doi: 10.3934/cpaa.2021199
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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 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 & Applied Analysis, 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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..  Google Scholar

[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.   Google Scholar

[8]

N. I. KavallarisA. A. LaceyC. 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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[12]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248.  Google Scholar

[13] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, New York, Plenum Press, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[16]

Q. Wang, On some touchdown behaviours of the generalized MEMS device equation, Commun. Pure Appl. Anal., 15 (2016), 2447-2456.   Google Scholar

[17]

Q. Wang, Quenching phenomenon for a parabolic MEMS equation, Chin. Ann. Math., 39 (2018), 129-144.  doi: 10.1007/s11401-018-1056-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. M. Zheng, Nonliear Evolution Equations, Boca Raton, Florida, Chapman & Hall/CRC, 2004. Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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..  Google Scholar

[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.   Google Scholar

[8]

N. I. KavallarisA. A. LaceyC. 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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[12]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248.  Google Scholar

[13] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, New York, Plenum Press, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[16]

Q. Wang, On some touchdown behaviours of the generalized MEMS device equation, Commun. Pure Appl. Anal., 15 (2016), 2447-2456.   Google Scholar

[17]

Q. Wang, Quenching phenomenon for a parabolic MEMS equation, Chin. Ann. Math., 39 (2018), 129-144.  doi: 10.1007/s11401-018-1056-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. M. Zheng, Nonliear Evolution Equations, Boca Raton, Florida, Chapman & Hall/CRC, 2004. Google Scholar

Figure 1.  The critical curve $ \Gamma $ in $ ( \lambda,\mu) $-plane
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