Wuhan Univ. J. Nat. Sci.
Volume 28, Number 2, April 2023
|Page(s)||99 - 105|
|Published online||23 May 2023|
CLC number: O 175
Global Existence and Extinction Behaviour for a Doubly Nonlinear Parabolic Equation with Logarithmic Nonlinearity
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China
This paper is mainly focused on the global existence and extinction behaviour of the solutions to a doubly nonlinear parabolic equation with logarithmic nonlinearity. By making use of energy estimates method and a series of ordinary differential inequalities, the global existence of the solution is obtained. Moreover, we give the sufficient conditions on the occurrence (or absence) of the extinction behaviour.
Key words: global existence / extinction behaviour / doubly nonlinear parabolic equation / logarithmic nonlinearity
Biography: LIU Dengming, male, Ph. D., Professor, research direction: nonlinear partial differential equations. E-mail: firstname.lastname@example.org
Fundation item: Supported by the Project of Education Department of Hunan Province (20A174)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 . 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 , 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 , 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
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
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  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.
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