doi: 10.3934/era.2021073
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Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

1. 

Department of Electronic Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China

2. 

College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China

* Corresponding author: liwenke@hrbeu.edu.cn

Received  July 2021 Revised  August 2021 Early access September 2021

Fund Project: The first author is supported by Jiangsu key R & D plan(BE2018007)

In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.

Citation: Xiaoqiang Dai, Wenke Li. Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem. Electronic Research Archive, doi: 10.3934/era.2021073
References:
[1]

G. Andrews, On the existence of solutions to the equation $u_tt-u_xxt = \sigma(u_x)_x$, J. Differential Equations, 35 (1980), 200-231.  doi: 10.1016/0022-0396(80)90040-6.  Google Scholar

[2]

G. Andrews and J. M. Ball, Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations, 44 (1982), 306-341.  doi: 10.1016/0022-0396(82)90019-5.  Google Scholar

[3]

D. D. Áng and A. Pham Ngoc Dinh, Strong solutions of a quasilinear wave equation with nonlinear damping, SIAM J. Math. Anal., 19 (1988), 337-347.  doi: 10.1137/0519024.  Google Scholar

[4]

Y. Cao, Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities, Elec. Res. Arch., 28 (2020), 27-46.  doi: 10.3934/era.2020003.  Google Scholar

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J. Clements, Existence theorems for a quasilinear evolution equation, SIAM. J. Appl. Math., 226 (1974), 745-752.  doi: 10.1137/0126066.  Google Scholar

[6]

J. C. Clements, On the existence and uniqueness of solutions of the equation $ {u_{tt}} - \frac{\partial }{{\partial {x_i}}}{\sigma _i}({u_x}_i) - {\Delta _N}{u_t} = f $, Canad. Math. Bull., 18 (1975), 181-187.  doi: 10.4153/CMB-1975-036-1.  Google Scholar

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C. M. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional visco-elasticity, J. Differential Equations, 6 (1969), 71-86.  doi: 10.1016/0022-0396(69)90118-1.  Google Scholar

[8]

X. DaiC. YangS. HuangY. Tao and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Elec. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.  Google Scholar

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P. L. Davis, A quasi-linear hyperbolic and related third-order equation, J. Math. Anal. Appl., 51 (1975), 596-606.  doi: 10.1016/0022-247X(75)90110-9.  Google Scholar

[10]

H. Engler, Strong solutions for strongly damped quasilinear wave equations, Contemp. Math., 64 (1987), 219-237.  doi: 10.1090/conm/064/881465.  Google Scholar

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J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Elec. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.  Google Scholar

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J. M. Greenberg and R. C. MacCamy, On the exponential stability of solutions of $E(u_x)u_xx+\lambda u_xtx = \rho u_tt$, J. Math. Anal. Appl., 31 (1970), 406-417.  doi: 10.1016/0022-247X(70)90034-X.  Google Scholar

[13]

J. M. GreenbergR. C. MacCamy and V. J. Mizel, On the existence, uniqueness and stability of the equation $\sigma(u_x)u_xx +\lambda u_xxt = \rho_0u_tt$, J. Math. Mech., 17 (1968), 707-728.   Google Scholar

[14]

H. A. Levine, Instablity and nonexistence of global solutions to nonlinear wave equations of the form $Pu = Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

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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

[16]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111.  Google Scholar

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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.  Google Scholar

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G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Elec. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar

[19]

Y. Liu and W. Li, A family of potential wells for a wave equation, Elec. Res. Arch., 28 (2020), 807-820.  doi: 10.3934/era.2020041.  Google Scholar

[20]

Y. Liu and J. Zhao, Multidimensional viscoelasticity equations with nonlinear damping and source terms, Nonlinear Anal., 56 (2004), 851-865.  doi: 10.1016/j.na.2003.07.021.  Google Scholar

[21]

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.  Google Scholar

[22]

R. XuW. 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.  Google Scholar

[23]

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.  Google Scholar

[24]

Z. Yang, Initial-boundary value problem and Cauchy problem for a quasilinear evolution equation, Acta Math. Sci., 19 (1999), 487-496.  doi: 10.1016/S0252-9602(17)30535-0.  Google Scholar

[25]

