Abstract
The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs.
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Alabau-Boussouira, F.: Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51(1), 61–105 (2005)
Al’shin, A.B.; Korpusov, M.O.; Sveshnikov, A.G.: Blow-up in nonlinear Sobolev type equations. In: Series in Nonlinear Analysis and Applications, vol. 15, 660 p. Walter de Gruyter (2011)
Antontsev, S.; Zhikov, V.: Higher integrability for parabolic equations of \(p(x, t)\)-Laplacian type. Adv. Differ. Equ. 10(9), 1053–1080 (2005)
Antontsev, S.; Shmarev, S.: Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math. 234(9), 2633–2645 (2010)
Antontsev, S.: Wave equation with \(p(x, t)\)-Laplacian and damping term: existence and blow-up. Differ. Equ. Appl. 3(4), 503–525 (2011)
Antontsev, S.: Wave equation with \(p(x, t)\)-Laplacian and damping term: blow-up of solutions. C. R. Mec. 339(12), 751–755 (2011)
Antontsev, S.; Ferreira, J.: Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions. Nonlinear Anal. Theory Methods Appl. 93, 62–77 (2013)
Antontsev, S.; Shmarev, S.: Evolution PDEs with nonstandard growth conditions: existence, uniqueness, localization, blow-up. In: Atlantis Studies in Differential Equations, vol. 4. Atlantis Press (2015)
Autuori, G.; Pucci, P.; Salvatori, M.: Global nonexistence for nonlinear Kirchhoff systems. Arch. Ration. Mech. Anal. 196(2), 489–516 (2010)
Ball, J.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. 28(4), 473–486 (1977)
Benaissa, A.; Messaoudi, S.A.: Blow up of solutions of a nonlinear wave equation. J. Appl. Math. 2(2), 105–108 (2002)
Benaissa, A.; Messaoudi, S.A.: Blow-up of solutions for Kirchhoff equation of \(q\)-Laplacian type with nonlinear dissipation. Colloq. Math. 94(1), 103–109 (2002)
Benaissa, A.; Messaoudi, S.A.: Blow-up of solutions of a quasilinear wave equation with nonlinear dissipation. J. Partial Differ. Equ. 15(3), 61–67 (2002)
Benaissa, A.; Mimouni, S.: Energy decay of solutions of a wave equation of \(p\)-Laplacian type with a weakly nonlinear dissipation. J. Inequal. Pure Appl. Math. 7(1), 1–8 (2006)
Benaissa, A.; Mimouni, S.: Energy decay of solutions of a wave equation of \(p\)-Laplacian type with a weakly nonlinear dissipation. JIPM. J. Inequal. Pure Appl. Math. 7(1), 8 (2006). Article 15
Benaissa, A.; Messaoudi, S.A.: Exponential decay of solutions of a nonlinearly damped wave equation. NoDEA Nonlinear Differ. Equ. Appl. 12(4), 391–399 (2006)
Benaissa, A.; Mokeddem, S.: Decay estimates for the wave equation of \(p\)-Laplacian type with dissipation of \(m\)-Laplacian type. Math. Methods Appl. Sci. 30(2), 237–247 (2007)
Benaissa, A.; Guesmia, A.: Global existence and general decay estimates of solutions for degenerate or nondegenerate Kirchhoff equation with general dissipation. Diff. Inte. Equ. 11(1), 1–35 (2011)
Cavalcanti, M.; Guesmia, A.: General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type. Differ. Integral Equ. 18(5), 583–600 (2005)
Cavalcanti, M.; Cavalcanti, V.D.; Tebou, L.: Stabilization of the wave equation with localized compensating frictional and Kelvin–Voigt dissipating mechanisms. Electron. J. Differ. Equ. 2017(83), 1–18 (2017)
Chen, C.; Yao, H.; Shao, L.: Global existence, uniqueness, and asymptotic behavior of solution for \(p\)-Laplacian type wave equation. J. Inequal. Appl. 2010(1), 216760 (2010)
Ferreira, J.; Messaoudi, S.A.: On the general decay of a nonlinear viscoelastic plate equation with a strong damping and \(\overrightarrow{p}(x, t)\)-Laplacian. Nonlinear Anal. Theory Methods Appl. 104, 40–49 (2014)
Galaktionov, V.A.; Pohozaev, S.I.: Blow-up and critical exponents for nonlinear hyperbolic equations. Nonlinear Anal. Theory Methods Appl. 53(3), 453–466 (2003)
Gao, H.; Ma, T.F.: Global solutions for a nonlinear wave equation with the \(p\)-Laplacian operator. Electron. J. Qual. Theory of Differ. Equ. 11, 1–13 (1999)
Gao, Y.; Gao, W.: Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents. Bound. Value Probl. 2013(1), 1–8 (2013)
Georgiev, V.; Todorova, G.: Existence of solutions of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 109(2), 295–308 (1994)
Guesmia, A.: On the decay estimates for elasticity systems with some localized dissipations. Asymptot. Anal. 22(1), 1–13 (2000)
Guesmia, A.: A new approach of stabilization of nondissipative distributed systems. SIAM J. Control Optim. 42(1), 24–52 (2003)
