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December  2021, 14(12): 4337-4366. doi: 10.3934/dcdss.2021121

## Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity

 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author

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

In this paper, we study the fractional pseudo-parabolic equations $u_{t} + \left(-\Delta\right)^{s} u + \left(-\Delta\right)^{s} u_{t} = u\log \left| u \right|$. Firstly, we recall the relationship between the fractional Laplace operator $\left(-\Delta\right)^{s}$ and the fractional Sobolev space $H^{s}$ and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of weak solution: for the low initial energy case (i.e., $J(u_{0}) < d$), the solution is global in time with $I(u_{0}) >0$ or $\Vert u_{0}\Vert_{{X_{0}(\Omega)}} = 0$ and blows up at $+\infty$ with $I(u_{0}) < 0$; for the critical initial energy case (i.e., $J(u_{0}) = d$), the solution is global in time with $I(u_{0}) \geq0$ and blows up at $+\infty$ with $I(u_{0}) < 0$. The decay estimate of the energy functional for the global solution is also given.

Citation: Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121
##### References:
 [1] A. Alsaedi, B. Ahmad, M. Kirane and B. T. Torebek, Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal., 10 (2021), 952-971.  doi: 10.1515/anona-2020-0153. [2] D. Applebaum, Lévy processes–-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [3] D. Applebaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [4] V. V. Au et al., On a final value problem for a nonlinear fractional pseudo-parabolic equation, Electron. Res. Arch., 29 (2021), 1709-1734.  doi: 10.3934/era.2020088. [5] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [6] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996. [7] G. M. Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397. [8] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20, Springer, 2016. doi: 10.1007/978-3-319-28739-3. [9] H. Chen, P. Luo and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030. [10] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038. [11] 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. [12] A. Cotsiolis and N. K. Tavoularis, On logarithmic Sobolev inequalities for higher order fractional derivatives, C. R. Math. Acad. Sci. Paris, 340 (2005), 205-208.  doi: 10.1016/j.crma.2004.11.030. [13] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017. [14] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408. [15] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [16] H. Di, Y. Shang and X. Peng, Blow-up phenomena for a pseudo-parabolic equation with variable exponents, Appl. Math. Lett., 64 (2017), 67-73.  doi: 10.1016/j.aml.2016.08.013. [17] H. Di, Y. Shang and X. Zheng, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 781-801.  doi: 10.3934/dcdsb.2016.21.781. [18] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 15, Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8. [19] A. A. Dubkov, B. Spagnolo and V. V. Uchaikin, Lévy flight superdiffusion: An introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649-2672.  doi: 10.1142/S0218127408021877. [20] Y. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 70, 17 pp. doi: 10.14232/ejqtde.2016.1.70. [21] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688. [22] N. E. Humphries et al., Environmental context explains Levy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069.  doi: 10.1038/nature09116. [23] J. D. Kečkić and P. M. Vasić, Some inequalities for the gamma function, Publ. Inst. Math., (Beograd) (N.S.), 11 (1971), 107–114. [24] M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources, Differ. Equ., 42 (2006), 431-443.  