# American Institute of Mathematical Sciences

March  2022, 21(3): 797-816. doi: 10.3934/cpaa.2021199

## Asymptotic and quenching behaviors of semilinear parabolic systems with singular nonlinearities

 1 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China 2 School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

*Corresponding author

Received  August 2021 Revised  October 2021 Published  March 2022 Early access  December 2021

Fund Project: The corresponding author is sponsored by "Chenguang Program" supported by Shanghai Educational Development Foundation and Shanghai Municipal Education Commission [grant number: 13CG20]; NSFC [grant number: 11431005]; and STCSM [grant number: 18dz2271000]

In this paper, we consider a family of parabolic systems with singular nonlinearities. We study the classification of global existence and quenching of solutions according to parameters and initial data. Furthermore, the rate of the convergence of the global solutions to the minimal steady state is given. Due to the lack of variational characterization of the first eigenvalue to the linearized elliptic problem associated with our parabolic system, some new ideas and techniques are introduced.

Citation: Qi Wang, Yanyan Zhang. Asymptotic and quenching behaviors of semilinear parabolic systems with singular nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (3) : 797-816. doi: 10.3934/cpaa.2021199
##### References:
 [1] Q. Y. Dai and Y. G. Gu, Quenching phenomena for systems of semilinear parabolic equations, I, Syst. Sci. Math. Sci., 10 (1997), 361-371. [2] J. M. do Ó and R. Clemente, On lane-emden systems with singular nonlinearities and applications to MEMS, Adv. Nonlinear Stud., 18 (2018), 41-53.  doi: 10.1515/ans-2017-6024. [3] P. Esposito, N. Ghoussoub and Y. J. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, New York, Courant Lect. Notes Math., Courant Institute of Mathematical Sciences, New York University, 2010. doi: 10.1090/cln/020. [4] S. Filippas and J. S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quart. Appl. Math., 51 (1993), 713-729.  doi: 10.1090/qam/1247436. [5] H. Fujita, On the nonlinear equations ${\Delta} u+e^u = 0$ and ${\partial v}/{\partial t} = {\Delta} v+e^v$, Bull. Amer. Math. Soc., 75 (1969), 132-135.  doi: 10.1090/S0002-9904-1969-12175-0. [6] Y. J. Guo, Y. Y. Zhang and F. Zhou, Singular behavior of an electrostatic-elastic membrane system with an external pressure, Nonlinear Anal., 190 (2020), 111611, 29 pp. doi: 10.1016/j.na.2019.111611.. [7] H. Kawarada, On solutions of initial-boundary problem for $u_t = u_xx+1/(1-u)$, Publ. Res. Inst. Math. Sci., 10 (1975), 729-736. [8] N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, On the quenching behaviour of a semilinear wave equation modelling MEMS technology, Discret. Contin. Dynam. Syst. A, 35 (2015), 1009-1037. [9] N. I. Kavallaris and T. Suzuki, Non-Local Partial Differential Equations for Engineering and Biology, Mathematical Modeling and Analysis, Mathematics for Industry, Springer Nature, 2018. [10] Z. Jia, Z. D. Yang and C. Y. Wang, Non-simultaneous quenching in a semi-linear parabolic system with multi-singular reaction terms, Electron. J. Differ. Equ., 100 (2019), 13 pp. [11] J. Y. Li and C. C. Liang, Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS, Discret. Contin. Dyn. Syst., 36 (2016), 833-849.  doi: 10.3934/dcds.2016.36.833. [12] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248. [13] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, New York, Plenum Press, 1992. [14] H. J. Pei and Z. P. Li, Quenching for a parabolic system with general singular terms, J. Nonlinear Sci. Appl., 9 (2016), 5281-5290.  doi: 10.22436/jnsa.009.08.14. [15] M. H. Protter and H. Weinberger, Maximum Principles in Differential Equations, Englewood Cliffs, New Jersey, Prentice-Hall, 1967. doi: 10.1007/978-1-4612-5282-5. [16] Q. Wang, On some touchdown behaviours of the generalized MEMS device equation, Commun. Pure Appl. Anal., 15 (2016), 2447-2456. [17] Q. Wang, Quenching phenomenon for a parabolic MEMS equation, Chin. Ann. Math., 39 (2018), 129-144.  doi: 10.1007/s11401-018-1056-6. [18] Q. Wang, Dynamical solutions of singular parabolic equations modeling electrostatic MEMS, Nonlinear Differ. Equ. Appl., 22 (2015), 629-650.  doi: 10.1007/s00030-014-0298-6. [19] D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. Partial Differ. Equ., 37 (2010), 259-274.  doi: 10.1007/s00526-009-0262-1. [20] S. N. Zheng and W. Wang, Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system, Nonlinear Anal., 69 (2008), 2274-2285.  doi: 10.1016/j.na.2007.08.007. [21] S. M. Zheng, Nonliear Evolution Equations, Boca Raton, Florida, Chapman & Hall/CRC, 2004.