Guesmia, A.; Messaoudi, S.A.: Decay estimates of solutions of a nonlinearly damped semilinear wave equation. In: Annales Polonici Mathematici, vol. 85, pp. 25–36. Instytut Matematyczny Polskiej Akademii Nauk (2005)
Guo, B.; Gao, W.: Blow-up of solutions to quasilinear hyperbolic equations with \(p(x, t)\)-Laplacian and positive initial energy. C. R. Mec. 342(9), 513–519 (2014)
Hamidi, A.E.; Vetois, J.: Sharp Sobolev asymptotics for critical anisotropic equations. Arch. Ration. Mech. Anal. 192(1), 1–36 (2009)
Ibrahim, S.; Lyaghfouri, A.: Blow-up solutions of quasilinear hyperbolic equations with critical Sobolev exponent. Math. Modell. Nat. Phenom. 7(2), 66–76 (2012)
Kafini, M.; Messaoudi, S.A.: A blow-up result in a nonlinear wave equation with delay. Mediterr. J. Math. 13(1), 237–247 (2016)
Kalantarov, V.K.; Ladyzhenskaya, O.A.: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type. J. Soviet Math. 10(1), 53–70 (1978)
Kirchhoff, G.: Vorlesungen uber Mechanik. Teubner, Leipzig (1883)
Komornik, V.; Zuazua, E.: A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69, 33–54 (1990)
Komornik, V.: Decay estimates for the wave equation with internal damping. Int. Ser. Numer. Math. 118, 253–266 (1994)
Komornik, V.: Exact Controllability and Stabilization. The Multiplier Method. Masson-John Wiley, Paris (1994)
Korpusov, M.O.: Non-existence of global solutions to generalized dissipative Klein–Gordon equations with positive energy. Electron. J. Differ. Equ. 2012(119), 1–10 (2012)
Kovacik, O.; Rakosnik, J.: On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\). Czechoslov. Math. J. 41(4), 592–618 (1991)
Lars, D.; Harjulehto, P.; Hasto, P.; Ruzicka, M.: Lebesgue and Sobolev spaces with variable exponents. In: Lecture Notes in Mathematics, vol. 2017 Springer-Verlag Berlin Heidelberg (2011)
Larson, S.; Thome, V.: Partial differential equations with numerical methods. In: Text in Applied Mathematics, vol 45. Springer (2009)
Lasiecka, I.: Stabilization of wave and plate-like equation with nonlinear dissipation on the boundary. J. Differ. Equ. 79, 340–381 (1989)
Lasiecka, I.; Tataru, D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6(3), 507–533 (1993)
Levine, H.A.: Instability and nonexistence of global solutions of nonlinear wave equation of the form \(Pu=Au+F(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974)
Levine, H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equation. SIAM J. Math. Anal. 5(1), 138–146 (1974)
Levine, H.A.; Pucci, P.; Serrin, J.: Some remarks on global nonexistence for nonautonomous abstract evolution equations. Contemp. Math. 208, 253–263 (1997)
Levine, H.A.; Serrin, J.: Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch. Ration. Mech. Anal. 137(4), 341–361 (1997)
Levine, H.A.; Park, S.R.: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl. 228(1), 181–205 (1998)
Lions, J.L.: Quelques Methodes de Resolution des Problemes Aux Limites Nonlineaires, 2nd edn. Dunod, Paris (2002)
Liu, W.: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. 73(6), 1890–1904 (2010)
Lacroix-Sonrier, M.-T.: Distrubutions, Espace de Sobolev, Applications. Ellipses (1998)
Martinez, P.: A new method to decay rate estimates for dissipative systems. ESIM Control Optim. Cal. Var. 4, 419–444 (1999)
Martinez, P.: A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12(1), 251–283 (1999)
Messaoudi, S.A.: Blow up in a nonlinearly damped wave equation. Math. Nachr. 231(1), 1–7 (2001)
Messaoudi, S.A.: Decay of the solution energy for a nonlinearly damped wave equation. Arab. J. Sci. Eng. 26(1), 63–68 (2001). Part A
Messaoudi, S.A.: Blow up in the Cauchy problem for a nonlinearly damped wave equation. Commun. Appl. Anal. 7(3), 379–386 (2003)
Messaoudi, S.A.; Houari, B.S.: Global non-existence of solutions of a class of wave equations with non-linear damping and source terms. Math. Methods Appl. Sci. 27(14), 1687–1696 (2004)
Messaoudi, S.A.: On the decay of solutions for a class of quasilinear hyperbolic equations with non-linear damping and source terms. Math. Methods Appl. Sci. 28(15), 1819–1828 (2005)
Messaoudi, S.A.; Soufyane, A.: General decay of solutions of a wave equation with a boundary control of memory type. Nonlinear Anal. Real World Appl. 11(4), 2896–2904 (2010)
Messaoudi, S.A.; Talahmeh, A.A.: A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities. Appl. Anal. 96(9), 1509–1515 (2017)
Messaoudi, S.A.; Talahmeh, A.A.: A blow-up result for a quasilinear wave equation with variable-exponent nonlinearities. Math. Methods Appl. Sci. 1–11 (2017). https://doi.org/10.1002/mma.4505
