Open Access
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 2, April 2023
Page(s) 99 - 105
Published online 23 May 2023

© Wuhan University 2023

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Let , and be three constants, be a bounded domain with smooth boundary , and be a bounded non-trivial function with . We focus our attention here on dealing with the global existence, extinction and non-extinction phenomenon of the following doubly nonlinear parabolic equation




Nonlinear evolutionary problems with logarithmic nonlinearity like model (1) came from inflation cosmology, super symmetric field theories, quantum mechanics and nuclear physics[1-6].

In the past few decades, many mathematical researchers have devoted themselves to investigating the doubly nonlinear parabolic equations, and obtained many meaningful results, such as local and global well-posedness, regularity, blow-up in a finite time and extinction singularity [7-12]. Especially, the authors [13-15] studied the problem


where , , , and is a non-negative bounded non-trivial function with . For the case , the authors [14,15] proved that the critical blow-up and extinction exponents are and , respectively. Compared with the case , the solution for the case exhibits completely different properties [13]. On the one hand, for any , the non-trivial solution to problem (3) will never become extinct at a finite time. On the other hand, the critical blow-up exponent becomes .

Recently, Le and Le [16,17] considered problem (1) with and obtained the existence and non-existence results of the global weak solutions. Precisely speaking, they concluded that if , then for any , problem (1) admits a global solution; if , then there exists a weak solution to problem (1) which is global provided that belongs to some specific stable sets, and the weak solution blows up in a finite time provided that belongs to some specific unstable sets.

According to our knowledge, there is no extinction result of the solution to problem (1) with . Inspired by the above works, we naturally take the following two questions into consideration. Does the solution to problem (1) with exist globally under some certain conditions? If so, is there a critical extinction exponent of the global solutions? In fact, in this paper, we work with the following equivalent formulation of problem (1), obtained by changing variable ,




It is clear that the first equation in problem (4) has degeneracy or singularity at the points where or , and hence problem (4) might not have classical solution in general. We introduce the definition of the weak solution to problem (4) as follows.

Definition 1   Let . A measurable function defined in is called a weak solution to problem (4) if , , and


holds for and any .

Similar to the proof of Theorem 3.3 [17], by Faedo-Galerkin method, we can prove the local existence result of the weak solution to problem (4). Now, we state the main results of this paper as follows.

Theorem 1   Assume that . Then the weak solution of problem (1) exists globally.

Theorem 2   Assume that . If


with , and .Then the weak solution to problem (1) will vanish in finite time, where and are two positive constants, given by (23) and (24), respectively.

Theorem 3   Assume that . If


Then the weak solution to problem (1) cannot vanish in finite time, where


Remark 1   From Theorems 2 and 3, we know that the critical extinction exponent of the global solutions to problem (1) is .

1 Proofs of the Main Results

Proof of Theorem 1   Multiplying both sides of the first equality in (4) by and integrating over , one gets


Remembering that and , we can select such that . For this chosen , we know . Then, from (10), it holds that


By using Hölder's inequality, (11) leads to


where . By a simple calculation, we get


It follows from (13) and Hölder's inequality that


On the other hand, multiplying both sides of the first equality in (4) by and integrating over , then with the help of Hölder's inequality and Cauchy's inequality with , we get


If and are sufficiently small such that , then from (15), it holds that


where and . Combining (14) and (16) tells us that


where . Integrating (17), we arrive at


which implies that is bounded for all . The proof of Theorem 1 is complete.

Proof of Theorem 2   Multiplying both sides of the first equality in (4) by , we find that


By virtue of Hölder's inequality and Sobolev embedding inequality, we have


which implies that


where is the optimal Sobolev embedding constant. Substituting (21) into (19) and using Hölder's inequality, we get






Recalling that and , we check that


On the other hand, our assumption (7) tells us that


Combining (22), (25), (26) and Lemma 1 [18], one can claim that there exist two positive constants and such that




Then it follows from (22) and (27) that


Integrating above inequality, one has


which suggests that there exists a


such that


Moreover, it can be concluded that


On the other hand, by a similar way, it can be shown that will vanish in finite time. Thus vanishes in finite time. Recalling that , one can claim that the solution to problem (1) possesses the extinction property. The proof of Theorem 2 is completed.

