Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models
-
Chao Yang
and
Mingyou Zhang
Abstract
For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invariant manifold via Nehari manifold. To establish the results of global well-posedness of solution, we consider the problem at three different initial energy levels, i.e., subcritical initial energy level, critical initial energy level, and arbitrarily high initial energy level. We first obtain the local existence of the solution by using the contraction mapping principle. Then, in the framework of potential well, we obtain global existence, nonexistence, and asymptotic behavior of solution for both subcritical initial energy level and critical initial energy level. In the end, we establish the global nonexistence of solution for the problem with linear weak damping and strong damping at the arbitrarily high initial energy level.
1 Introduction
In this study, we consider the initial boundary value problem (IBVP) of nonlinear extensible beam equations with linear strong damping term, nonlinear weak damping term and nonlinear source term
where
The extensible beam model describes the evolution of the transverse deflection of an extensible beam obeying continuous dynamics. A key feature that the extensible beam model affords a description during the vibrations is the dynamic buckling of a hinged extensible beam under an axial force, while it often depends on the fixing manner and distance of the two ends of the beam. The original version of extensible beam equation, proposed by Woinowsky-Krieger [49] in 1950, reads
where
which is called the Berger plate model [12], as a generalization of the Woinowsky-Krieger model to describe the large deflection of the plate, where the parameter
In (1.1), the nonlinear term
under the same variational framework. In addition, the full use of the Nehari tools allows us to learn more about the qualitative properties of the solution, for example, in proving the existence of at least one minimizer of some variational problem [13] or in the search for nodal solutions to the elliptic equations [14]. It seems impossible to list the works by exploiting the advanced Nehari manifold; hence, we just recommend [15] and [47] and the references herein as examples.
Equations such as (1.8) pose a general form of the multidimensional Woinowsky-Krieger model, and the study of well-posedness theory attracts a lot of interest. See [7,17,23,34,40,51] and below for detail.
Without damping. For equation (1.8) without damping term and nonlinear source term, i.e.,
Weak damping: linear or nonlinear. For the linear weak damping case, i.e.,
Strong damping. For a damping of fractionary order
where
Motivated by the above works, we consider problem (1.1)–(1.3) to discuss the global well-posedness of the solutions and the dynamical behaviors as time approaches infinity. By recalling the established results, the main contributions of the present paper can be summarized in the following two aspects.
We conduct a comprehensive study on the global well-posedness of solution at three initial energy levels, i.e., subcritical initial energy level
The present paper deals with the problem with both the nonlinear weak damping term and the linear strong damping term. Considering that there are a lot of established results [28,29,53,54] about the model with the single linear weak damping term or the single nonlinear strong damping term or sometimes the combinations of two of them, in the present paper, we shall consider a more general case with both the nonlinear weak damping term and the linear strong damping term, which will be an extension of the known results. But when we discuss the high energy problem, i.e., the case
2 Preliminaries
In this section, we show some notations, assumptions, and preliminary results, which will be used later. Throughout the present paper, we denote the Sobolev space
Let
For problem (1.1)–(1.3), we introduce the following notations [19,31],
where
In this paper, for simplicity, we denote by
We use the following lemmas throughout this paper.
Lemma 2.1
(Sobolev-Poincaré inequality). Let
with some positive constant C holds.
For the initial boundary value problems (1.1)–(1.3), we introduce the total energy functional
the potential functional
and the Nehari functional
From (2.3)–(2.5), we can easily see that
By
the outer space of the potential well (unstable set)
and the depth of the potential well (the so-called mountain pass level)
where the Nehari manifold is
Next, we give some properties of the aforementioned manifolds and functionals as follows.
Lemma 2.2
Let
On the interval
Proof
This conclusion comes directly from
(2.10)An easy calculation shows that
(2.11)which leads to the conclusion.
By a direct calculation, relation (2.11) gives
and
which proves the conclusion of (iii).
The conclusion follows from
For the depth of the potential well
Lemma 2.3
(Depth of the potential well). The potential well depth is
where
is the imbedding constant form H into
Proof
By the definition of
Noting
then
And, from (2.4) and (2.5), it follows that
Combining (2.16) with
In turn, this implies that
Here, we need to ensure that the equal sign of (2.18) is true, that is, there exists an extremal of the variational problem (2.9). The proof of this part is standard and similar to that in [42]; hence, we give this proof in the Appendix.□
Lemma 2.4
(Nonincreasing energy). Let
Proof
The energy identity can be obtained by testing (1.1) with
which proves (2.19).□
We give the definition of the weak solution for problem (1.1)–(1.3).
Definition 2.1
(Weak solution). Function
and
for all test functions
3 Local existence
To prove the local existence of the solution to problem (1.1)–(1.3), we first give the following lemma, which can be proved by the Banach fixed point theory (see [10,22,38,39]).
