Blowup phenomena for a fourth order wave equation at high initial energy level

Tiantian Pang; Lanlan Qin; Xingchang Wang

doi:10.3934/dcdss.2023178

Article Contents

2023, Volume 16, Issue 11: 3258-3269. Doi: 10.3934/dcdss.2023178

This issue Previous Article On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes Next Article The Brézis–Nirenberg equation for the $ {(m,p)} $ Laplacian in the whole space

Blowup phenomena for a fourth order wave equation at high initial energy level

1.
College of Mathematical Sciences, Harbin Engineering University, 150001, China
2.
Department of Mathematics, University of Craiova, 200585, Craiova, Romania
3.
China-Romania Research Center in Applied Mathematics, Craiova, Romania

^*Corresponding author: Xingchang Wang
^*Corresponding author: Xingchang Wang

This paper is dedicated to Romanian mathematician Professor Vicentiu D. Rǎdulescu on the occasion of his 65th birthday.

Received: April 2023

Revised: August 2023

Early access: September 25, 2023

Published online: September 25, 2023

The first author is supported by the financial support of the China Scholarship Council (No. 202106680001) and the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (No. 3072021CF2405). The third author is supported by the National Natural Science Foundation of China (No. 12271122) and the Fundamental Research Funds for the Central Universities.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

This paper deals with the initial boundary value problem of a class of fourth order wave equations with nonlinear strain and linear weak damping terms. By establishing the invariance of the unstable set and a delicate auxiliary function, we give a sufficient condition such that the solution blows up in a finite time at high initial energy level.

Keywords:

Mathematics Subject Classification: Primary: 35L05, 35L35; Secondary: 35A01.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	L. An and A. Peirce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155. doi: 10.1137/S0036139993255327.
[2]	G. Chen and Z. Yang, Existence and non-existence of global solutions for a class of nonlinear wave equations, Math. Meth. Appl. Sci., 23 (2000), 615-631.
[3]	S. Chen, R. Xu and C. Yang, Improved blowup time estimates for fourth-order damped wave equations with strain term and arbitrary positive initial energy, Electron. J. Differential Equations, 2022 (2022), Paper No. 70, 13 pp. doi: 10.58997/ejde.2022.70.
[4]	X. Dai, J. Han, Q. Lin and X. Tian, Anomalous pseudo-parabolic Kirchhoff-type dynamical model, Adv. Nonlinear Anal., 11 (2022), 503-534. doi: 10.1515/anona-2021-0207.
[5]	J. A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal., 63 (2005), 331-343. doi: 10.1016/j.na.2005.02.108.
[6]	F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207. doi: 10.1016/j.anihpc.2005.02.007.
[7]	J. Han, R. Xu and Y. Yang, Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation, Asymptot. Anal., 122 (2021), 349-369. doi: 10.3233/ASY-201621.
[8]	H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. doi: 10.2307/1996814.
[9]	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.
[10]	W. Lian, V. D. Rǎdulescu, R. Xu, Y. Yang and N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589-611. doi: 10.1515/acv-2019-0039.
[11]	W. Lian, J. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959. doi: 10.1016/j.jde.2020.03.047.
[12]	W. Lian and R. 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. doi: 10.1515/anona-2020-0016.
[13]	Y. Liu and R. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228. doi: 10.1016/j.jde.2007.10.015.
[14]	Y. Liu and R. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607. doi: 10.1016/j.jmaa.2006.09.010.
[15]	N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovs, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.
[16]	X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288. doi: 10.1515/anona-2020-0141.
[17]	R. Xu, W. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356. doi: 10.1007/s11425-017-9280-x.
[18]	R. Xu, Q. Lin, S. Chen, G. Wen and W. Lian, Difficulties in obtaining finite time blowup for fourth-order semilinear Schrödinger equations in the variational method frame, Electron, J. Differential Equations, (2019), Paper No. 83, 22 pp.
[19]	R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763. doi: 10.1016/j.jfa.2013.03.010.
[20]	Z. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540. doi: 10.1016/S0022-0396(02)00042-6.
[21]	H. Zhang and G. Chen, Potential well method for a class of nonlinear wave equations of fourth-order, Acta Math. Sci. Ser. A, 23 (2003), 758-768.