Proof of Theorem 3   Denoting


with , then a direct calculation shows that


which implies




A direct calculation tells us that


By virtue of (36), (38) and Hölder's inequality, one derives


If . Then from (39), one can see that, for any ,


Keeping in mind that




then (40) gives us that , which means that the solution to problem (1) cannot vanish in finite time.

If . Remembering that




combining (39) and Lemma 1.2 [19] gives us that, for any ,


which means that the solution to problem (1) cannot vanish in finite time, where



The proof of Theorem 3 is completed.


  1. Barrow J D, Parsons P. Inflationary models with logarithmic potentials [J]. Physical Review D, 1995, 52(10): 576-587. [Google Scholar]
  2. Bialynicki-Birula I, Mycielski J. Nonlinear wave mechanics [J]. Annals of Physics, 1976, 100(1-2): 62-93. [Google Scholar]
  3. Bialynicki-Birula I, Mycielski J. Gaussons: Solitons of the logarithmic Schrödinger equation [J]. Physica Scripta, 1979, 20(3-4): 539-544. [Google Scholar]
  4. Deng X M, Zhou J. Extinction and non-extinction of solutions to a fast diffusion p-Laplace equation with logarithmic non-linearity [J]. Journal of Dynamical and Control Systems, 2022, 28(4): 757-769. [Google Scholar]
  5. Ding H, Zhou J. Global existence and blow-up for a parabolic problem of Kirchhoff type with logarithmic nonlinearity [J]. Applied Mathematics & Optimization, 2021, 83(3): 1651-1707. [Google Scholar]
  6. Enqvist K, McDonald J. Q-balls and baryogenesis in the MSSM [J]. Physics Letters B, 1998, 425(3-4): 309-321. [Google Scholar]
  7. Han Y Z, Liu X. Global existence and extinction of solutions to a fast diffusion p-Laplace equation with special medium void [J]. Rocky Mountain Journal of Mathematics, 2021, 51(3): 869-881. [Google Scholar]
  8. Liu D M, Liu C Y. On the global existence and extinction behavior for a polytropic filtration equation with variable coefficients [J]. Electronic Research Archive, 2022, 30(2): 425-439. [Google Scholar]
  9. Shang H F. Doubly nonlinear parabolic equations with measure data [J]. Annali di Matematica Pura ed Applicata, 2013, 192(2): 273-296. [Google Scholar]
  10. Tian Y, Mu C L. Extinction and non-extinction for a p-Laplacian equation with nonlinear source [J]. Nonlinear Analysis: Theory, Methods & Applications, 2008, 69(8): 2422-2431. [Google Scholar]
  11. Li H L, Wu Z Q, Yin J X, et al. Nonlinear Diffusion Equations [M]. Singapore: World Scientific, 2001. [Google Scholar]
  12. Xu X H, Cheng T Z. Extinction and decay estimates of solutions for a non-Newton polytropic filtration system [J]. Bulletin of the Malaysian Mathematical Sciences Society, 2020, 43(3): 2399-2415. [Google Scholar]
  13. Yin J X, Jin C H. Non-extinction and critical exponent for a polytropic filtration equation [J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(1-2): 347-357. [Google Scholar]
  14. Jin C H, Yin J H, Ke Y Y. Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources [J]. Proceedings of the Edinburgh Mathematical Society, 2009, 52(2): 419-444. [Google Scholar]
  15. Zhou J, Mu C L. Critical blow-up and extinction exponents for non-Newton polytropic filtration equation with source [J]. Bulletin of the Korean Mathematical Society, 2009, 46(6): 1159-1173. [Google Scholar]
  16. Le C N, Le X T. Global solution and blow-up for a class of p-Laplacian evolution equations with logarithmic nonlinearity [J]. Acta Applicandae Mathematicae, 2017, 151(1): 149-169. [Google Scholar]
  17. Le N C, Le T X. Existence and nonexistence of global solutions for doubly nonlinear diffusion equations with logarithmic nonlinearity [J]. Electronic Journal of Qualitative Theory of Differential Equations, 2018, 67: 1-25. [Google Scholar]
  18. Liu W J, Wu B. A note on extinction for fast diffusive p-Laplacian with sources [J]. Mathematical Methods in the Applied Sciences, 2008, 31(12): 1383-1386. [Google Scholar]
  19. Guo B, Gao W J. Non-extinction of solutions to a fast diffusive p-Laplace equation with Neumann boundary conditions [J]. Journal of Mathematical Analysis and Applications, 2015, 422(2): 1527-1531. [Google Scholar]

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