Lemma 3.1
Assume
Next, we establish the local existence and uniqueness for solutions of problems (1.1)–(1.3) by using the Banach fixed point theorem.
Theorem 3.1
(Local existence) Assume
and
Proof
The method to prove the local existence theorem is similar to the method of [39]. Throughout the proof, we denote by
where
Then
From Lemma 3.1, we consider the map
Step I:
We multiply (3.1) by
Next, we estimate the first term in the second line of (3.2) as follows:
By Hölder’s inequality and the Sobolev-Poincaré inequality, for the second term in the second line of (3.2), we obtain
Thus, form (3.3) and (3.4), (3.2) gives
Integrating (3.5) over
where
By the Gronwall inequality, we obtain
Then, from the definition of
Form (3.6), we have
and
which implies
then, from (3.6) and (3.7), we have
which shows
Step II:
Now, by taking
By setting
Next, we estimate the right-hand side of (3.10). First, from the mean value theorem and definition of
where
Thus, for
Then thanks to Hölder’s inequality, (3.11), the Sobolev-Poincaré inequality and (3.12), we have
Next, from Hölder’s inequality and the Sobolev-Poincaré inequality, we obtain
Then, by the mean value theorem and the Hölder, Minkowski, and Sobolev inequalities, we have
where
and
Hence, combining inequalities (3.13)–(3.15), (3.10) becomes
where
Then, integrating (3.16) over
we obtain
where
Hence,
Thus, by the definition of
Hence, if
and then
By using the Banach fixed point theorem, we prove the local existence result.□
4 Global existence, asymptotic behavior, and blowup of solutions at subcritical initial energy level
E
(
0
)
<
d
4.1 Global existence for the subcritical initial energy
E
(
0
)
<
d
In this subsection, we consider the global existence of weak solution to problem (1.1)–(1.3). First, we discuss the invariance of the stable set
Lemma 4.1
(Invariant stable set
Now, we prove the global existence of a solution to problem (1.1)–(1.3) for
Theorem 4.1
(Global existence when
Proof
Let
satisfying
Multiplying (4.1) by
Integrating (4.4) over
From (4.2) and (4.3), we derive
Hence, combining (4.5) and
Further, (2.16) and (4.6) allow us to see that
From
Again using
From inequalities (4.8) and (4.9), we arrive at
Then, integrating (4.1) with respect to
Therefore, on the one hand, up to a subsequence, from (4.10) to (4.13), we may pass to the limit in (4.14) and obtain a weak solution
4.2 Asymptotic behavior for the subcritical initial energy
E
(
0
)
<
d
In this section, we state the asymptotic behavior of solutions to problem (1.1)–(1.3) emanating from the initial data satisfying the conditions required by that for the global solutions in Theorem 4.1.
Theorem 4.2
(Asymptotic behavior when
Proof
To prove Theorem 4.2, we consider
for
for two positive constants
Using (4.15), (2.19), and (4.17), we have
Next, we will estimate the right-hand side of (4.18). For the term
From (4.19), (2.3), (2.4), and Lemma 2.4, we obtain
For the term
where
Moreover, by
where
Applying Hölder and Young inequalities
to estimate
where
Hence, from (4.24), (4.23) gives
Moreover, from Poincaré inequality
Note that
where
to make
and
Then from (4.27) and (4.20), inequality (4.26) gives
For the term
where
and
Since
and
From (4.28), we easily have
Then integration of (4.32) (Gronwall inequality) leads to
where
where
4.3 Global nonexistence for the subcritical initial energy
E
(
0
)
<
d
In this subsection, we prove the finite time blowup of solutions to problem (1.1)–(1.3). First, by the same argument as Lemma 4.1, we can obtain the following invariance of the unstable set
Lemma 4.2
(Invariant unstable set
Now, we give some relations of the depth of the potential well
Corollary 4.1
If
Proof
The condition
Noting
i.e.,
By the expression of potential well depth, i.e., (2.12), we obtain (4.33).□
Theorem 4.3
(Global nonexistence when
Proof
As we need different strategies for the cases
Case I:
and
Then
Testing equation (1.1) by
Next, we estimate the last two terms of (4.35). By Lemma 4.2, we obtain
Again using Hölder’s inequality, interpolation inequality, and (4.36), we have
where
Note that
and hence,
Using Hölder and Young inequalities,
where we set
where
We set
where
From (2.5), (4.38), and (4.39), (4.35) becomes
By Lemma 2.4, (2.3), and (2.5), we obtain
where the constant
We now estimate the right-hand side of (4.42). From Lemma 4.2, we obtain
Then by (4.43) and (4.42), we have
As
which guarantees
Combining (4.44) and (4.46), we obtain
By Corollary 4.1 and (4.45), we have
Then, from (4.48) and (4.47), we arrive at
Due to
and
Therefore, by (4.43) and Corollary 4.1, (4.49) becomes
Integrating (4.50) over the time interval
Note that (2.20) gives
Then by (4.52), (4.51) becomes
where
Integrating (4.53) over the time interval
Thus, the norm
On the other side, we estimate
To estimate
From (4.56) and (4.55), we obtain
From (4.52) and (4.57), we have
where
Case II:
It is clear that
Hence, we have
and
From (4.61), we have
Using the Schwarz inequality, we estimate each term in (4.63) as follows
and
Then, (4.63) becomes
By (4.62) and (4.64), we obtain
Setting
Note that
then
Hence, we obtain
Now, we set
Moreover, by Lemma 4.2 and Corollary 4.1, we conclude that there exists
hence,
Let
which tells that
This proves that
5 Global existence, asymptotic behavior, and blowup of solutions at critical initial energy level
E
(
0
)
=
d
5.1 Global existence for the critical initial energy level
E
(
0
)
=
d
This subsection proves the global existence of weak solution to problem (1.1)–(1.3) for the critical initial energy level
Theorem 5.1
(Global existence when
Proof
Let
As
Case I.
