Nonexistence of Global Solutions to Semilinear Moore-Gibson-Thompson Equations With Space-Dependent Coefficients and Source Terms

OUYANG Baiping

doi:10.21656/1000-0887.420094

Volume 43 Issue 3

Mar. 2022

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OUYANG Baiping. Nonexistence of Global Solutions to Semilinear Moore-Gibson-Thompson Equations With Space-Dependent Coefficients and Source Terms[J]. Applied Mathematics and Mechanics, 2022, 43(3): 353-362. doi: 10.21656/1000-0887.420094

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Nonexistence of Global Solutions to Semilinear Moore-Gibson-Thompson Equations With Space-Dependent Coefficients and Source Terms

doi: 10.21656/1000-0887.420094

OUYANG Baiping

Guangzhou Huashang College, Guangzhou 511300, P.R.China

Received Date: 2021-04-13
Rev Recd Date: 2021-10-27

Available Online: 2021-11-09

Publish Date: 2022-03-08

Abstract

Abstract

Blow-up of solutions to semilinear Moore-Gibson-Thompson (MGT) equations with space-dependent coefficients and source terms was studied. Under subcritical conditions, through selection of suitable energy functionals and test functions, and with an iteration method and some differential inequality techniques, the nonexistence of global solutions to the Cauchy problem was obtained. Furthermore, the upper bound estimate of the solutions of the lifespan was derived.
- space-dependent coefficient and source term,
- Moore-Gibson-Thompson equation,
- blow-up

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References

[1]	ALVES M O, CAIXETA A H, SILVA M A J, et al. Moore-Gibson-Thompson equation with memory in a history framework: a semigroup approach[J]. Zeitschrift fur Angewandte Mathematik und Physik, 2018, 69(4): 106. doi: 10.1007/s00033-018-0999-5
[2]	CAIXETA A H, LASIECKA I, DOMINGOS CAVALCANTI V N. On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation[J]. Evolution Equations and Control Theory, 2016, 5(4): 661-676. doi: 10.3934/eect.2016024
[3]	DELL’ORO F, LASIECKA I, PATA V. A note on the Moore-Gibson-Thompson equation with memory of type Ⅱ[J]. Journal of Evolution Equations, 2020, 20(4): 1251-1268. doi: 10.1007/s00028-019-00554-0
[4]	DELL’ORO F, LASIECKA I, PATA V. The Moore-Gibson-Thompson equation with memory in the critical case[J]. Journal of Differential Equations, 2016, 261(7): 4188-4222. doi: 10.1016/j.jde.2016.06.025
[5]	DELL’ORO F, PATA V. On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity[J]. Applied Mathematics and Optimization, 2017, 76(3): 641-655. doi: 10.1007/s00245-016-9365-1
[6]	LASIECKA I. Global solvability of Moore-Gibson-Thompson equation with memory arising in nonlinear acoustics[J]. Journal of Evolution Equations, 2017, 17(1): 411-441. doi: 10.1007/s00028-016-0353-3
[7]	LASIECKA I, WANG X. Moore-Gibson-Thompson equation with memory, part Ⅰ: exponential decay of energy[J]. Zeitschrift fur Angewandte Mathematik und Physik, 2016, 67(2). DOI: 10.1007/s00033-015-0597-8.
[8]	PELLICER M, SAID-HOUARI B. Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound[J]. Applied Mathematics and Optimization, 2019, 80(2): 447-478. doi: 10.1007/s00245-017-9471-8
[9]	PELLICER M, SOLÁ-MORALES J. Optimal scalar products in the Moore-Gibson-Thompson equation[J]. Evolution Equations and Control Theory, 2019, 8(1): 203-220. doi: 10.3934/eect.2019011
[10]	KALTENBACHER B, MARCHAND R. Well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound[J]. Control and Cybernetics, 2011, 40(4): 971-988.
[11]	LINDBLAD H, SOGGE C D. Long-time existence for small amplitude semilinear wave equations[J]. American Journal of Mathematics, 1996, 118(5): 1047-1135. doi: 10.1353/ajm.1996.0042
[12]	TAKAMURA H, WAKASA K. The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions[J]. Journal of Differential Equations, 2011, 251(4/5): 1157-1171. doi: 10.1016/j.jde.2011.03.024
[13]	ZHOU Y. Cauchy problem for semilinear wave equations in four space dimensions with small initial data[J]. Journal of Partial Differential Equations, 1995, 8(2): 135-144.
[14]	ZHOU Y. Blow up of solutions to semilinear wave equations with critical exponent in high dimensions[J]. Chinese Annals of Mathematics (Series B), 2007, 28(2): 205-212. doi: 10.1007/s11401-005-0205-x
[15]	DI POMPONIO S, GEORGIEV V. Life-span of subcritical semilinear wave equation[J]. Asymptotic Analysis, 2001, 28(2): 91-114.
[16]	王虎生, 孙海霞. 一类非线性项的二维波动方程解的生命跨度研究[J]. 应用数学, 2020, 33(3): 620-633. (WANG Husheng, SUN Haixia. The life span of classical solutions to nonlinear wave equations with weighted function in two space dimensions[J]. Mathematica Applicata, 2020, 33(3): 620-633.(in Chinese)
[17]	CHEN W, PALMIERI A. Nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case[J]. Discrete and Continuous Dynamical Systems, 2020, 40(9): 5513-5540. doi: 10.3934/dcds.2020236
[18]	CHEN W, PALMIERI A. A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case[J]. Evolution Equations and Control Theory, 2020, 10(4): 673-687.
[19]	CHEN W. Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms[J]. Nonlinear Analysis, 2021, 202: 112160. doi: 10.1016/j.na.2020.112160
[20]	CHEN W, REISSIG M. Blow-up of solutions to Nakao’s problem via an iteration argument[J]. Journal of Differential Equations, 2021, 275: 733-756. doi: 10.1016/j.jde.2020.11.009
[21]	LAI N A, TAKAMURA H. Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey’s conjecture[J]. Differential and Integral Equations, 2019, 32(1/2): 37-48.
[22]	LAI N A, TAKAMURA H, WAKASA K. Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent[J]. Journal of Differential Equations, 2017, 263(9): 5377-5394. doi: 10.1016/j.jde.2017.06.017
[23]	PALMIERI A, TAKAMURA A. Blow-up for a weekly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities[J]. Nonlinear Analysis, 2019, 187: 467-492. doi: 10.1016/j.na.2019.06.016
[24]	YORDANOV B T, ZHANG Q S. Finite time blow up for critical wave equations in high dimensions[J]. Journal of Functional Analysis, 2006, 231(2): 361-374. doi: 10.1016/j.jfa.2005.03.012