Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 2, April 2024



Page(s)  95  105  
DOI  https://doi.org/10.1051/wujns/2024292095  
Published online  14 May 2024 
Mathematics
CLC number: O175
Blowup for a Porous Medium Equation with Local Linear Boundary Dissipation
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China
^{†} Corresponding author. Email: 1070099@hnust.edu.cn
Received:
15
September
2023
This article investigates the blowup behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation. Making use of the concavity method, we establish sufficient conditions to guarantee the occurrence of the finite time blowup phenomenon. Meanwhile, we show the existence of the finite time blowup solutions for arbitrarily high initial energy. Finally, we derive the life span bounds (i.e., the lower and upper bounds of the blowup time).
Key words: porous medium equation / blowup behavior / life span bounds
Cite this article: YANG Jichen, LIU Dengming. Blowup for a Porous Medium Equation with Local Linear Boundary Dissipation[J]. Wuhan Univ J of Nat Sci, 2024, 29(2): 95105.
Biography: YANG Jichen, male, Master candidate, research direction: differential equations and their applications. Email: 2193822875@qq.com
Fundation item: Supported by Scientific Research Fund of Hunan Provincial Education Department (23A0361)
© Wuhan University 2024
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.
0 Introduction
In the present article, our attention is focused on the blowup behavior of the following porous medium equation with superline source and local linear boundary dissipation
where , , is a bounded open subset of with boundary and represents the unit outer normal vector to . is a partition of the boundary satisfying
Moreover, and are measurable over , endowed with dimensional surface measure and .
Problem (1) with can be regarded as a mathematical model to depict a heat reactiondiffusion process that occurs inside a solid body surrounded by a fluid, with contact and having an internal cavity with a contact boundary In this physics background, the quantity of heat produced by the reaction is proportional to a superlinear power of the temperature. To avoid an internal explosion in , a refrigeration system is installed in the fluid. The operational mechanism of this refrigeration system lies in the fact that the heat absorbed from the fluid is proportional to the power of the rate of change of the temperature, which can be expressed as: where stands for the heat flux from to the fluid.
Evolution equations^{ [18]} with boundary damping have attracted the attention of mathematicians in the past period. For instance, Fiscella and Vitillaro ^{[9]} studied the following problem with local nonlinear boundary dissipation
where and denotes the Sobolev conjugate of 2. Using the monotonicity method of Lions^{[9]} and a contraction argument, they proved the local wellposedness in the Hadamard sense. Moreover, in the case of a superlinear source, i.e. , under the condition
they gave the global existence and finite time blowup results. To be precise, if
then the weak solution is global, while if
and then the weak solution will blow up in some finite time. In a recent work, the authors^{ [10]} considered problem (1) with , and obtained the finite time blowup result for arbitrary high initial energy. Moreover, under some additional conditions, the authors also gave estimates of the blowup time. In addition, using some differential inequality techniques, the authors^{ [11, 12]} considered the lower bounds for the blowup time of blowup solutions to some porous medium equations with null Dirichlet boundary conditions or homogeneous Neumann boundary conditions.
To the best of our knowledge, there is no previous work on the blowup behavior of the solution to the problem (1). Building on the aforementioned work, we will analyze the effects of the nonlinear diffusion and the local linear boundary dissipation on the blowup phenomenon of problem (1). In order to deal with the difficulties caused by the nonlinear diffusion term better and more effectively, throughout this paper, we work with the following equivalent formulation of the problem (1) obtained by changing variables ,
First, we obtain the finite time blowup criterion of the solution to the problem (3) by using a modified concavity method (Theorem 1). Second, for any we prove that there exists a with initial energy that leads to a finite time blowup solution (Corollary 1). Finally, the lower bounds of the blowup time are derived by combining the interpolation inequality for norms, the Sobolev embedding theorem, with some differential inequality techniques (Theorem 2).
The article is organized as follows: In Section 1, we introduce some notations and state some useful lemmas. In Section 2, we give the finite time blowup criterion and the lower and upper estimates of the blowup time.
1 Preliminaries
In this section, we first introduce some notations, definitions, and some known results. Throughout this article, we denote , for some , and the Hilbert space
and stand for the inner products on the Hilbert spaces and , respectively. From the trace theorem, one knows that there exists a continuous trace mapping . Moreover, since then, Theorem 6.75^{ [13]} tells us that a Poincarétype inequality holds. Therefore, is equivalent to the norm in the space On the other hand, since , one can define the following positive optimal constants
The definition of the weak solution to the problem (3) is given as follows.