Case II.
Hence, for the aforementioned two cases, we have
Therefore, it follows from Theorem 4.1 that for sufficiently large
5.2 Asymptotic behavior for the critical initial energy level
E
(
0
)
=
d
This subsection shows the asymptotic behavior of the global solution to problem (1.1)–(1.3) for the critical initial energy level
Lemma 5.1
Let
Proof
Let
which gives
which gives
Next, we prove the asymptotic behavior for the case
Theorem 5.2
(Asymptotic behavior when
for some positive constants C and k.
Proof
First, Theorem 5.1 gives the existence of the global solution
Hence, from (2.20), it follows that
From Theorem 5.1, we have
then
where
5.3 Global nonexistence for the critical initial energy level
E
(
0
)
=
d
In this section, to prove the finite time blowup result for the case
Lemma 5.2
Let
Proof
Let
Therefore, we obtain
which implies
Next, we display the finite time blowup result at the critical initial energy level
Theorem 5.3
(Global nonexistence when
Proof
First, Theorem 3.1 gives the existence and uniqueness of the local solution
From (2.20) and
In addition, from Lemma 5.2, it follows
6 Global nonexistence of solutions for the high arbitrarily initial energy
E
(
0
)
>
0
when
r
=
1
In the following, we show the global nonexistence of solution to problem (1.1)–(1.3) with strong and linear weak damping (
First, we present the following two lemmas to prove the global nonexistence stated in Theorem 6.1.
Lemma 6.1
(Increasing map). Let
is strictly increasing as long as
Proof
Noting
Testing equation (1.1) by
then
Hence, by
Lemma 6.2
(Invariant sign of
where
Proof
We prove
and
From Lemma 6.1, we obtain that function
Moreover, from the continuity of
On the other hand, by (2.19), (2.6), and (6.3), we can obtain
Using Sobolev, Young, and Cauchy-Schwarz inequalities, we have
Then by (6.5) and (6.6), we obtain
which contradicts (6.4). Hence, this lemma is proved.□
Next, we show the blowup result with arbitrarily positive initial energy
Theorem 6.1
(Global nonexistence when
Proof
Suppose by contradiction that the solution
where
and the constant
Substituting (6.9) into
Set
which together with
Then by Lemmas 6.2, 6.1 and (6.2), we have
which means there exists a constant
Hence,
Letting
which tells
It proves that
Remark 6.1
Here, we give an example to show that the set of the initial data satisfying the following condition (Theorem 6.1):
For the initial data
and
where
Taking any
Note that
and
Once
For condition (i), from (6.13), we see that
For condition (ii), from (6.12), for the aforementioned
which tells that
For condition (iii), from (6.13), we obtain
Then, from (6.14) and
So this example shows one of such initial data satisfying the conditions required in Theorem 6.1, which makes the solution blow up.
Open problems. For the high energy case, i.e.,
Acknowledgment
The authors would like to appreciate the referees for taking their valuable time to provide constructive and valuable comments, which led to the improvement of our paper. This work is supported by the National Natural Science Foundation of China (11871017) and the Fundamental Research Funds for the Central Universities. Part of the work was done while Runzhang Xu was visiting the Mathematical Institute, University of Oxford. Chao Yang is supported by the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072022GIP2403).
-
Conflict of interest: The authors state no conflict of interest.
Appendix
In this section, by following the ideas in [42], we prove that there exists an extremal of the variational problem (2.9), which was used to prove Lemma 2.3.
