Lower and upper bounds for the blow-up time to a viscoelastic Petrovsky wave equation with variable sources and memory term

References

• Wenying C, Yong Z. Global non existence for a semilinear Petrovsky equation. Nonlinear Anal. 2009;70:3203–3208.
   Web of Science ®Google Scholar
• Han X, Wang M. Asymptotic behavior for Petrovsky equation with localized damping. Acta Appl Math. 2010;110(3):1057–1076.
   Web of Science ®Google Scholar
• Li G, Sun Y, Liu W. Global existence, uniform decay and blow-up of solutions for a system of Petrovsky equations. Nonlinear Anal. 2011;74(4):1523–1538.
   Web of Science ®Google Scholar
• Li G, Sun Y, Liu W. Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping. Appl Anal. 2012;91(3):575–586.
   Web of Science ®Google Scholar
• Yaojun Y. Global existence and blow-up of solutions for a system of Petrovsky equations. Appl Anal. 2017;96(16):2869–2890.
   Web of Science ®Google Scholar
• Zhou J. Lower bounds for blow-up time of two nonlinear wave equations. Appl Math Lett. 2015;45:64–68.
   Web of Science ®Google Scholar
• Wang Y, Wang Y. On the initial-boundary problem for fourth order wave equations with damping, strain and source terms. J Math Anal Appl. 2013;405(1):116–127.
   Web of Science ®Google Scholar
• Jun Z. Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping. Appl Math Comput. 2015;265:807–818.
   Web of Science ®Google Scholar
• Li F, Bai Y. Uniform rates of decay for nonlinear viscoelastic Marguerrevon Karman shallow shell system. J Math Anal Appl. 2009;351:522–535.
   Web of Science ®Google Scholar
• Li F. Limit behavior of the solution to nonlinear viscoelastic Marguerre–von Karman shallow shells system. J Differ Equ. 2010;249:1241–1257.
   Web of Science ®Google Scholar
• Park S-H, Kang J-R. Blow-up of solutions for a viscoelastic wave equation with variable exponents. Math Meth Appl Sci. 2019;42:2083–2097.
   Web of Science ®Google Scholar
• Park S-H. Blow-up for a viscoelastic von Karman equation with strong damping and variable exponent source terms. Bound Value Probl.2021;63(2021. doi:10.1186/s13661-021-01537-2.
   Web of Science ®Google Scholar
• Lagnese JE. Asymptotic energy estimates for Kirchoff plates subject to weak viscoelastic damping. Bassel:Birkhcauser-Verlag;1989. (International Series of Numerical Mathematics; vol. 91).
   Google Scholar
• Munoz Rivera JE, Lapa EC, Barreto R. Decay rates for viscoelastic plates with memory. J Elast. 1996;44:61–87.
   Web of Science ®Google Scholar
• Alabau-Boussouira F, Cannarsa P, Sforza D. Decay estimates for the second order evaluation equation with memory. J Funct Anal. 2008;254:1342–1372.
   Web of Science ®Google Scholar
• Liu L, Sun F, Wu Y. Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level. Bound Value Probl. 2019;15(2019). doi:10.1186/s13661-019-1136-x.
   Web of Science ®Google Scholar
• Fushan L, Qingyong G. Blow-up of solution for a nonlinear Petrovsky type equation with memory. Appl Math Comput. 2016;274:383–392.
   Web of Science ®Google Scholar
• Tahamtani F, Shahrouzi M. Existence and blow up of solutions to a Petrovsky equation withmemory and nonlinear source term. Bound Value Probl. 2012;2012:50.
   Web of Science ®Google Scholar
• Li G, Sun Y, Liu W. On a symptotic behavior and blow-up of solutions for a nonlinear viscoelastic Petrovsky equation with positive initial energy. J Funct Spaces Appl. 2013;2013:905867.
   Google Scholar
• Aboulaich R, Meskine D, Souissi A. New diffusion models in image processing. Comput Math Appl. 2008;56:874–882.
   Web of Science ®Google Scholar
• Kbiri Alaoui M, Nabil T, Altanji M. On some new nonlinear diffusion model for the image filtering. Appl Anal. 2013;93(2):269–280.
