Taiwanese Journal of Mathematics

A Class of Fourth-order Parabolic Equations with Logarithmic Nonlinearity

Menglan Liao and Qingwei Li

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Abstract

In this paper, we apply the modified potential well method and the logarithmic Sobolev inequality to study the fourth-order parabolic equation with $p$-Laplacian and logarithmic nonlinearity. Some results are obtained under the different initial data conditions. More precisely, we give the global existence of weak solution by combining the classical Galerkin's method with the modified potential well method, decay estimates, and blow-up in finite time when the initial energy is subcritical and critical, respectively. In addition, sufficient conditions for the global existence and blow-up of the weak solution are also provided for supercritical initial energy. These results extend and improve many results in the literature.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 29 pages.

Dates
First available in Project Euclid: 13 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1565683225

Digital Object Identifier
doi:10.11650/tjm/190801

Subjects
Primary: 35A01: Existence problems: global existence, local existence, non-existence 35B44: Blow-up 35K30: Initial value problems for higher-order parabolic equations

Keywords
logarithmic nonlinearity $p$-Laplacian potential well global existence uniqueness decay estimates blow-up

Citation

Liao, Menglan; Li, Qingwei. A Class of Fourth-order Parabolic Equations with Logarithmic Nonlinearity. Taiwanese J. Math., advance publication, 13 August 2019. doi:10.11650/tjm/190801. https://projecteuclid.org/euclid.twjm/1565683225


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References

  • Y. Cao and C. Liu, Initial boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differential Equations 2018 (2018), no. 116, 19 pp.
  • 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), no. 1, 84–98.
  • 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), no. 12, 4424–4442.
  • M. Del Pino and J. Dolbeault, Asymptotic behavior of nonlinear diffusions, Math. Res. Lett. 10 (2003), no. 4, 551–557.
  • Z. Dong and J. Zhou, Global existence and finite time blow-up for a class of thin-film equation, Z. Angew. Math. Phys. 68 (2017), no. 4, Art. 89, 17 pp.
  • Y. Han, A class of fourth-order parabolic equation with arbitrary initial energy, Nonlinear Anal. Real World Appl. 43 (2018), 451–466.
  • Y. He, H. Gao and H. Wang, Blow-up and decay for a class of pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl. 75 (2018), no. 2, 459–469.
  • S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations 261 (2016), no. 10, 5446–5464.
  • V. K. Kalantarov and O. A. Ladyženskaja, Formation of collapses in quasilinear equations of parabolic and hyperbolic types, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 77–102.
  • B. B. King, O. Stein and M. Winkler, A fourth-order parabolic equation modeling epitaxial thin film growth, J. Math. Anal. Appl. 286 (2003), no. 2, 459–490.
  • C. N. Le and X. T. Le, Global solution and blow-up for a class of $p$-Laplacian evolution equations with logarithmic nonlinearity, Acta. Appl. Math. 151 (2017), 149–169.
  • H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au + \mathscr{F}(u)$, Arch. Rational Mech. Anal. 51 (1973), 371–386.
  • Q. Li, W. Gao and Y. Han, Global existence blow up and extinction for a class of thin-film equation, Nonlinear Anal. 147 (2016), 96–109.
  • M. Liao and W. Gao, Blow-up phenomena for a nonlocal $p$-Laplace equation with Neumann boundary conditions, Arch. Math. (Basel) 108 (2017), no. 3, 313–324.
  • J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.
  • Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations 192 (2003), no. 1, 155–169.
  • Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. 64 (2006), no. 12, 2665–2687.
  • P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var. 4 (1999), 419–444.
  • L. C. Nhan and L. X. Truong, Global solution and blow-up for a class of pseudo $p$-Laplacian evolution equations with logarithmic nonlinearity, Comput. Math. Appl. 73 (2017), no. 9, 2076–2091.
  • K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci. 20 (1997), no. 2, 151–177.
  • M. Ortiz, E. A. Repetto and H. Si, A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids 47 (1999), no. 4, 697–730.
  • L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273–303.
  • 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), no. 2, 796–809.
  • D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148–172.
  • J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.
  • M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci. 8 (1972), no. 2, 211–229.
  • G. Xu and J. Zhou, Global existence and finite time blow-up of the solution for a thin-film equation with high initial energy, J. Math. Anal. Appl. 458 (2018), no. 1, 521–535.
  • R. Xu, T. Chen, C. Liu and Y. Ding, Global well-posedness and global attractor of fourth order semilinear parabolic equation, Math. Methods Appl. Sci. 38 (2015), no. 8, 1515–1529.
  • 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), no. 12, 2732–2763.
  • A. Zangwill, Some causes and a consequence of epitaxial roughening, J. Cryst. Growth 163 (1996), no. 1-2, 8–21.