Global existence and asymptotic behaviour of solutions for a class of fourth order strongly damped nonlinear wave equations

Runzhang, Xu; Yanbing, Yang

doi:10.1090/S0033-569X-2012-01295-6

Authors: Xu Runzhang and Yang Yanbing
Journal: Quart. Appl. Math. 71 (2013), 401-415
MSC (2010): Primary 35L25, 35A01, 35L30
DOI: https://doi.org/10.1090/S0033-569X-2012-01295-6
Published electronically: October 23, 2012
MathSciNet review: 3112820
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the initial boundary value problem for a class of fourth order strongly damped nonlinear wave equations $u_{tt}-\Delta u+ \Delta ^2 u-\alpha \Delta u_t=f(u)$. By introducing a family of potential wells we prove the existence of global weak solutions and global strong solutions under some weak growth conditions on $f(u)$. Furthermore we give the asymptotic behaviour of solutions.

References

Đ\hckudot{a}ng Điñh Áng and Alain Pham Ngoc Dinh, On the strongly damped wave equation: $u_{tt}-\Delta u-\Delta u_t+f(u)=0$, SIAM J. Math. Anal. 19 (1988), no. 6, 1409–1418. MR 965260, DOI https://doi.org/10.1137/0519103
José Arrieta, Alexandre N. Carvalho, and Jack K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations 17 (1992), no. 5-6, 841–866. MR 1177295, DOI https://doi.org/10.1080/03605309208820866
Veronica Belleri and Vittorino Pata, Attractors for semilinear strongly damped wave equations on $\Bbb R^3$, Discrete Contin. Dynam. Systems 7 (2001), no. 4, 719–735. MR 1849655, DOI https://doi.org/10.3934/dcds.2001.7.719
Alexandre N. Carvalho and Jan W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc. 66 (2002), no. 3, 443–463. MR 1939206, DOI https://doi.org/10.1017/S0004972700040296
Caisheng Chen and Lei Ren, Weak solution for a fourth-order nonlinear wave equation, J. Southeast Univ. (English Ed.) 21 (2005), no. 3, 369–374 (English, with English and Chinese summaries). MR 2174130

J.A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal. 63 (2005), 331–343.

Filippo Gazzola and Marco Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 2, 185–207. MR 2201151, DOI https://doi.org/10.1016/j.anihpc.2005.02.007

S. Gerbi and B. Said-Houari, Exponential decay for solutions to semilinear damped wave equation, arXiv:0812.3637v3.

Howard A. Levine, Patrizia Pucci, and James Serrin, Some remarks on global nonexistence for nonautonomous abstract evolution equations, Harmonic analysis and nonlinear differential equations (Riverside, CA, 1995) Contemp. Math., vol. 208, Amer. Math. Soc., Providence, RI, 1997, pp. 253–263. MR 1467010, DOI https://doi.org/10.1090/conm/208/02743

Howard A. Levine and James Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal. 137 (1997), no. 4, 341–361. MR 1463799, DOI https://doi.org/10.1007/s002050050032

Qun Lin, Yong Hong Wu, and Shaoyong Lai, On global solution of an initial boundary value problem for a class of damped nonlinear equations, Nonlinear Anal. 69 (2008), no. 12, 4340–4351. MR 2467236, DOI https://doi.org/10.1016/j.na.2007.10.057

J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.

Ya Cheng Liu, Feng Wang, and Da Cheng Liu, Strongly damped nonlinear wave equations in arbitrary dimensions. I. Initial-boundary value problems, Math. Appl. (Wuhan) 8 (1995), no. 3, 262–266 (Chinese, with English and Chinese summaries). MR 1359437

Ya Cheng Liu and Da Cheng Liu, The initial-boundary value problem for the equation $u_{tt}-a\Delta u_t-\Delta u=f(u)$, J. Huazhong Univ. Sci. Tech. 16 (1988), no. 6, 169–173 (Chinese, with English summary). MR 1120240

Ya Cheng Liu and Ping Liu, On potential wells and application to strongly damped nonlinear wave equations, Acta Math. Appl. Sin. 27 (2004), no. 4, 710–722 (Chinese, with English and Chinese summaries). MR 2126318

Yacheng Liu and Runzhang Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl. 331 (2007), no. 1, 585–607. MR 2306025, DOI https://doi.org/10.1016/j.jmaa.2006.09.010

Masahito Ohta, Remarks on blowup of solutions for nonlinear evolution equations of second order, Adv. Math. Sci. Appl. 8 (1998), no. 2, 901–910. MR 1657188

Kosuke Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci. 20 (1997), no. 2, 151–177. MR 1430038, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819970125%2920%3A2%3C151%3A%3AAID-MMA851%3E3.3.CO%3B2-S

Vittorino Pata and Marco Squassina, On the strongly damped wave equation, Comm. Math. Phys. 253 (2005), no. 3, 511–533. MR 2116726, DOI https://doi.org/10.1007/s00220-004-1233-1

Vittorino Pata and Sergey Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), no. 7, 1495–1506. MR 2229785, DOI https://doi.org/10.1088/0951-7715/19/7/001

Patrizia Pucci and James Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations 150 (1998), no. 1, 203–214. MR 1660250, DOI https://doi.org/10.1006/jdeq.1998.3477

Enzo Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999), no. 2, 155–182. MR 1719145, DOI https://doi.org/10.1007/s002050050171

G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canadian J. Math. 32 (1980), no. 3, 631–643. MR 586981, DOI https://doi.org/10.4153/CJM-1980-049-5

Runzhang Xu and Yacheng Liu, Asymptotic behavior of solutions for initial-boundary value problems for strongly damped nonlinear wave equations, Nonlinear Anal. 69 (2008), no. 8, 2492–2495. MR 2446346, DOI https://doi.org/10.1016/j.na.2007.08.027