   Web of Science ®Google Scholar
• Songzhe L, Gao W, Cao C, et al. Study of the solutions to a model porousmedium equation with variable exponent of nonlinearity. J Math Anal Appl. 2008;342:27–38.
   Web of Science ®Google Scholar
• Ting TW. Certain non-steady flows of second-order fluids. Arch Ration Mech Anal. 1963;14:1–26.
   Web of Science ®Google Scholar
• Korpusov MO, Sveshnikov AG. Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources. J Differ Equ. 2006;42:431–443.
   Web of Science ®Google Scholar
• Diening L, Růžička M. Calderó n–Zygmund operators on generalized Lebesgue spaces Lp(x)(Ω) and problems related to fluid dynamics. J Reine Angew Math. 2003;563:197–220.
   Web of Science ®Google Scholar
• Samarskii AA, Galaktionov VA, Kurdyumov SP, et al. Blow-up in quasilinear parabolic equations. Berlin (NY): Walter de Gruyter; 1995.
   Google Scholar
• Pinasco JP. Blow-up for parabolic and hyperbolic problems with variable exponents. Nonlinear Anal Theory Methods Appl. 2009;71:1094–1099.
   Web of Science ®Google Scholar
• Halsey TC. Electrorheological fluids. Science. 1992;258;761–766.
   PubMed Web of Science ®Google Scholar
• Acerbi E, Mingione G. Regularity results for electrorheological fluids, the stationary case. C R Acad Sci Paris. 2002;334:817–822.
   Web of Science ®Google Scholar
• Růžička M. Electrorheological fluids, modeling and mathematical theory. in: Lecture Notes in Mathematics, Vol. 1748. New York: Springer; 2000.
   Google Scholar
• Diening L, Hästo P, Harjulehto P, et al. Lebesgue and Sobolev spaces with variable exponents. Berlin: Springer-Verlag; 2011. (Springer Lecture Notes; vol. 2017, ).
   Google Scholar
• Michael R, Barry S. Scattering theory. Vol. III. London, New York:Academic Press;1979. (Methods of modern mathematical physics).
   Google Scholar
• Erich Z. Partial differential equations of applied mathematics, Pure and Applied Mathematics, seconded. New York (NY): A Wiley-Interscience Publication, John Wiley and Sons, Inc; 1989.
   Google Scholar
• Abita R. Existence and asymptotic stability for the semilinear wave equation with variable-exponent nonlinearities. J Math Phys. 2019;60:122701.
   Web of Science ®Google Scholar
• Lions JL. Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; 1966.
   Google Scholar
• Lions JL, Magenes E. Problemes aux limites nonhomog ènes et applications. Paris: Dunod; 1968.
   Google Scholar
• Abita R. Existence and asymptotic behavior of solutions for degenerate nonlinear Kirchhoff strings with variable-exponent nonlinearities. Acta Math Vietnam. 2021;46:613–643.
   Web of Science ®Google Scholar
• Kalantarov V, Ladyzhenskaya OA. The occurence of collapse for quasilinear equation of paprabolic and hyperbolic types. J Sov Math. 1978;10:53–70.
   Google Scholar
• Payne LE. Improperly posed problems in partial differential equations, Regional Conference Series in Applied Mathematics. Pennsylvania (PA): Society for industrial and applied mathematics; 1975.
   Google Scholar
• Levin HA. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+F(u). Arch Ration Mech Anal. 1973;51:371–386.
   Web of Science ®Google Scholar
• Ali K, Khadijeh B. Blow-up in a semilinear parabolic problem with variable source under positive initial energy. Appl Anal. 2015;94(9):1888–1896.
   Web of Science ®Google Scholar
• Abita R. Blow-up phenomenon for a semilinear pseudo-parabolic equation involving variable source. Appl Anal. 2021;1–16.
   Web of Science ®Google Scholar
• Abita R. Blow-up phenomenon for a semilinear pseudo-parabolic equation involving variable source. Appl Anal. 2020;101(6):1871–1879.
   Google Scholar