doi: 10.1134/S001226610603013X. [25] G. Li, J. Yu and W. Liu, Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 8 (2017), 629–660. doi: 10.1007/s11868-017-0216-x. [26] M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066. [27] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016. [28] W. Liu, Y. Sun and G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term, Topol. Methods Nonlinear Anal., 49 (2017), 299-323.  doi: 10.12775/tmna.2016.077. [29] W. Liu and J. Yu, A note on blow-up of solution for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 274 (2018), 1276-1283.  doi: 10.1016/j.jfa.2018.01.005. [30] W. Liu, B. Zhu and G. Li, Upper and lower bounds for the blow-up time for a viscoelastic wave equation with dynamic boundary conditions, Quaest. Math., 43 (2020), 999-1017.  doi: 10.2989/16073606.2019.1595768. [31] W. Liu and H. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 67, 35 pp. doi: 10.1007/s00030-017-0491-5. [32] Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171. [33] 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. [34] A. Massaccesi and E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, J. Math. Biol., 74 (2017), 113-147.  doi: 10.1007/s00285-016-1019-z. [35] S. A. Messaoudi, B. Said-Houari and N. Tatar, Global existence and asymptotic behavior for a fractional differential equation, Appl. Math. Comput., 188 (2007), 1955-1962.  doi: 10.1016/j.amc.2006.11.105. [36] R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161–R208. doi: 10.1088/0305-4470/37/31/R01. [37] V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3. [38] N. Pan, P. Pucci and B. Zhang, Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.  doi: 10.1007/s00028-017-0406-2. [39] 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. [40] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942. [41] J.-C. Saut and Y. Wang, Long time behavior of the fractional Korteweg–de Vries equation with cubic nonlinearity, Discrete Contin. Dyn. Syst., 41 (2021), 1133-1155.  doi: 10.3934/dcds.2020312. [42] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032. [43] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105. [44] D. W. Sims et al., Scaling laws of marine predator search behavior, Nature, 451 (2008), 1098-1102.  doi: 10.1038/nature06518. [45] D. Stan, F. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with nonlocal pressure, J. Differential Equations, 260 (2016), 1154-1199.  doi: 10.1016/j.jde.2015.09.023. [46] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA No., 49 (2009), 33-44. [47] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857. [48] J. L. Vázquez, Asymptotic behaviour for the fractional heat equation in the Euclidean space, Complex Var. Elliptic Equ., 63 (2018), 1216-1231.  doi: 10.1080/17476933.2017.1393807. [49] M. Xiang, D. Vicentiu 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. [50] 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. [51] C. Zhang, F. Li and J. Duan, Long-time behavior of a class of nonlocal partial differential equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 749-763.  doi: 10.3934/dcdsb.2018041. [52] X. Zhang, Stochastic Lagrangian particle approach to fractal Navier-Stokes equations, Comm. Math. Phys., 311 (2012), 133–155. doi: 10.1007/s00220-012-1414-2. [53] W. Zhao and W. Liu, A note on blow-up of solutions for a class of fourth-order wave equation with viscous damping term, Appl. Anal., 97 (2018), 1496-1504.  doi: 10.1080/00036811.2017.1313410.

