Initial boundary value problem for a class of $ p $-Laplacian equations with logarithmic nonlinearity

Fugeng Zeng; Yao Huang; Peng Shi; Fugeng Zeng; Yao Huang; Peng Shi

doi:10.3934/mbe.2021198

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 3957-3976. doi: 10.3934/mbe.2021198

Previous Article Next Article

Research article Special Issues

Initial boundary value problem for a class of $ p $-Laplacian equations with logarithmic nonlinearity

1.
Department of Artificial Intelligence and Big Data, Yibin University, Yibin 644000, China
2.
School of Date Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China

Academic Editor: Juan Luis García Guirao

Received: 07 March 2021 Accepted: 25 April 2021 Published: 07 May 2021

In this paper, we discuss global existence, boundness, blow-up and extinction properties of solutions for the Dirichlet boundary value problem of the $ p $-Laplacian equations with logarithmic nonlinearity $ u_{t}-{\rm{div}}(|\nabla u|^{p-2}\nabla u)+\beta|u|^{q-2}u = \lambda |u|^{r-2}u\ln{|u|} $, where $ 1 < p < 2 $, $ 1 < q\leq2 $, $ r > 1 $, $ \beta, \lambda > 0 $. Under some appropriate conditions, we obtain the global existence of solutions by means of the Galerkin approximations, then we prove that weak solution is globally bounded and blows up at positive infinity by virtue of potential well theory and the Nehari manifold. Moreover, we obtain the decay estimate and the extinction of solutions.
- $ p $-Laplacian,
- global existence,
- blow-up,
- extinction,
- decay estimate
Citation: Fugeng Zeng, Yao Huang, Peng Shi. Initial boundary value problem for a class of $ p $-Laplacian equations with logarithmic nonlinearity[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 3957-3976. doi: 10.3934/mbe.2021198

Related Papers:

Abstract

In this paper, we discuss global existence, boundness, blow-up and extinction properties of solutions for the Dirichlet boundary value problem of the $ p $-Laplacian equations with logarithmic nonlinearity $ u_{t}-{\rm{div}}(|\nabla u|^{p-2}\nabla u)+\beta|u|^{q-2}u = \lambda |u|^{r-2}u\ln{|u|} $, where $ 1 < p < 2 $, $ 1 < q\leq2 $, $ r > 1 $, $ \beta, \lambda > 0 $. Under some appropriate conditions, we obtain the global existence of solutions by means of the Galerkin approximations, then we prove that weak solution is globally bounded and blows up at positive infinity by virtue of potential well theory and the Nehari manifold. Moreover, we obtain the decay estimate and the extinction of solutions.