Messaoudi, S.A.; Talahmeh, A.A.; Al-Smail, J.H.: Nonlinear damped wave equation: existence and blow-up. Comput. Math. Appl. https://doi.org/10.1016/j.camwa.2017.07.048
Mokeddem, S.; Mansour, K.B.W.: The rate at which the energy of solutions for a class of-Laplacian wave equation decays. Int. J. Differ. Equ. 2015, 721503-1–721503-5 (2015)
Mustafa, M.I.; Messaoudi, S.A.: General energy decay rates for a weakly damped wave equation. Commun. Math. Anal. 9(2), 67–76 (2010)
Nakano, H.: Modulared Semi-ordered Linear Spaces. Maruzen Co., Ltd, Tokyo (1950)
Nakano, H.: Topology and Topological Linear Spaces. Maruzen Co., Ltd, Tokyo (1951)
Nakao, M.: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. 60, 542–549 (1977)
Nakao, M.: A difference inequality and its applications to nonlinear evolution equations. J. Math. Soc. Jpn. 30, 747–762 (1978)
Nakao, M.: Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations. Math Z. 206, 265–275 (1991)
Nakao, M.; Ono, K.: Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation. Funkc. Ekvacioj 38, 417–431 (1995)
Nakao, M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305(3), 403–417 (1996)
Nishihara, K.; Yamada, Y.: On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkc. Ekvacioj 33(1), 151–159 (1990)
Ono, K.: On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type. Math. Methods Appl. Sci. 20, 151–177 (1997)
Ono, K.: On global solutions and blow-up solutions of nonlinear Kirchhoff string with nonlinear dissipation. J. Math. Anal. Appl. 216, 321–342 (1997)
Ono, K.: Global solvability for degenerate Kirchhoff equations with weak dissipation. Math. Jpn. 50(3), 409–413 (1999)
Orlicz, W.: Uber konjugierte Exponentenfolgen. Stud. Math. 3(1), 200–212 (1931)
Pucci, P.; Serrin, J.: Asymptotic stability for nonautonomous dissipative wave system. Commun. Pure Appl. Math. 49, 177–216 (1996)
Quarteroni, A.; Sacco, R.; Saleri, F.: Numerical Mathematics. Springer, New-York, 2000, TAM Series n., vol. 37, 2 edn (2007)
Quarteroni, A.; Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Heidelberg (1994). SCM Series n. 23
Rammaha, M.A.; Toundykov, D.: Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources. J. Math. Phys. 56(8), 081503 (2015)
Sun, L.; Ren, Y.; Gao, W.: Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources. Comput. Math. Appl. 71(1), 267–277 (2016)
Taniguchi, T.: Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Commun. Pure Appl. Anal. 16(5), 1571–1585 (2017)
Todorova, G.: Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms. J. Math. Anal. Appl. 239(2), 213–226 (1999)
Todorova, G.; Vitillaro, E.: Blow-up for nonlinear dissipative wave equations in \(\mathbb{R}^n\). J. Math. Anal. Appl. 303(1), 242–257 (2005)
Tsenov, I.V.: Generalization of the problem of best approximation of a function in the space \(L^s\), (Russian) Uch. Zap. Dagest. Gos. Univ. 7, 25–37 (1961)
Vitillaro, E.: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149(2), 155–182 (1999)
Vitillaro, E.: Global existence for the wave equation with nonlinear boundary damping and source terms. J. Differ. Equ. 186(1), 259–298 (2002)
Wang, Y.: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl. Math. Lett. 22(9), 1394–1400 (2009)
Wu, S.T.: Blow-up of solutions for an integro-differential equation with a nonlinear source. Electron. J. Differ. Equ. 2006(45), 1–9 (2006)
Wu, Y.; Xue, X.: Decay rate estimates for a class of quasilinear hyperbolic equations with damping terms involving \(p\)-Laplacian. J. Math. Phys. 55(12), 121504 (2014)
Yang, Z.: Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms. Math. Methods Appl. Sci. 25(10), 795–814 (2002)
Ye, Y.: Global nonexistence of solutions for systems of quasilinear hyperbolic equations with damping and source terms. Bound. Value Probl. 2014(1), 251 (2014)
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR Izv. 29(1), 33–66 (1987)
Zuazua, E.: Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. P.D.E no. 15, 205–235 (1990)
Zuazua, E.: Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl. 70(4), 513–529 (1991)
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Messaoudi, S.A., Talahmeh, A.A. On wave equation: review and recent results. Arab. J. Math. 7, 113–145 (2018). https://doi.org/10.1007/s40065-017-0190-4
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DOI: https://doi.org/10.1007/s40065-017-0190-4