Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
-
Yanchao Gao
and
Bingbai Pan
Abstract
In this article, we consider the initial-boundary value problem for a class of viscoelastic extensible beam equations with logarithmic source term, strong damping term, and weak damping term. We establish the local existence and uniqueness of weak solutions by applying the Faedo-Galerkin method and the contraction mapping principle. Under the framework of potential wells, for subcritical initial energy, we derive the sharp condition for initial data classification, i.e., the global existence and decay estimates of weak solutions in the stable set and the finite-time blow-up of weak solutions in the unstable set. Moreover, we derive sufficient conditions for the finite-time blow-up of weak solutions with negative initial energy and null initial energy. And we obtain an upper bound estimate for the blow-up time.
1 Introduction
In this article, we investigate the following initial-boundary value problems governing the viscoelastic beam equation, which incorporates both strong and weak damping mechanisms alongside logarithmic nonlinear source terms
where
In the framework of continuous dynamics, the evolution of transverse deflections in extendable beam structures can be characterized by their governing equations. The core characteristics of this equation are reflected in the dynamic buckling behavior of articulated beams under axial forces, where the vibration modes are significantly regulated by the beam length and boundary constraint conditions. Woinowsky-Krieger [27] first formulated the governing equations for extensible beams in their seminal work, presenting the following mathematical model:
where
where
where
The system parameters are defined as follows.
With a growing understanding of the physical features of extensible beam models, such as stress distributions and boundary constraint conditions, and the engineering application scenarios of these models, these beam equations rooted in practical physics attract many mathematicians due to their rich mathematical connotations and potential application values. Dickey [7] obtained the existence of solutions for standard beam model (1.2) by Fourier sine series. Ma [21] further extended the study to the global existence of weak solutions and decay estimates. Besides, Ma and Narciso [22] derived results on the global attractors and the existence of bounded absorbing sets for standard beam model in high dimensions (1.3) when
Patcheu [24] considered the Cauchy problem for beam equation
The equation addresses the physical origin of the problem by studying the dynamic buckling of a hinged, beam under axial tensile or compressive forces. Using the Faedo-Galerkin method, the authors established the local existence and uniqueness of weak solutions. Furthermore, they obtained exponential decay of energy through a special integral inequality. For linear damping, where
The authors considered the issue under three distinct initial energy scenarios, including subcritical critical and arbitrarily high levels. Using the contraction mapping principle, they first proved the local existence of solutions. Within the potential well framework, the global existence, non-existence, and decay estimate were deduced for subcritical and critical initial energy levels. Additionally, the global non-existence of solutions was analyzed in both weak and strong damping cases under arbitrarily high initial energy levels.
While researching the equations for extensible beams with damping terms, viscoelastic terms are also taken into consideration. Viscoelastic materials exhibit characteristics that lie between those of elastic solids and those of viscous fluids with differential properties, and their behaviors can be modeled through partial differential equation models. Liu et al. [20] considered the IBVP for viscoelastic Kirchhoff-like beam equations with weak and strong dampings, and power source terms
Through the utilization of linearization techniques and contraction mapping theorems, they first demonstrated the local existence and uniqueness of solutions to the problem. Subsequently, by implementing the potential well theory, the investigation of global solution existence was conducted under subcritical and critical initial energy conditions. Additionally, the decay estimates of globally existing solutions were analyzed for cases where positive initial energies lie strictly beneath the potential well depth. Finally, a thorough examination of finite-time blow-up scenarios was carried out. This analysis systematically addressed solutions with negative initial energy, zero initial energy, and positive initial energy values strictly below the potential wells depth, alongside solutions characterized by arbitrary positive initial energies. Each case was treated separately to ensure a comprehensive understanding of the dynamical behaviors under varying energy configurations.
