 203-214. | MR 1660250 | Zbl 0915.35012
, ,[26] Some new results on global nonexistence for abstract evolution with positive initial energy, Topol. Methods Nonlinear Anal. 10 (1997) 241-247. | MR 1634571 | Zbl 0911.35035
, ,[27] On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968) 148-172. | MR 227616 | Zbl 0159.39102
,[28] On solutions of semilinear differential equations in a Hilbert space, Math. Japon. 17 (1972) 173-193. | MR 355247 | Zbl 0273.34044
,[29] Global existence theorems for a class of evolution equations with dissipation, Arch. Rational Mech. Anal. 149 (1999) 155-182. | MR 1719145 | Zbl 0934.35101
,[30] Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979) 19-33. | MR 549869 | Zbl 0414.34042
,[31] Minimax Theorems, Progress Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Boston, MA, 1996, 162 p. | MR 1400007 | Zbl 0856.49001
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