Proof
Let
The space
and
From (2.14) and (2.15), we know that
Therefore,
If
If
where
For
For
From
which combining (1.5) gives
In addition,
which combining (A2) and (1.5) gives
Hence,
Then, we obtain,
which contradicts the definition of the potential well depth
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Articles in the same Issue
- Research Articles
- Editorial
- A sharp global estimate and an overdetermined problem for Monge-Ampère type equations
- Non-degeneracy of bubble solutions for higher order prescribed curvature problem
- On fractional logarithmic Schrödinger equations
- Large solutions of a class of degenerate equations associated with infinity Laplacian
- Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings
- Asymptotic mean-value formulas for solutions of general second-order elliptic equations
- Weighted critical exponents of Sobolev-type embeddings for radial functions
- Existence and asymptotic behavior of solitary waves for a weakly coupled Schrödinger system
- On the Lq-reflector problem in ℝn with non-Euclidean norm
- Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity
- Normalized solutions for a class of scalar field equations involving mixed fractional Laplacians
- Multiplicity and concentration of semi-classical solutions to nonlinear Dirac-Klein-Gordon systems
- Multiple solutions to multi-critical Schrödinger equations
- Existence of solutions to contact mean-field games of first order
- The regularity of weak solutions for certain n-dimensional strongly coupled parabolic systems
- Uniform stabilization for a strongly coupled semilinear/linear system
- Existence of nontrivial solutions for critical Kirchhoff-Poisson systems in the Heisenberg group
- Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic field
- Least energy sign-changing solutions for Schrödinger-Poisson systems with potential well
- Lp Hardy's identities and inequalities for Dunkl operators
- Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models
- Gradient estimate of the solutions to Hessian equations with oblique boundary value
- Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry
- A Liouville theorem for the Hénon-Lane-Emden system in four and five dimensions
- Regularity of degenerate k-Hessian equations on closed Hermitian manifolds
- Principal eigenvalue problem for infinity Laplacian in metric spaces
- Concentrations for nonlinear Schrödinger equations with magnetic potentials and constant electric potentials
- A general method to study the convergence of nonlinear operators in Orlicz spaces
- Existence of ground state solutions for critical quasilinear Schrödinger equations with steep potential well
- Global existence of the two-dimensional axisymmetric Euler equations for the Chaplygin gas with large angular velocities
- Existence of two solutions for singular Φ-Laplacian problems
- Existence and multiplicity results for first-order Stieltjes differential equations
- Concentration-compactness principle associated with Adams' inequality in Lorentz-Sobolev space
Articles in the same Issue
- Research Articles
- Editorial
- A sharp global estimate and an overdetermined problem for Monge-Ampère type equations
- Non-degeneracy of bubble solutions for higher order prescribed curvature problem
- On fractional logarithmic Schrödinger equations
- Large solutions of a class of degenerate equations associated with infinity Laplacian
- Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings
- Asymptotic mean-value formulas for solutions of general second-order elliptic equations
- Weighted critical exponents of Sobolev-type embeddings for radial functions
- Existence and asymptotic behavior of solitary waves for a weakly coupled Schrödinger system
- On the Lq-reflector problem in ℝn with non-Euclidean norm
- Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity
- Normalized solutions for a class of scalar field equations involving mixed fractional Laplacians
- Multiplicity and concentration of semi-classical solutions to nonlinear Dirac-Klein-Gordon systems
- Multiple solutions to multi-critical Schrödinger equations
- Existence of solutions to contact mean-field games of first order
- The regularity of weak solutions for certain n-dimensional strongly coupled parabolic systems
- Uniform stabilization for a strongly coupled semilinear/linear system
- Existence of nontrivial solutions for critical Kirchhoff-Poisson systems in the Heisenberg group
- Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic field
- Least energy sign-changing solutions for Schrödinger-Poisson systems with potential well
- Lp Hardy's identities and inequalities for Dunkl operators
- Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models
- Gradient estimate of the solutions to Hessian equations with oblique boundary value
- Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry
- A Liouville theorem for the Hénon-Lane-Emden system in four and five dimensions
- Regularity of degenerate k-Hessian equations on closed Hermitian manifolds
- Principal eigenvalue problem for infinity Laplacian in metric spaces
- Concentrations for nonlinear Schrödinger equations with magnetic potentials and constant electric potentials
- A general method to study the convergence of nonlinear operators in Orlicz spaces
- Existence of ground state solutions for critical quasilinear Schrödinger equations with steep potential well
- Global existence of the two-dimensional axisymmetric Euler equations for the Chaplygin gas with large angular velocities
- Existence of two solutions for singular Φ-Laplacian problems
- Existence and multiplicity results for first-order Stieltjes differential equations
- Concentration-compactness principle associated with Adams' inequality in Lorentz-Sobolev space
