October  2017, 10(5): 1175-1185. doi: 10.3934/dcdss.2017064

Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  September 2016 Revised  February 2017 Published  June 2017

Fund Project: This work is partially supported by the the Basic and Advanced Research Project of CQCSTC grant cstc2016jcyjA0018, NSFC grant 11201380, Fundamental Research Funds for the Central Universities grant XDJK2015A16, XDJK2016E120, Project funded by China Postdoctoral Science Foundation grant 2014M550453,2015T80948

This paper deals with a higher-order wave equation with general nonlinear dissipation and source term
$u''+(-Δ)^mu+g(u')=b|u|^{p-2}u, $
which was studied extensively when
$m=1, 2$
and the nonlinear dissipative term
$g(u')$
is a polynomial, i.e.,
$g(u')=a|u'|^{q-2}u'$
. We obtain the global existence of solutions and show the energy decay estimate when
$m≥1$
is a positive integer and the nonlinear dissipative term
$g$
does not necessarily have a polynomial grow near the origin.
Citation: Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
References:
[1]

M. Aassila, Global existence of solutions to a wave equation with damping and source terms, Diff. Inte. Equations, 14 (2001), 1301-1314. Google Scholar

[2]

Q. GaoF. Li and Y. Wang, Blow up of solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. Euro. J. Math., 9 (2011), 686-698. doi: 10.2478/s11533-010-0096-2. Google Scholar

[3]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308. doi: 10.1006/jdeq.1994.1051. Google Scholar

[4]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491. Google Scholar

[5]

R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27 (1996), 1165-1175. doi: 10.1016/0362-546X(95)00119-G. Google Scholar

[6]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Du_{tt}=Au+f(u)$, Trans. Am. Math. Soc., 192 (1974), 1-21. doi: 10.2307/1996814. Google Scholar

[7]

H. A. 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. Google Scholar

[8]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM. Cont. Opt. Cal. Var., 4 (1999), 419-444. doi: 10.1051/cocv:1999116. Google Scholar

[9]

S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308. doi: 10.1006/jmaa.2001.7697. Google Scholar

[10]

M. Nako, Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term, J. Math. Anal. Appl., 58 (1977), 336-343. doi: 10.1016/0022-247X(77)90211-6. Google Scholar

[11]

K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216 (1997), 321-342. doi: 10.1006/jmaa.1997.5697. Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, in: Scattering Theiry, vol Ⅲ, Academic Press, New York, London, 1979. Google Scholar

[13]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172. doi: 10.1007/BF00250942. Google Scholar

[14]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226. doi: 10.1006/jmaa.1999.6528. Google Scholar

[15]

S. T. Wu and L. Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math., 13 (2009), 545-558. Google Scholar

[16]

Y. Ye, Existence and asymptotic behavior of gobal solutions for aclass of nonlinear higher-order wave equation, J. Ineq. Appl. , 2010 (2010), Art. ID 394859, 14 pp. doi: 10.1155/2010/394859. Google Scholar

[17]

E. Zauderer, Partial Differential Equations of Applied Mathematics, in: Pure and Applied Mathematics, second edition, A Wiley-interscience Publication, Johu Wiely & Sons, Inc. , New York, 1989. Google Scholar

[18]

J. ZhouX. R. WangX. J. Song and C. L. Mu, Global existence and blowup of solutions for a class of nonlinear higher-order wave equations, Z. Angew. Math. Phys., 63 (2012), 461-473. doi: 10.1007/s00033-011-0165-9. Google Scholar

show all references

References:
[1]

M. Aassila, Global existence of solutions to a wave equation with damping and source terms, Diff. Inte. Equations, 14 (2001), 1301-1314. Google Scholar

[2]

Q. GaoF. Li and Y. Wang, Blow up of solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. Euro. J. Math., 9 (2011), 686-698. doi: 10.2478/s11533-010-0096-2. Google Scholar

[3]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308. doi: 10.1006/jdeq.1994.1051. Google Scholar

[4]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491. Google Scholar

[5]

R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27 (1996), 1165-1175. doi: 10.1016/0362-546X(95)00119-G. Google Scholar

[6]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Du_{tt}=Au+f(u)$, Trans. Am. Math. Soc., 192 (1974), 1-21. doi: 10.2307/1996814. Google Scholar

[7]

H. A. 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. Google Scholar

[8]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM. Cont. Opt. Cal. Var., 4 (1999), 419-444. doi: 10.1051/cocv:1999116. Google Scholar

[9]

S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308. doi: 10.1006/jmaa.2001.7697. Google Scholar

[10]

M. Nako, Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term, J. Math. Anal. Appl., 58 (1977), 336-343. doi: 10.1016/0022-247X(77)90211-6. Google Scholar

[11]

K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216 (1997), 321-342. doi: 10.1006/jmaa.1997.5697. Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, in: Scattering Theiry, vol Ⅲ, Academic Press, New York, London, 1979. Google Scholar

[13]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172. doi: 10.1007/BF00250942. Google Scholar

[14]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226. doi: 10.1006/jmaa.1999.6528. Google Scholar

[15]

S. T. Wu and L. Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math., 13 (2009), 545-558. Google Scholar

[16]

Y. Ye, Existence and asymptotic behavior of gobal solutions for aclass of nonlinear higher-order wave equation, J. Ineq. Appl. , 2010 (2010), Art. ID 394859, 14 pp. doi: 10.1155/2010/394859. Google Scholar

[17]

E. Zauderer, Partial Differential Equations of Applied Mathematics, in: Pure and Applied Mathematics, second edition, A Wiley-interscience Publication, Johu Wiely & Sons, Inc. , New York, 1989. Google Scholar

[18]

J. ZhouX. R. WangX. J. Song and C. L. Mu, Global existence and blowup of solutions for a class of nonlinear higher-order wave equations, Z. Angew. Math. Phys., 63 (2012), 461-473. doi: 10.1007/s00033-011-0165-9. Google Scholar

[1]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[2]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358

[3]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[4]

Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100

[5]

Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019

[6]

Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67

[7]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001

[8]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181

[9]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

[10]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013

[11]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731

[12]

Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361

[13]

Jibin Li, Weigou Rui, Yao Long, Bin He. Travelling wave solutions for higher-order wave equations of KDV type (III). Mathematical Biosciences & Engineering, 2006, 3 (1) : 125-135. doi: 10.3934/mbe.2006.3.125

[14]

Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609

[15]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

[16]

Adam Larios, E. S. Titi. On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 603-627. doi: 10.3934/dcdsb.2010.14.603

[17]

Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503

[18]

Sihong Shao, Huazhong Tang. Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 623-640. doi: 10.3934/dcdsb.2006.6.623

[19]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81

[20]

Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (16)
  • HTML views (23)
  • Cited by (0)

Other articles
by authors

[Back to Top]