Blow up and asymptotic behavior in a nondissipative nonlinear wave equation

References

• Willem M. Minimax theorems. Vol. 24, Progress in Nonlinear Differential Equations and Applications. Boston, MA: Birkhäuser; 1996.
   Google Scholar
• Payne LE, Sattinger DH. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 1975;22:273–303.
   Web of Science ®Google Scholar
• Gazzola F, Squassina M. Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2006;23:185–207.
   Web of Science ®Google Scholar
• Li K, Zhang Q. Existence and nonexistence of global solutions for the equation of dislocation of crystals. J. Differ. Equ. 1998;146:5–21.
   Web of Science ®Google Scholar
• Runzhang X. Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Quart. Appl. Math. 2010;68:459–468.
   Web of Science ®Google Scholar
• Wang Y. A sufficient condition for finite time blow up of the nonlinear Klein–Gordon equations with arbitrarily positive initial energy. Proc. Am. Math. Soc. 2008;136:3477–3482.
   Web of Science ®Google Scholar
• Haraux A. Quelques petites questions qui m’ont résisté jusqu’a aujourd’hui; 2009. Available from: http://www.math.univ-metz.fr/harauxconference/, https://www.ljll.math.upmc.fr/modules/resources/download/ljll/Seminaires/Systemes_Dynamiques/ProblemesouvertsAH.pdf.
   Google Scholar
• Cazenave T, Haraux A, Weissler FB. Global behavior for some conservative nonlinear equations. Mat. Contemp. 1995;8:89–106.
   Google Scholar
• Rabinowitz PH. Free vibrations for a semilinear wave equation. Comm. Pure Appl. Math. 1978;31:31–68.
   Web of Science ®Google Scholar
• Brezis H, Coron J-M, Nirenberg L. Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz. Comm. Pure Appl. Math. 1980;33:667–684.
   Web of Science ®Google Scholar
• Berti M, Bolle P. Multiplicity of periodic solutions of nonlinear wave equations. Nonlinear Anal. 2004;56:1011–1046.
   Web of Science ®Google Scholar
• Ji S, Li Y. Time periodic solutions to the one-dimensional nonlinear wave equation. Arch. Ration. Mech. Anal. 2011;199:435–451.
   Web of Science ®Google Scholar
• Ton B. An time-periodic solutions of a nonlinear wave equation. Nonlinear Anal. 2011;74:5088–5096.
   Web of Science ®Google Scholar
• Nakanishi K, Schlag W. Invariant manifolds and dispersive Hamiltonian evolution equations. Zürich Lectures in Advanced Mathematics. Zürich: European Mathematical Society; 2011.
   Google Scholar
• Georgiev V, Todorova G. Existence of a solution of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 1994;109:295–308.
   Web of Science ®Google Scholar
• Ikehata R. Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Anal. 1996;27:1165–1175.
   Web of Science ®Google Scholar
• Ikehata R, Suzuki T. Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math. J. 1996;26:475–491.
   Google Scholar
• Esquivel-Avila JA. The dynamics of a nonlinear wave equation. J. Math. Anal. Appl. 2003;279:135–150.
   Web of Science ®Google Scholar
• Esquivel-Avila JA. Qualitative analysis of a nonlinear wave equation. Discrete Contin. Dyn. Syst. Ser. A. 2004;10:787–804.
   Web of Science ®Google Scholar
• Alves CO, Cavalcanti MM. On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source. Calc. Var. Partial Differ. Equ. 2009;34:377–411.
   Web of Science ®Google Scholar
• Alves CO, Cavalcanti MM, Domingos Cavalcanti VN, Rammaha MA, Toundykov D. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete Contin. Dyn. Syst. Ser. S. 2009;2:583–608.
   Web of Science ®Google Scholar
• Bociu L, Lasiecka I. Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping. J. Differ. Equ. 2010;249:654–683.
   Web of Science ®Google Scholar
• Cavalcanti MM, Domingos Cavalcanti VN, Lasiecka I. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differ. Equ. 2007;236:407–459.
