### Electronic Research Archive

2021, Issue 6: 4087-4098. doi: 10.3934/era.2021073

# Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

• Received: 01 July 2021 Revised: 01 August 2021 Published: 22 September 2021
• Primary: 35L05

• 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[J]. Electronic Research Archive, 2021, 29(6): 4087-4098. doi: 10.3934/era.2021073

### Related Papers:

• 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.

 [1] On the existence of solutions to the equation $u_tt-u_xxt = \sigma(u_x)_x$. J. Differential Equations (1980) 35: 200-231. [2] Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity. J. Differential Equations (1982) 44: 306-341. [3] Strong solutions of a quasilinear wave equation with nonlinear damping. SIAM J. Math. Anal. (1988) 19: 337-347. [4] Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities. Elec. Res. Arch. (2020) 28: 27-46. [5] Existence theorems for a quasilinear evolution equation. SIAM. J. Appl. Math. (1974) 226: 745-752. [6] 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. (1975) 18: 181-187. [7] The mixed initial-boundary value problem for the equations of nonlinear one-dimensional visco-elasticity. J. Differential Equations (1969) 6: 71-86. [8] Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Elec. Res. Arch. (2020) 28: 91-102. [9] A quasi-linear hyperbolic and related third-order equation. J. Math. Anal. Appl. (1975) 51: 596-606. [10] Strong solutions for strongly damped quasilinear wave equations. Contemp. Math. (1987) 64: 219-237. [11] Blow-up in damped abstract nonlinear equations. Elec. Res. Arch. (2020) 28: 347-367. [12] On the exponential stability of solutions of $E(u_x)u_xx+\lambda u_xtx = \rho u_tt$. J. Math. Anal. Appl. (1970) 31: 406-417. [13] On the existence, uniqueness and stability of the equation $\sigma(u_x)u_xx +\lambda u_xxt = \rho_0u_tt$. J. Math. Mech. (1968) 17: 707-728. [14] Instablity and nonexistence of global solutions to nonlinear wave equations of the form $Pu = Au+F(u)$. Trans. Amer. Math. Soc. (1974) 192: 1-21. [15] Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. (1974) 5: 138-146. [16] Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity. Opuscula Math. (2020) 40: 111-130. [17] Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. (2020) 9: 613-632. [18] The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Elec. Res. Arch. (2020) 28: 263-289. [19] A family of potential wells for a wave equation. Elec. Res. Arch. (2020) 28: 807-820. [20] Multidimensional viscoelasticity equations with nonlinear damping and source terms. Nonlinear Anal. (2004) 56: 851-865. [21] Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv. Nonlinear Anal. (2021) 10: 261-288. [22] Global well-posedness of coupled parabolic systems. Sci. China Math. (2020) 63: 321-356. [23] Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. (2013) 264: 2732-2763. [24] Initial-boundary value problem and Cauchy problem for a quasilinear evolution equation. Acta Math. Sci. (1999) 19: 487-496. [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 [26] Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Elec. Res. Arch. (2020) 28: 369-381.
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