A function with is called a weak solution of problem (3) on if , and
holds for a.e. and any Moreover, the spatial trace of has a distributional time derivative on , belonging to .
In what follows, we introduce the energyfunctional and Nehari functional
in related to problem (3). Evidently, one knows that and are continuous on , and for a.e.
which implies that
Now, we introduce the definition of the potential welldepth where is the Nehari manifold Indeed, also can be characterized as
We are now able to give some lemmas, which will play a key role in our proof of the main results.
Thereupon, one has . Furthermore, according to the definition of the potential well depth , one can arrive at
which results in
The proof of the Lemma 1 is completed.
Suppose that , and the weak solution of problem (3) blows up in finite time . Then there is a such that .
which contradicts the assumption that is a finitetime blowup weak solution. The proof of Lemma 2 is completed.
Lemma 3^{ [14]} Suppose that a positive function on satisfies the following conditions: is differentiable on and is absolutely continuous on with ; there exists a positive constant such that
holds for a.e. . Then
2 The Finite Time Blowup Results
Recalling that one can easily check that
For any , we define an auxiliary function as the form
where and are two positive parameters which will be determined later. It is clear that is positive on . By a simple calculation, one has
and . Moreover, from (6) and (7), it follows that
On the other hand, employing the Young's inequality and CauchySchwartz inequality, we obtain:
Now, we are in the position to estimate with . Combining (10), (11), and (12), one can arrive at:
Noticing that, , then (14) leads to
Up to now, from the above discussion, one can summarize the following lemma.
holds for any . Then
Choosing to guarantee , then (16) tells us that,
By a series of calculations, one can verify that the right side of (17) takes its minimum at the point
In other words, one has
The proof of Lemma 4 is completed.
Now, we give the finite time blowup criteria as follows.
Then, the weak solution to the problem (3) blows up in a finite time. Moreover, one has
for and
for .
Suppose that there exists a such that and for any . Thereupon, Lemma 1 and the continuity of the mapping are applicable to produce
which implies that . And hence, one has which contradicts (18). That is to say, for any , one can claim that provided that .
Selecting
and keeping Lemma 1 in mind, one has, for any ,
A direct application of Lemma 4 tells us that,
If . Then from (4) and (9), it follows that,
where,
Putting and combining (6) with (19), one has, for a.e. ,
Integrating the above inequality from 0 to results in:
On the other hand, the assumption implies that,
Combining (19), (20) with (21) yields that, for a.e.
which means that is increasing in . Therefore, one can see that,
provided that
According to Lemma 4, one can obtain the following estimate,
namely,
The proof of Theorem 1 is completed.
In fact, Theorem 1 tells us the sets and are invariant under the semiflow associated with problem (3). Namely, if then , while , then . On the other hand, with the help of (4), for any , one has
Therefore, one can claim that for any provided initial data . Based on the above arguments, it is natural to ask whether or not the condition is sufficient enough for a finitetime blowup. This is a difficult task, and the authors^{ [15]} conducted a similar study.
In addition, one can know that both and are nonempty sets. Moreover, Corollary 1 implies that for any , there exists a with initial energy , which leads to finite time blowup solution.
Then .
From the proof of Theorem 3.7^{ [16]}, one knows that there is a sequence such that,
On the other hand, choosing an arbitrary nonzero function with supp , then,
and there exists a such that
holds for any . By (23) and (24), there are and such that
Let , where,
It is not difficult to show that and in . From (25) and (26), it follows that,
which means that . The proof of Corollary 1 is completed.
Suppose that ,, the weak solution of problem (3) blows up in finite time and for any . Then
where and are two constants given by (28) and (33), respectively.
Furthermore, from the assumption , it follows that,
and
Noticing that for any , the interpolation inequality for norms and the embedding can be used to obtain:
where denotes the optimal constant of the embedding . Keeping in mind, one can show that . And hence, (29) implies that
From (9), (29) and (30), it follows that,
where,
It is not difficult to verify that,
From (31), it follows that,
Letting , then (32) results in:
which means that,
where,
The proof of Theorem 2 is completed.
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