December  2021, 29(6): 3833-3851. doi: 10.3934/era.2021064

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Yang Cao

Received  January 2021 Revised  July 2021 Published  December 2021 Early access  September 2021

Fund Project: The first author is supported by NSFC grant 11871134, 12171166

In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy $ J(u_0)\leq d $. When the initial energy $ J(u_0)>d $, we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.

Citation: Yang Cao, Qiuting Zhao. Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations. Electronic Research Archive, 2021, 29 (6) : 3833-3851. doi: 10.3934/era.2021064
References:
[1]

G. AutuoriP. Pucci and M. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

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C. BuddB. Dold and A. Stuart, Blow-up in a partial differential equation with conserved first integral, SIAM J. Appl. Math., 53 (1993), 718-742.  doi: 10.1137/0153036.  Google Scholar

[4]

Y. Cao and Q. Zhao, Asymptotic behavior of global solutions to a class of mixed pseudo-parabolic Kirchhoff equations, Appl. Math. Lett., 118 (2021), 107119, 6 pp. doi: 10.1016/j.aml.2021.107119.  Google Scholar

[5]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness, Nonlinear Dynam., 84 (2016), 35-50.  doi: 10.1007/s11071-015-2200-4.  Google Scholar

[6]

A. S. CarassoJ. G. Sanderson and J. M. Hyman, Digital removal of random media image degradations by solving the diffusion equation backwards in time, SIAM J. Numer. Anal., 15 (1978), 344-367.  doi: 10.1137/0715023.  Google Scholar

[7]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.  Google Scholar

[8]

S. ChenB. Zhang and X. Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9 (2020), 148-176.  doi: 10.1515/anona-2018-0147.  Google Scholar

[9]

Y. ChenS. Levine and M. Rao, Variable exponent linear growth functionals in image restoration, SIMA. J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.  Google Scholar

[10]

H. Ding and J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393-420.  doi: 10.1016/j.jmaa.2019.05.018.  Google Scholar

[11]

Y. Fu and M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl. Anal., 95 (2016), 524-544.  doi: 10.1080/00036811.2015.1022153.  Google Scholar

[12]

Y. HanW. GaoZ. Sun and H. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. with Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar

[13]

Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Comput. Math. Appl., 75 (2018), 3283-3297.  doi: 10.1016/j.camwa.2018.01.047.  Google Scholar

[14]

W. HeD. Qin and Q. Wu, Existence, multiplicity and nonexistence results for Kirchhoff type equations, Adv. Nonlinear Anal., 10 (2021), 616-635.  doi: 10.1515/anona-2020-0154.  Google Scholar

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W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.  Google Scholar

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Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.  Google Scholar

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Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.  Google Scholar

[21]

V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Commun. in Partial Differential Equations, 23 (1998), 457-486.  doi: 10.1080/03605309808821353.  Google Scholar

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L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[23]

C. Qu and W. Zhou, Asymptotic analysis for a pseudo-parabolic equation with nonstandard growth conditions, Appl. Anal., (2021). doi: 10.1080/00036811.2020.1869941.  Google Scholar

[24]

C. Qu and W. Zhou, Blow-up and extinction for a thin-film equation with initial-boundary value conditions, J. Math. Anal. Appl., 436 (2016), 796-809.  doi: 10.1016/j.jmaa.2015.11.075.  Google Scholar

[25]

K. R. Rajagopal and M. $R\overset{\circ}{u}\breve{z}i\breve{c}ka$, Mathematical modeling of electrorheological materials, Continu. Mech. Thermodyn., 13 (2001), 59-78.  doi: 10.1007/s001610100034.  Google Scholar

[26]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar

[27]

R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1975), 25-42.  doi: 10.1137/0506004.  Google Scholar

[28]

N. H. Tuan, On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5465-5494.  doi: 10.3934/dcdsb.2020354.  Google Scholar

[29]

X. WangY. ChenY. YangJ. Li and R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019), 475-499.  doi: 10.1016/j.na.2019.06.019.  Google Scholar

[30]

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

[31]

M. XiangV. D. Rǎdulescu and B. Zhang, Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.  Google Scholar

[32]

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

[33]

C. YangY. Cao and S. Zheng, Second critical exponent and life span for pseudo-parabolic equation, J. Differential Equations, 253 (2012), 3286-3303.  doi: 10.1016/j.jde.2012.09.001.  Google Scholar

[34]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675–710,877.  Google Scholar

[35]

V. V. Zhikov, On Lavrentiev's phenomenon, Russ. J. Math. Phys., 3 (1995), 249-269.   Google Scholar

[36]

V. V. Zhikov, Solvability of the three-dimensional thermistor problem, Proc. Stekolov Inst. Math., 261 (2008), 98-111.  doi: 10.1134/S0081543808020090.  Google Scholar

[37]

J. Zhou, Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth, Nonlinear Anal. Real World Appl., 48 (2019), 54-70.  doi: 10.1016/j.nonrwa.2019.01.001.  Google Scholar

show all references

References:
[1]