   Web of Science ®Google Scholar
• Cavalcanti MM, Domingos Cavalcanti VN, Nascimento FAF. Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation. Proc. Am. Math. Soc. 2013;141:3183–3193.
   Web of Science ®Google Scholar
• Gerbi S, Said-Houari B. Exponential decay for solutions to semilinear damped wave equation. Discrete Contin. Dyn. Syst. Ser. S. 2012;5:559–566.
   Web of Science ®Google Scholar
• Ikehata R, Todorova G, Yordanov B. Wave equations with strong damping in Hilbert spaces. J. Differ. Equ. 2013;254:3352–3368.
   Web of Science ®Google Scholar
• Runzhang X. Global existence, blow up and asymptotic behavior of solutions for nonlinear Klein–Gordon equation with dissipative term. Math. Methods Appl. Sci. 2010;33:831–844.
   Web of Science ®Google Scholar
• Said-Houari B. Global existence and decay of solutions of a nonlinear system of wave equations. Appl. Anal. 2012;91:475–489.
   Web of Science ®Google Scholar
• Said-Houari B. Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms. Differ. Integral Equ. 2010;23:79–92.
   Web of Science ®Google Scholar
• Serrin J, Todorova G, Vitillaro E. Existence for a nonlinear wave equation with damping and source terms. Differ. Integral Equ. 2013;16:13–50.
   Google Scholar
• Sun F, Wang M, Li H. Global and blow-up solutions for a quasilinear hyperbolic equation with strong damping. Nonlinear Anal. 2010;73:1408–1425.
   Web of Science ®Google Scholar
• Todorova G. Dynamics of non-linear wave equations. Math. Methods Appl. Sci. 2004;27:1831–1841.
   Web of Science ®Google Scholar
• Toundykov D. Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions. Nonlinear Anal. 2007;67:512–544.
   Web of Science ®Google Scholar
• Vazquez JL, Vitillaro E. Wave equation with second-order non-standard dynamical boundary conditions. Math. Models Methods Appl. Sci. 2008;18:2019–2054.
   Web of Science ®Google Scholar
• Yacheng L, Junsheng Z. On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. 2006;64:2665–2687.
   Web of Science ®Google Scholar
• Yacheng L, Runzhang X. A class of fourth order wave equations with dissipative and nonlinear strain terms. J. Differ. Equ. 2008;244:200–228.
   Web of Science ®Google Scholar
• Ball J. Finite blow up in nonlinear problems. In: Crandall MG, editor. Nonlinear evolution equations. New York, NY: Academic Press; 1978. p. 189–205.
   Google Scholar
• Ball J. Remarks on blow up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. Oxford. 1977;28:473–486.
   Web of Science ®Google Scholar
• Cazenave T, Haraux A. An introduction to semilinear evolution equations. Oxford: Clarendon Press; 1998.
   Google Scholar
• Dancer EN. The efect of domain shape on the number of positive solutions of certain nonlinear equations. J. Differ. Equ. 1988;74:120–156.
   Web of Science ®Google Scholar
• Gazzola F, Weth T. Finite time blow up and global solutions for semilinear parabolic equations with initial data at high energy level. Differ. Integral Equ. 2005;18:961–990.
   Web of Science ®Google Scholar
• Wang H-C, Wu T-F. Symmetry breaking in a bounded symmetry domain. NoDEA Nonlinear Differ. Equ. Appl. 2004;11:361–377.
   Web of Science ®Google Scholar
• Cazenave T. Uniform estimates for solutions of nonlinear Klein–Gordon equations. J. Funct. Anal. 1985;60:36–55.
   Web of Science ®Google Scholar
• Ball J. On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations. J. Differ. Equ. 1978;27:224–265.
   Web of Science ®Google Scholar
• Galaktionov A, Pohozaev SI. Blow up and critical exponents for nonlinear hyperbolic equations. Nonlinear Anal. 2006;53:453–466.
   Web of Science ®Google Scholar