When we discuss beam equations with damping terms, power nonlinearity and logarithmic nonlinearity are two important forms of nonlinear source terms. Fang and Li [8] considered the following plate equation:
where
By analyzing different beam models, readers can clearly identify the unsolved problems in each beam equation. Nonetheless, we believe it remains essential to describe these issues in a clearer manner. To address this, we provide Table 1. In this table, each black dot denotes a specific term within the model, and the description on the far right indicates the conclusions already derived. From this table, we can identify an overarching framework that aims to derive finite-time blow-up of weak solutions for each beam model under different energy levels and asymptotic behavior for solutions. For the beam model considered in [1], which includes viscoelastic, logarithmic source, and strong damping terms, Pereira et al. proved that the model admits a unique weak solution. They also obtained decay estimates and finite-time blow-up results for solutions under the energy condition
Mathematical physics models and the constituent elements of the results
| Terms included in the function | Results | ||||||
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Gloabal existence and exponential decay and finite time blow-up for
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Global existence and infinite time blow-up for
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Gloabal existence and exponential decay [9]. | |||
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Gloabal existence and exponential decay and finite time blow-up for
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Gloabal existence and asymptotic behavior and finite time blow-up for
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Gloabal existence and exponential decay and infinite time blow-up for
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Gloabal existence and exponential decay and finite time blow-up for
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The following is the organizational framework of this article. In Section 2, we introduce fundamental assumptions, lemmas, and key definitions necessary for the subsequent analysis. Section 3 demonstrates the local existence of weak solutions through a combined application of the Galerkin method and the contraction mapping principle. In Section 4, we established the global existence of solution and decay estimates with subcritical energy condition. In Section 5, we proved the blow-up of solutions in finite time by using the convex method and obtained an upper bound estimate for the blow-up time.
2 Preliminaries
In this article, we assume that
and
Through simple calculation, we can prove that there exist two positive numbers
We define
Regarding Problem (1.1), the following assumptions are put forward for analysis:
and
To facilitate our discussion, we define the energy functional
where
And the potential energy functional is defined as
Besides, the Nehari functional is defined as
According to the definitions of
We also define the Nehari manifold
and
Additionally, we define the well depth
which is alternatively written as
The proof of
Subsequently, we present the definition of the weak solution of Problem (1.1).
Definition 2.1
For a given positive
and for every test function
Definition 2.2
For a weak solution
If
If
Next, we will introduce several important lemmas used in subsequent proofs.
Lemma 2.1
Suppose that
there exist a unique positive real number
the function
Proof
From (2.2), we have
then it is evident that the conclusion of (1) is valid. By differentiating
Let
As a result, through taking
such that
then clearly, the conclusion of (4) holds. Lemma 2.1 demonstrates that the set
Lemma 2.2
[16] Let
with
Lemma 2.3
[12] If
where
Lemma 2.4
[19] Let X be a Banach space, if
Lemma 2.5
From the definition of
Proof
Combining (1.1) and (2.1), we arrive at
From (2.9), we receive
Lemma 2.6
[15] Suppose that
where
and the constant
3 Local existence of weak solutions
Within this section, the local existence of weak solutions is established via the Faedo-Galerkin technique and the contraction mapping principle. For a given positive number
Theorem 3.1
Assume that
satisfying
Proof
Let
and
for any
Next, we define the finite dimensional subspace
satisfying
and when
Thus, we have
and
where
where
By Peano’s existence theorem, for any
Taking the first equation (3.3), we multiply both sides by
By integrating (3.6) over the interval
Our assessment focuses on the final term situated on the right-hand side of equation (3.7). By Hölder’s inequality, we obtain
Let
we have
where
where
where
From the definition of dual norm, we acquire
In view of Hölder’s inequality, hypothesis
where
Taking
Next, we prove the initial value condition. According to Aubin Lion’s lemma (see [26]) and (3.12), (3.15), we have