M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894. doi: 10.1515/anona-2020-0031.  Google Scholar

[26]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Elec. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.  Google Scholar

show all references

References:
[1]

G. Andrews, On the existence of solutions to the equation $u_tt-u_xxt = \sigma(u_x)_x$, J. Differential Equations, 35 (1980), 200-231.  doi: 10.1016/0022-0396(80)90040-6.  Google Scholar

[2]

G. Andrews and J. M. Ball, Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations, 44 (1982), 306-341.  doi: 10.1016/0022-0396(82)90019-5.  Google Scholar

[3]

D. D. Áng and A. Pham Ngoc Dinh, Strong solutions of a quasilinear wave equation with nonlinear damping, SIAM J. Math. Anal., 19 (1988), 337-347.  doi: 10.1137/0519024.  Google Scholar

[4]

Y. Cao, Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities, Elec. Res. Arch., 28 (2020), 27-46.  doi: 10.3934/era.2020003.  Google Scholar

[5]

J. Clements, Existence theorems for a quasilinear evolution equation, SIAM. J. Appl. Math., 226 (1974), 745-752.  doi: 10.1137/0126066.  Google Scholar

[6]

J. C. Clements, On the existence and uniqueness of solutions of the equation $ {u_{tt}} - \frac{\partial }{{\partial {x_i}}}{\sigma _i}({u_x}_i) - {\Delta _N}{u_t} = f $, Canad. Math. Bull., 18 (1975), 181-187.  doi: 10.4153/CMB-1975-036-1.  Google Scholar

[7]

C. M. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional visco-elasticity, J. Differential Equations, 6 (1969), 71-86.  doi: 10.1016/0022-0396(69)90118-1.  Google Scholar

[8]

X. DaiC. YangS. HuangY. Tao and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Elec. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.  Google Scholar

[9]

P. L. Davis, A quasi-linear hyperbolic and related third-order equation, J. Math. Anal. Appl., 51 (1975), 596-606.  doi: 10.1016/0022-247X(75)90110-9.  Google Scholar

[10]

H. Engler, Strong solutions for strongly damped quasilinear wave equations, Contemp. Math., 64 (1987), 219-237.  doi: 10.1090/conm/064/881465.  Google Scholar

[11]

J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Elec. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.  Google Scholar

[12]

J. M. Greenberg and R. C. MacCamy, On the exponential stability of solutions of $E(u_x)u_xx+\lambda u_xtx = \rho u_tt$, J. Math. Anal. Appl., 31 (1970), 406-417.  doi: 10.1016/0022-247X(70)90034-X.  Google Scholar

[13]

J. M. GreenbergR. C. MacCamy and V. J. Mizel, On the existence, uniqueness and stability of the equation $\sigma(u_x)u_xx +\lambda u_xxt = \rho_0u_tt$, J. Math. Mech., 17 (1968), 707-728.   Google Scholar

[14]

H. A. Levine, Instablity and nonexistence of global solutions to nonlinear wave equations of the form $Pu = Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

[15]

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

[16]

W. LianM. S. Ahmed and R. Xu, Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity, Opuscula Math., 40 (2020), 111-130.  doi: 10.7494/OpMath.2020.40.1.111.  Google Scholar

[17]

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.  Google Scholar

[18]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Elec. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar

[19]

Y. Liu and W. Li, A family of potential wells for a wave equation, Elec. Res. Arch., 28 (2020), 807-820.  doi: 10.3934/era.2020041.  Google Scholar

[20]

Y. Liu and J. Zhao, Multidimensional viscoelasticity equations with nonlinear damping and source terms, Nonlinear Anal., 56 (2004), 851-865.  doi: 10.1016/j.na.2003.07.021.  Google Scholar

[21]

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.  Google Scholar

[22]

R. XuW. 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.  Google Scholar

[23]

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.  Google Scholar

[24]

Z. Yang, Initial-boundary value problem and Cauchy problem for a quasilinear evolution equation, Acta Math. Sci., 19 (1999), 487-496.  doi: 10.1016/S0252-9602(17)30535-0.  Google Scholar

[25]

M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894. doi: 10.1515/anona-2020-0031.  Google Scholar

[26]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Elec. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.  Google Scholar

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