Global existence of weak solutions to a class of higher-order nonlinear evolution equations

Li-ming Xiao; Cao Luo; Jie Liu; Li-ming Xiao; Cao Luo; Jie Liu

doi:10.3934/era.2024248

Electronic Research Archive

2024, Volume 32, Issue 9: 5357-5376. doi: 10.3934/era.2024248

Previous Article Next Article

Research article Special Issues

Global existence of weak solutions to a class of higher-order nonlinear evolution equations

1.
Teaching Department of Mathematics and Physics, Guangzhou Institute of Science and Technology, Guangzhou 510540, China
2.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China

Received: 14 April 2024 Revised: 23 August 2024 Accepted: 06 September 2024 Published: 19 September 2024

This paper deals with the initial boundary value problem for a class of n-dimensional higher-order nonlinear evolution equations that come from the viscoelastic mechanics and have no positive definite energy. Through the analysis of functionals containing higher-order energy of motion, a modified potential well with positive depth is constructed. Then, using the potential well method, and Galerkin method, it has been shown that when the initial data starts from the stable set, there exists a global weak solution to such an evolution problem.
- potential well method,
- higher-order $ n $-dimensional nonlinear evolution equations,
- initial boundary value problem,
- global weak solution,
- Galerkin method
Citation: Li-ming Xiao, Cao Luo, Jie Liu. Global existence of weak solutions to a class of higher-order nonlinear evolution equations[J]. Electronic Research Archive, 2024, 32(9): 5357-5376. doi: 10.3934/era.2024248

Related Papers:

Abstract

This paper deals with the initial boundary value problem for a class of n-dimensional higher-order nonlinear evolution equations that come from the viscoelastic mechanics and have no positive definite energy. Through the analysis of functionals containing higher-order energy of motion, a modified potential well with positive depth is constructed. Then, using the potential well method, and Galerkin method, it has been shown that when the initial data starts from the stable set, there exists a global weak solution to such an evolution problem.