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##### References:
 [1] A. Alsaedi, B. Ahmad, M. Kirane and B. T. Torebek, Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal., 10 (2021), 952-971.  doi: 10.1515/anona-2020-0153. [2] D. Applebaum, Lévy processes–-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [3] D. Applebaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [4] V. V. Au et al., On a final value problem for a nonlinear fractional pseudo-parabolic equation, Electron. Res. Arch., 29 (2021), 1709-1734.  doi: 10.3934/era.2020088. [5] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [6] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996. [7] G. M. Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397. [8] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20, Springer, 2016. doi: 10.1007/978-3-319-28739-3. [9] H. Chen, P. Luo and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030. [10] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038. [11] 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. [12] A. Cotsiolis and N. K. Tavoularis, On logarithmic Sobolev inequalities for higher order fractional derivatives, C. R. Math. Acad. Sci. Paris, 340 (2005), 205-208.  doi: 10.1016/j.crma.2004.11.030. [13] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017. [14] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408. [15] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [16] H. Di, Y. Shang and X. Peng, Blow-up phenomena for a pseudo-parabolic equation with variable exponents, Appl. Math. Lett., 64 (2017), 67-73.  doi: 10.1016/j.aml.2016.08.013. [17] H. Di, Y. Shang and X. Zheng, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 781-801.  doi: 10.3934/dcdsb.2016.21.781. [18] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 15, Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8. [19] A. A. Dubkov, B. Spagnolo and V. V. Uchaikin, Lévy flight superdiffusion: An introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649-2672.  doi: 10.1142/S0218127408021877. [20] Y. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 70, 17 pp. doi: 10.14232/ejqtde.2016.1.70. [21] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688. [22] N. E. Humphries et al., Environmental context explains Levy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069.  doi: 10.1038/nature09116. [23] J. D. Kečkić and P. M. Vasić, Some inequalities for the gamma function, Publ. Inst. Math., (Beograd) (N.S.), 11 (1971), 107–114. [24] M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources, Differ. Equ., 42 (2006), 431-443.  doi: 10.1134/S001226610603013X. [25] G. Li, J. Yu and W. Liu, Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 8 (2017), 629–660. doi: 10.1007/s11868-017-0216-x. [26] M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066. [27] G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016. [28] W. Liu, Y. Sun and G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term, Topol. Methods Nonlinear Anal., 49 (2017), 299-323.  doi: 10.12775/tmna.2016.077. [29] W. Liu and J. Yu, A note on blow-up of solution for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 274 (2018), 1276-1283.  doi: 10.1016/j.jfa.2018.01.005. [30] W. Liu, B. Zhu and G. Li, Upper and lower bounds for the blow-up time for a viscoelastic wave equation with dynamic boundary conditions, Quaest. Math., 43 (2020), 999-1017.  doi: 10.2989/16073606.2019.1595768. [31] W. Liu and H. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 67, 35 pp. doi: 10.1007/s00030-017-0491-5. [32] Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171. [33] 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. [34] A. Massaccesi and E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, J. Math. Biol., 74 (2017), 113-147.  doi: 10.1007/s00285-016-1019-z. [35] S. A. Messaoudi, B. Said-Houari and N. Tatar, Global existence and asymptotic behavior for a fractional differential equation, Appl. Math. Comput., 188 (2007), 1955-1962.  doi: 10.1016/j.amc.2006.11.105. [36] R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161–R208. doi: 10.1088/0305-4470/37/31/R01. [37] V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3. [38] N. Pan, P. Pucci and B. Zhang, Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.  doi: 10.1007/s00028-017-0406-2. [39] 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. [40] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942. [41] J.-C. Saut and Y. Wang, Long time behavior of the fractional Korteweg–de Vries equation with cubic nonlinearity, Discrete Contin. Dyn. Syst., 41 (2021), 1133-1155.  doi: 10.3934/dcds.2020312. [42] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032. [43] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105. [44] D. W. Sims et al., Scaling laws of marine predator search behavior, Nature, 451 (2008), 1098-1102.  doi: 10.1038/nature06518. [45] D. Stan, F. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with nonlocal pressure, J. Differential Equations, 260 (2016), 1154-1199.  doi: 10.1016/j.jde.2015.09.023. [46] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA No., 49 (2009), 33-44. [47] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857. [48] J. L. Vázquez, Asymptotic behaviour for the fractional heat equation in the Euclidean space, Complex Var. Elliptic Equ., 63 (2018), 1216-1231.  doi: 10.1080/17476933.2017.1393807. [49] M. Xiang, D. Vicentiu 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. [50] 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. [51] C. Zhang, F. Li and J. Duan, Long-time behavior of a class of nonlocal partial differential equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 749-763.  doi: 10.3934/dcdsb.2018041. [52] X. Zhang, Stochastic Lagrangian particle approach to fractal Navier-Stokes equations, Comm. Math. Phys., 311 (2012), 133–155. doi: 10.1007/s00220-012-1414-2. [53] W. Zhao and W. Liu, A note on blow-up of solutions for a class of fourth-order wave equation with viscous damping term, Appl. Anal., 97 (2018), 1496-1504.  doi: 10.1080/00036811.2017.1313410.
$J(\lambda u)$ curve about $\lambda$
Space partition diagram with $\gamma(\delta)$ and $I_{\delta}(u)$
$d(\delta)$ curve about $\delta$
Space partition diagram with $\sqrt{\frac{d(\delta)}{a(\delta)}}$ and $I_{\delta}(u)$
Vacuum region $U_{e}$
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