References

[1]	J. N. Zhao, Existence and nonexistence of solution for $u_t = {\text {div}}(\|\nabla u\|^{p-2}\nabla u)+f(\nabla u, u, x, t)$, J. Math. Anal. Appl., 172 (1993), 130–146. doi: 10.1006/jmaa.1993.1012
[2]	P. Pucci, M. Q. Xiang, B. L. Zhang, A diffusion problem of Kirchhoff type involving the nonlocal fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 37 (2017), 4035–4051. doi: 10.3934/dcds.2017171
[3]	C. N. Le, X. T. Le, Global solution and blow-up for a class of pseudo $p$-Laplacian evolution equations with logarithmic nonlinearity, Acta. Math. Appl., 151 (2017), 149–169. doi: 10.1007/s10440-017-0106-5
[4]	T. Sarra, A. Zarai, B. Salah, Decay estimate and nonextinction of solutions of $p$-Laplacian nonlocal heat equations, AIMS Math., 5 (2020), 1663–1679. doi: 10.3934/math.2020112
[5]	N. Mezouar, S. M. Boulaaras, A. Allahem, Global existence of solutions for the viscoelastic Kirchhoff equation with logarithmic source terms, Complexity, 2020 (2020), 1–25.
[6]	Y. L. Li, D. B. Wang, J. L. Zhang, Sign-changing solutions for a class of $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity, AIMS Math., 5 (2020), 2100–2112. doi: 10.3934/math.2020139
[7]	D. Han, J. Zhou, Global existence and blow-up for a parabolic problem of Kirchhoff type with logarithmic nonlinearity, Appl. Math. Optim., 2019 (2019), 1–57.
[8]	J. Zhou, Ground state solution for a fourth-order elliptic equation with logarithmic nonlinearity modeling epitaxial growth, Comput. Math. Appl., 78 (2019), 1878–1886. doi: 10.1016/j.camwa.2019.03.025
[9]	H. Ding, J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 98 (2019), 1–35.
[10]	F. Sun, L. Liu, Y. Wu, Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term, Appl. Anal., 98 (2017), 735–755.
[11]	L. M. Song, Positive Solutions for Fractional Differential Equation with $p$-Laplacian Operator and Sign-changing Nonlinearity, J. Math. Pract., 19 (2015), 254–258.
[12]	S. Y. Chung, J. H. Park, A complete characterization of extinction versus positivity of solutions to a parabolic problem of $p$-Laplacian type in graphs, J. Math. Anal. Appl., 1 (2017), 226–245.
[13]	W. J. Liu, K. W. Chen, J. Yu, Extinction and asymptotic behavior of solutions for the $\omega$-heat equation on graphs with source and interior absorption, J. Math. Anal. Appl., 435 (2016), 112–132. doi: 10.1016/j.jmaa.2015.10.024
[14]	W. J. Liu, Extinction properties of solutions for a class of fast diffusive $p$-Laplacian equations, Nonlinear Anal. Theory Methods Appl., 74 (2011), 4520–4532.
[15]	Y. Cao, C. Liu, Initial boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differ. Equ., 116 (2018), 1–19.
[16]	P. Dai, C. Mu, G. Xu, Blow-up phenomena for a pseudo-parabolic equation with $p$-Laplacian and logarithmic nonlinearity terms, J. Math. Anal. Appl., 481 (2019), 123439.
[17]	M. Q. Xiang, D. Yang, Nonlocal Kirchhoff problems: Extinction and non-extinction of solutions, J. Math. Anal. Appl., 477 (2019), 133–152. doi: 10.1016/j.jmaa.2019.04.020
[18]	F. Zeng, P. Shi, M. Jiang, Global existence and finite time blow-up for a class of fractional $p$-Laplacian Kirchhoff type equations with logarithmic nonlinearity, AIMS Math., 6 (2021), 2559–2578.
[19]	D. Hang, J. Zhou, Comments on blow-up and decay for a class of pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl., 75 (2018), 459–469. doi: 10.1016/j.camwa.2017.09.027
[20]	E. Piskin, S. Boulaaras, N. Irkil, Qualitative analysis of solutions for the $p$-Laplacian hyperbolic equation with logarithmic nonlinearity, Math. Meth. Appl. Sci., 44 (2020), 4654–4672.
[21]	A. Choucha, S. Boulaaras, D. Ouchenane, S. Beloul, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, logarithmic nonlinearity and distributed delay terms, Math. Meth. Appl. Sci., 44 (2020), 5436–5457.
[22]	M. Q. Xiang, D. Hu, D. Yang, Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity, Nonlinear Anal., 198 (2020), 111899.
[23]	M. Pino, J. Dolbeault, I. Gentil, Nonlinear diffusions, hypercontractivity and the optimal $L^p$-Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl., 293 (2004), 375–388. doi: 10.1016/j.jmaa.2003.10.009
[24]	T. Boudjeriou, On the diffusion $p(x)$-Laplacian with logarithmic nonlinearity, J. Ell. Par. Equ., 146 (2020), 1–22.
[25]	S. Chen, The extinction behavior of solutions for a class of reaction diffusion equations, Appl. Math. Mech., 11 (2001), 122–126.
[26]	T. Boudjeriou, Global Existence and blow-up for the fractional $p$-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 17 (2020), 1–24. doi: 10.1007/s00009-019-1430-y

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)