References

[1]	J. L. Bogolubsky, Some examples of inelastic soliton interaction, Comput. Phys. Commun., 13 (1997), 149–155. https://doi.org/10.1016/0010-4655(77)90009-1 doi: 10.1016/0010-4655(77)90009-1
[2]	P. A. Clarkson, R. J. Leveque, R. A. Saxton, Solitary-wave interaction in elastic rods, Stud. Appl. Math., 75 (1986), 95–121. https://doi.org/10.1002/sapm198675295 doi: 10.1002/sapm198675295
[3]	M. Hayes, G. Saccomandi, Finite amplitude transverse waves in special incompressible viscoelastic solids, J. Elast. Phys. Sci. Solids, 59 (2000), 213–225. https://doi.org/10.1023/A:1011081920910 doi: 10.1023/A:1011081920910
[4]	Y. D. Shang, Initial boundary value problem of equation $u_tt-\Delta u-\Delta u_{t}-\Delta u_tt = f(u)$, Acta Math. Appl. Sin., Chinese edition, 23 (2000), 385–393.
[5]	R. Z. Xu, X. R. Zhao, J. H. Shen, Asymptotic behavior of solution for fourth order wave equation with dispersive and dissipative terms, Appl. Math. Mech., 29 (2008), 259–262. https://doi.org/10.1007/s10483-008-0213-y doi: 10.1007/s10483-008-0213-y
[6]	F. Gazzola, M. Squassina, Global solutions and finite time blowup for damped semilinear wave equations, Ann. Inst. Henri Poincare, 23 (2006), 185–207. https://doi.org/10.1016/j.anihpc.2005.02.007 doi: 10.1016/j.anihpc.2005.02.007
[7]	D. H. Sattinger, On global solutions of nonlinear hyperbolic equalitions, Arch. Ration. Mech. Anal., 30 (1968), 148–172. https://doi.org/10.1007/BF00250942 doi: 10.1007/BF00250942
[8]	W. Lian, R. Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613–632. https://doi.org/10.1515/anona-2020-0016 doi: 10.1515/anona-2020-0016
[9]	R. Z. Xu, Y. B. Yang, Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level, Int. J. Math., 23 (2012), 1250060. https://doi.org/10.1142/S0129167X12500607 doi: 10.1142/S0129167X12500607
[10]	P. G. Ciarlet, Mathematical Elasticity, Vol. II: theory of plates, 1997.
[11]	J. E. Lagnese, Boundary Stabilization of Thin Plates, Philadephia: SIAM, 1989. https://doi.org/10.1137/1.9781611970821
[12]	R. Z. Xu, X. C. Wang, Y. B. Yang, S. H. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503. https://doi.org/10.1063/1.5006728 doi: 10.1063/1.5006728
[13]	Q. Lin, Y. H. Wu, S. Y. Lai, On global solution of an initial boundary value problem for a class of damped nonlinear equations, Nonlinear Anal. Theory Methods Appl., 69 (2008), 4340–4351. https://doi.org/10.1016/j.na.2007.10.057 doi: 10.1016/j.na.2007.10.057
[14]	A. C. Biazutti, On a nonlinear evolution equation and its applications, Nonlinear Anal. Theory Methods Appl., 24 (1995), 1221–1234. https://doi.org/10.1016/0362-546X(94)00193-L doi: 10.1016/0362-546X(94)00193-L
[15]	J. A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal. Theory Methods Appl., 63 (2005), e331–e343. https://doi.org/10.1016/j.na.2005.02.108 doi: 10.1016/j.na.2005.02.108
[16]	Y. C. Liu, R. Z. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585–607. https://doi.org/10.1016/j.jmaa.2006.09.010 doi: 10.1016/j.jmaa.2006.09.010
[17]	Y. C. Liu, R. Z. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differ. Equations, 244 (2008), 200–228. https://doi.org/10.1016/j.jde.2007.10.015 doi: 10.1016/j.jde.2007.10.015
[18]	Y. Wang, Y. Wang, On the initial-boundary problem for fourth order wave equations with damping, strain and source terms, J. Math. Anal. Appl., 405 (2013), 116–127. https://doi.org/10.1016/j.jmaa.2013.03.060 doi: 10.1016/j.jmaa.2013.03.060
[19]	W. Lian, V. D. Rǎdulescu, R. Z. Xu, Y. B. Yang, N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589–611. https://doi.org/10.1515/acv-2019-0039 doi: 10.1515/acv-2019-0039
[20]	T. Y. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications, 2$^{nd}$ edition, Beijing: Science Press, 2017. https://doi.org/10.1007/978-3-642-14643-5
[21]	L. E. Payne, D. H. Sattinger, Saddle points and instability on nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273–303. https://doi.org/10.1007/BF02761595 doi: 10.1007/BF02761595
[22]	P. Pucci, J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differ. Equations, 150 (1998), 203–214. https://doi.org/10.1006/jdeq.1998.3477 doi: 10.1006/jdeq.1998.3477
[23]	H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave eqaution of the form $Pu_tt = Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1–21. https://doi.org/10.1090/S0002-9947-1974-0344697-2 doi: 10.1090/S0002-9947-1974-0344697-2
[24]	H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138–146. https://doi.org/10.1137/0505015 doi: 10.1137/0505015
[25]	E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155–182. https://doi.org/10.1007/s002050050171 doi: 10.1007/s002050050171
[26]	X. H. Yang, H. X. Zhang, Q. Zhang, G. W. Yuan, Z. Q. Sheng, The finite volume scheme preserving maximum principle for two-dimensional time-fractional Fokker-Planck equations on distorted meshes, Appl. Math. Lett., 97 (2019), 99–106. https://doi.org/10.1016/j.aml.2019.05.030 doi: 10.1016/j.aml.2019.05.030
[27]	X. H. Yang, H. X. Zhang, The uniform l1 long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data, Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644 doi: 10.1016/j.aml.2021.107644
[28]	R. A. Adams, J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Springer, Amsterdam, 2003.
[29]	Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101–123. https://doi.org/10.1090/S0002-9947-1960-0111898-8 doi: 10.1090/S0002-9947-1960-0111898-8
[30]	M. Willem, Minimax Theorems, Birkhäuser Inc., Boston, 1996.
[31]	K. Yosida, Functional Analysis, Spring-Verlag, 1978. https://doi.org/10.1007/978-3-642-96439-8
[32]	G. B. Folland, Introduction to Partial Differential Equations, 2$^{nd}$ edition, Princeton University Press, 2020.
[33]	M. Yan, L. M. Xiao, The initial boundary value problems for a class of mutidimensional nonlinear pseudo-hyperbolic system of higher order, Adv. Appl. Math., 2 (2013), 20–33.
[34]	X. L. Fu, J. J. Fan, Nonlinear Differential Equations, Chinese edition, Beijing: Science Press, 2011.
[35]	J. L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Chinese edition, Sun Yat-Sen University Press, 1992.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)