Restricted accessResearch articleFirst published online 2022-9-7
Global existence,nonexistence,and decay of solutions for a wave equation of p -Laplacian type with weak and p -Laplacian damping,nonlinear boundary delay and source terms
In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases and . To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term.
N.Boumaza and B.Gheraibia, On the existence of a local solution for an integro-differential equation with an integral boundary condition, Bol. Soc. Mat. Mex26 (2020), 521–534. doi:10.1007/s40590-019-00266-y.
2.
M.M.Cavalcanti, V.N.D.Cavalcanti and I.Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equ.236 (2007), 407–459. doi:10.1016/j.jde.2007.02.004.
3.
M.M.Cavalcanti, V.N.D.Cavalcanti and P.Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differ. Equ.203 (2004), 119–158. doi:10.1016/j.jde.2004.04.011.
4.
A.Chahtou, M.Abdelli and A.Hakem, Well-posedness and energy decay of solutions for a quasilinear Petrovsky with a localized nonlinear dissipation involving the p-Laplacian, Nonlinear Studies27(4) (2020), 1091–1104.
5.
H.Chen and G.Liu, Global existence, uniform decay and exponential growth for a class of semilinear wave equation with strong damping, Acta. Math. Sci. Ser. B33(1) (2013), 41–58. doi:10.1016/S0252-9602(12)60193-3.
6.
Q.Dai and Z.Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys.65(5) (2014), 885–903. doi:10.1007/s00033-013-0365-6.
7.
R.Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks, SIAM J Control Optim.26(3) (1988), 697–713. doi:10.1137/0326040.
8.
H.F.Di, Y.D.Shang and J.Yu, Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source, Electronic Research Archive28(1) (2020), 221–261. doi:10.3934/era.2020015.
9.
M.Ferhat and A.Hakem, Global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term, Comput. Math. Appl.71 (2016), 779–804. doi:10.1016/j.camwa.2015.12.039.
10.
M.Ferhat and A.Hakem, Asymptotic behavior for a weak viscoelastic wave equations with a dynamic boundary and time varying delay term, J. Appl. Math. Comput.51 (2016), 509–526. doi:10.1007/s12190-015-0917-3.
11.
V.Georgiev and G.Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Diff. Equ.109(2) (1994), 295–308. doi:10.1006/jdeq.1994.1051.
12.
S.Gerbi and B.Houari, Exponential decay for solutions to semilinear damped wave equation, Discrete Contin. Dyn. Syst. Ser. B5(3) (2012), 559–566.
13.
J.Jeong, J.Park and Y.H.Kang, Global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions, Bound. Value. Probl.2017 (2017), 42.
14.
J.Jeong, J.Park and Y.H.Kang, Global nonexistence of solutions for a nonlinear wave equation with time delay and acoustic boundary conditions, Comput. Math. Appl.76 (2018), 661–671. doi:10.1016/j.camwa.2018.05.006.
15.
M.Kafini and S.A.Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math.13 (2016), 237–247. doi:10.1007/s00009-014-0500-4.
16.
N.J.Kass and M.A.Rammaha, Local and global existence of solutions to a strongly damped wave equation of the p-Laplacian type, Commun. Pure Appl. Anal.17(4) (2018), 1449–1478. doi:10.3934/cpaa.2018070.
17.
H.Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal.5 (1974), 138–146. doi:10.1137/0505015.
18.
G.Liu and H.Zhang, Well-posedness for a class of wave equation with past history and a delay, Z Angew. Math. Phys.67(1) (2016), 1–14. doi:10.1007/s00033-015-0604-0.
19.
S.A.Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr.231 (2001), 105–111. doi:10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I.
20.
S.A.Messaoudi, On the decay of solutions for a class of quasilinear hyperbolic equations with nonlinear damping and source terms, Math Meth. Appl. Sci.28 (2005), 1819–1828. doi:10.1002/mma.641.
21.
S.A.Messaoudi and B.S.Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms, Math. Meth. Appl. Sci.27 (2004), 1687–1696. doi:10.1002/mma.522.
22.
S.Mokeddem and K.B.Mansour, Asymptotic behaviour of solutions for p-Laplacian wave equation with m-Laplacian dissipation, Z. Anal. Anwend.33(3) (2014), 259–269. doi:10.4171/ZAA/1510.
23.
S.Nicaise and C.Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim.45 (2006), 1561–1585. doi:10.1137/060648891.
24.
S.Nicaise and C.Pignotti, Interior feedback stabilization of wave equations with time dependence delay, Electron J. Differ. Equ.41 (2011), 1.
25.
D.Pereira, C.A.Raposo and C.H.M.Maranhão, Global solution and asymptotic behaviour for a wave equation type p-Laplacian with p-Laplacian damping, MathLAB Journal5 (2020), 35–45.
26.
E.Pişkin, On the decay and blow up of solutions for a quasilinear hyperbolic equations with nonlinear damping and source terms, Bound. Value. Probl.2015 (2015), 127.
27.
E.Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal.149 (1999), 155–182. doi:10.1007/s002050050171.
28.
E.Vitillaro, Global existence for wave equation with nonlinear boundary damping and source terms, J. Differ. Equ.186 (2002), 259–298. doi:10.1016/S0022-0396(02)00023-2.
29.
E.Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J.44 (2002), 375–395. doi:10.1017/S0017089502030045.
30.
G.F.Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math.23(3) (1980), 631–643. doi:10.4153/CJM-1980-049-5.
31.
Y.Wu and X.Xue, Uniform decay rate estimates for a class of quasilinear hyperbolic equations with nonlinear damping and source terms, Appl. Anal.92(6) (2013), 1169–1178. doi:10.1080/00036811.2012.661043.
32.
H.Zhang and Q.Hu, Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition, Commun. Pure Appl. Anal.4 (2005), 861–869. doi:10.3934/cpaa.2005.4.861.