The uniqueness of solutions is established through a proof by contradiction. Assume that
By multiplying the derived equation by
Combining this equation with the boundary conditions of (3.1), we can obtain
Theorem 3.2
Under the assumption that conditions
Proof
For sufficiently large positive constants
where
Then,
We take
we select
By (3.20)–(3.22), we deduce that
Taking the first equation in (3.23), we multiply both sides by
Furthermore, by the mean value theorem, we acquire
where
Through the application of Hölder’s inequality and Young’s inequality, we can derive
where
Let
where
By similar argument to (3.9), we can obtain
where
the notation
Let
where
Combining (3.24), (3.26), and (3.30), we have
By selecting
we conclude
Thus, the contraction mapping principe ensures the uniqueness and the existence of weak solution for Problem (1.1). The proof is complete.□
4 Global solution and decay estimation
Within this part, we analyze the global existence and decay estimations for the weak solutions of Problem (1.1). Initially, we examine the invariance characteristic of the stable set
Lemma 4.1
Suppose
Proof
Let
Next, we assert if
Then, we derive a result that contradicts equation (4.1). Consequently,
Therefore,
In this section, our focus is to demonstrate the global existence of weak solutions for Problem (1.1), under the scenario where the initial energy condition
Theorem 4.1
Assume
Proof
By integrating the results from (2.9) and (4.1), we derive
Through a derivation analogous to that in (3.9), it can be deduced that
By applying Lemma 2.6 together with Young’s inequality, for
where
We choose
then
and
We combine (4.3)–(4.5) to obtain
The aforementioned inequality becomes
From Lemma 4.1, we obtain
Substituting (4.7) into (4.6), we have
where
Using an estimation similar to (3.9), we can obtain
By the definition of the dual norm and the first equation of Problem (1.1), we can derive
In view of Hölder’s inequality, hypothesis
This completes the proof for
In the subsequent discussion, we demonstrate the decay characteristics of the weak solution as outlined in the following.
Define function
where
Lemma 4.2
Let
Proof
According to Hölder’s, Young’s, and Poincaré’s inequality and (2.2), we obtain
where
then
so we acquire
where
Theorem 4.2
Assume
then there are two positive numbers
where
Proof
Differentiating the function
By making use of Young’s and Hölder’s inequality, and referring to Lemma 2.3, we are able to infer that
and
where
From (2.1), we can obtain
where
By (2.9) and (4.2), we can obtain
So we obtain
in which
in which
Since
we have
By choosing
and
Thus, we arrive
where
By taking each side of (4.20) and multiplying by
Define
Based on the definition of
then
Let
so
Through simple calculation, we obtain
which means
where
5 Blow-up of weak solutions
In this section, we utilize the convexity method to establish the finite-time blow-up of weak solutions to Problem (1.1) under varying energy-level conditions and derive an upper bound estimation for the blow-up time.
Theorem 5.1
Under the assumptions that conditions
the initial energy
the kernel function
where
with the function
Proof
We define the function
where
Through straightforward computation, one can derive
and
We can derive the following result from the Cauchy-Schwarz inequality and (5.3)
Through an analysis of (5.4) and (5.5), it can be deduced that
Let
then (5.6) can be rewritten as
By combining (2.1), (2.9), and (5.7) and then applying Young’s inequality, given an arbitrary positive number
After further simplification, we acquire
Let us now examine the classification of the initial energy
For the first case, let
For the second case, let
Alternatively, when
Since
then
By combining (5.10) and (5.13), we acquire
By selecting a sufficiently small positive number
By integrating the information from (5.9) and (5.15), it can be deduced that
In both scenarios where
Thus, by choosing
where
So
The proof is complete.□
Finally, we state the conclusion on the finite-time blow-up of weak solutions with null initial energy in the form of a remark.
Remark 5.1
When
Proof
Taking
The condition
Consequently, we arrive at a similar result that
Acknowledgments
The authors express sincere gratitude to the anonymous referees for their insightful feedback and constructive suggestions, which have greatly contributed to the improvement of this article.
-
Funding information: This work was supported by Scientific Research Project of Jilin Provincial Department of Education (JJKH20250464KJ).
-
Author contributions: All authors made equal contributions to the writing of this article. All authors read and approved the final manuscript.
-
Conflict of interest: The authors declare no conflicts of interest.
-
Data availability statement: No data were used for the research described in the article.
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