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##### References:
 [1] Q. Y. Dai and Y. G. Gu, Quenching phenomena for systems of semilinear parabolic equations, I, Syst. Sci. Math. Sci., 10 (1997), 361-371. [2] J. M. do Ó and R. Clemente, On lane-emden systems with singular nonlinearities and applications to MEMS, Adv. Nonlinear Stud., 18 (2018), 41-53.  doi: 10.1515/ans-2017-6024. [3] P. Esposito, N. Ghoussoub and Y. J. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, New York, Courant Lect. Notes Math., Courant Institute of Mathematical Sciences, New York University, 2010. doi: 10.1090/cln/020. [4] S. Filippas and J. S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quart. Appl. Math., 51 (1993), 713-729.  doi: 10.1090/qam/1247436. [5] H. Fujita, On the nonlinear equations ${\Delta} u+e^u = 0$ and ${\partial v}/{\partial t} = {\Delta} v+e^v$, Bull. Amer. Math. Soc., 75 (1969), 132-135.  doi: 10.1090/S0002-9904-1969-12175-0. [6] Y. J. Guo, Y. Y. Zhang and F. Zhou, Singular behavior of an electrostatic-elastic membrane system with an external pressure, Nonlinear Anal., 190 (2020), 111611, 29 pp. doi: 10.1016/j.na.2019.111611.. [7] H. Kawarada, On solutions of initial-boundary problem for $u_t = u_xx+1/(1-u)$, Publ. Res. Inst. Math. Sci., 10 (1975), 729-736. [8] N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, On the quenching behaviour of a semilinear wave equation modelling MEMS technology, Discret. Contin. Dynam. Syst. A, 35 (2015), 1009-1037. [9] N. I. Kavallaris and T. Suzuki, Non-Local Partial Differential Equations for Engineering and Biology, Mathematical Modeling and Analysis, Mathematics for Industry, Springer Nature, 2018. [10] Z. Jia, Z. D. Yang and C. Y. Wang, Non-simultaneous quenching in a semi-linear parabolic system with multi-singular reaction terms, Electron. J. Differ. Equ., 100 (2019), 13 pp. [11] J. Y. Li and C. C. Liang, Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS, Discret. Contin. Dyn. Syst., 36 (2016), 833-849.  doi: 10.3934/dcds.2016.36.833. [12] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248. [13] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, New York, Plenum Press, 1992. [14] H. J. Pei and Z. P. Li, Quenching for a parabolic system with general singular terms, J. Nonlinear Sci. Appl., 9 (2016), 5281-5290.  doi: 10.22436/jnsa.009.08.14. [15] M. H. Protter and H. Weinberger, Maximum Principles in Differential Equations, Englewood Cliffs, New Jersey, Prentice-Hall, 1967. doi: 10.1007/978-1-4612-5282-5. [16] Q. Wang, On some touchdown behaviours of the generalized MEMS device equation, Commun. Pure Appl. Anal., 15 (2016), 2447-2456. [17] Q. Wang, Quenching phenomenon for a parabolic MEMS equation, Chin. Ann. Math., 39 (2018), 129-144.  doi: 10.1007/s11401-018-1056-6. [18] Q. Wang, Dynamical solutions of singular parabolic equations modeling electrostatic MEMS, Nonlinear Differ. Equ. Appl., 22 (2015), 629-650.  doi: 10.1007/s00030-014-0298-6. [19] D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. Partial Differ. Equ., 37 (2010), 259-274.  doi: 10.1007/s00526-009-0262-1. [20] S. N. Zheng and W. Wang, Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system, Nonlinear Anal., 69 (2008), 2274-2285.  doi: 10.1016/j.na.2007.08.007. [21] S. M. Zheng, Nonliear Evolution Equations, Boca Raton, Florida, Chapman & Hall/CRC, 2004.
The critical curve $\Gamma$ in $( \lambda,\mu)$-plane
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