G. AutuoriP. Pucci and M. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[3]

C. BuddB. Dold and A. Stuart, Blow-up in a partial differential equation with conserved first integral, SIAM J. Appl. Math., 53 (1993), 718-742.  doi: 10.1137/0153036.  Google Scholar

[4]

Y. Cao and Q. Zhao, Asymptotic behavior of global solutions to a class of mixed pseudo-parabolic Kirchhoff equations, Appl. Math. Lett., 118 (2021), 107119, 6 pp. doi: 10.1016/j.aml.2021.107119.  Google Scholar

[5]

T. CaraballoM. Herrera-Cobos and P. Marín-Rubio, Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness, Nonlinear Dynam., 84 (2016), 35-50.  doi: 10.1007/s11071-015-2200-4.  Google Scholar

[6]

A. S. CarassoJ. G. Sanderson and J. M. Hyman, Digital removal of random media image degradations by solving the diffusion equation backwards in time, SIAM J. Numer. Anal., 15 (1978), 344-367.  doi: 10.1137/0715023.  Google Scholar

[7]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.  Google Scholar

[8]

S. ChenB. Zhang and X. Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9 (2020), 148-176.  doi: 10.1515/anona-2018-0147.  Google Scholar

[9]

Y. ChenS. Levine and M. Rao, Variable exponent linear growth functionals in image restoration, SIMA. J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.  Google Scholar

[10]

H. Ding and J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393-420.  doi: 10.1016/j.jmaa.2019.05.018.  Google Scholar

[11]

Y. Fu and M. Xiang, Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent, Appl. Anal., 95 (2016), 524-544.  doi: 10.1080/00036811.2015.1022153.  Google Scholar

[12]

Y. HanW. GaoZ. Sun and H. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. with Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar

[13]

Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Comput. Math. Appl., 75 (2018), 3283-3297.  doi: 10.1016/j.camwa.2018.01.047.  Google Scholar

[14]

W. HeD. Qin and Q. Wu, Existence, multiplicity and nonexistence results for Kirchhoff type equations, Adv. Nonlinear Anal., 10 (2021), 616-635.  doi: 10.1515/anona-2020-0154.  Google Scholar

[15]

M. Jazar and R. Kiwan, Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 215-218.  doi: 10.1016/j.anihpc.2006.12.002.  Google Scholar

[16]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[17]

J. Li and Y. Han, Global existence and finite time blow-up of solutions to a nonlocal $p$-Laplace equation, Math. Model. Anal., 24 (2019), 195-217.  doi: 10.3846/mma.2019.014.  Google Scholar

[18]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.  Google Scholar

[19]

Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.  Google Scholar

[20]

Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.  Google Scholar

[21]

V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Commun. in Partial Differential Equations, 23 (1998), 457-486.  doi: 10.1080/03605309808821353.  Google Scholar

[22]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[23]

C. Qu and W. Zhou, Asymptotic analysis for a pseudo-parabolic equation with nonstandard growth conditions, Appl. Anal., (2021). doi: 10.1080/00036811.2020.1869941.  Google Scholar

[24]

C. Qu and W. Zhou, Blow-up and extinction for a thin-film equation with initial-boundary value conditions, J. Math. Anal. Appl., 436 (2016), 796-809.  doi: 10.1016/j.jmaa.2015.11.075.  Google Scholar

[25]

K. R. Rajagopal and M. $R\overset{\circ}{u}\breve{z}i\breve{c}ka$, Mathematical modeling of electrorheological materials, Continu. Mech. Thermodyn., 13 (2001), 59-78.  doi: 10.1007/s001610100034.  Google Scholar

[26]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar

[27]

R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1975), 25-42.  doi: 10.1137/0506004.  Google Scholar

[28]

N. H. Tuan, On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5465-5494.  doi: 10.3934/dcdsb.2020354.  Google Scholar

[29]

X. WangY. ChenY. YangJ. Li and R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019), 475-499.  doi: 10.1016/j.na.2019.06.019.  Google Scholar

[30]

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

[31]

M. XiangV. D. Rǎdulescu and B. Zhang, Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.  Google Scholar

[32]

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

[33]

C. YangY. Cao and S. Zheng, Second critical exponent and life span for pseudo-parabolic equation, J. Differential Equations, 253 (2012), 3286-3303.  doi: 10.1016/j.jde.2012.09.001.  Google Scholar

[34]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675–710,877.  Google Scholar

[35]

V. V. Zhikov, On Lavrentiev's phenomenon, Russ. J. Math. Phys., 3 (1995), 249-269.   Google Scholar

[36]

V. V. Zhikov, Solvability of the three-dimensional thermistor problem, Proc. Stekolov Inst. Math., 261 (2008), 98-111.  doi: 10.1134/S0081543808020090.  Google Scholar

[37]

J. Zhou, Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth, Nonlinear Anal. Real World Appl., 48 (2019), 54-70.  doi: 10.1016/j.nonrwa.2019.01.001.  Google Scholar

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