Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
-
Yuxuan Chen
Abstract
In this work, we examine the initial boundary value problem for the thermoelastic system with
1 Introduction
This article is concerned with the dynamical behavior of solutions for the following initial boundary value problem of the generalized thermoelastic system with
where
The initial data
The nonlinearity
For
(1.2)and
(1.3)where
Thermoelasticity has always been an extremely important subject in both theoretical and practical fields. As Dafermos mentioned [8], the theory of thermoelasticity is an appropriate model to explain the attenuation of the amplitude of free vibrations in certain elastic bodies. From a physical perspective, the theory of thermoelasticity is a combination of the theory of elasticity and the theory of heat conduction [15]. It involves the effect of heat on the deformation of an elastic medium, as well as the reverse effect of deformation on the thermal state of the medium. Thermal stresses arise when the time rate of change of the heat sources in the medium or the time rate of change of the thermal boundary conditions on the medium is compared with the characteristics of structural vibrations. In this case, the solution for the temperature and stress fields should be derived from the coupled thermoelastic equations [10]. From an application standpoint, thermoelasticity involves the interaction between the thermal and mechanical fields. In these fields, the phenomenon of thermomechanical coupling is extremely common in design and performance improvement, and often brings about many interesting phenomena. In fact, many new structures and devices often involve complex elastic materials and require operation in highly nonlinear states. For example, beams, plates, alloys, and composite materials are typical examples of such media. Therefore, the application needs of these materials in various fields, such as thermal protection systems, mechanical component design, heat treatment processes, and piezoelectric materials, have attracted widespread attention [23]. In classical thermodynamics, extensive research on nonlinear elasticity and thermoelasticity models can be found in the literature. This includes reviews, expositions, general models, and applications aimed at studying the effects of nonlinear coupling, such as those in [4,5,12,46,48].
Since the pioneering work of Dafermos [8] on linear thermoelasfticity, significant progress has been made on the mathematical aspect of thermoelasticity. In the beginning, people mainly considered the dynamical problems of classical thermoelastic systems [33], the 1D linear model of which is given as follows:
where
and obtained the existence of global solutions. Moreover, decay rates of Sobolev norms of the solutions as time tends to infinity are also investigated. Compared with the linear case, the study of existence of solution for nonlinear thermoelastic systems requires more sophisticated methods. We refer to the relevant work done by [31,32,37].
The asymptotic behavior of solutions to the thermoelastic equations as
Pereira and Menzala [41] introduced a linear internal damping mechanism effective throughout the domain and established a uniform decay rate. Oliveira and Charão [38] improved upon the results in [41] by including a weak localized dissipative term effective only in a neighborhood of part of the boundary. They proved an exponential decay result when the damping term is linear and a polynomial decay result for a nonlinear damping term. For more studies focusing on the existence, regularity, and asymptotic behavior of solutions to the equations of nonlinear thermoelasticity, we refer the reader to books by Jiang and Racke [18] and Zheng [51].
Regarding the nonexistence results in the literature, there are numerous studies on this topic. These include the blowup rate of solutions to initial-boundary value problems for thermoelastic systems, in both one-dimensional [20,42] and multidimensional settings (see [1,36] and the references therein). Specifically, for thermoelasticity with second sound, Messaoudi [36] investigated the
He established a local existence result and demonstrated that solutions with negative energy blow up in finite time. Later, Qin and Rivera [42] focused on the Cauchy problem in nonautonomous nonlinear one-dimensional thermoelastic models that follow both Fourier’s law of heat flux and the theory proposed by Gurtin and Pipkin,
They studied the blow-up phenomena of solutions in finite time when the initial energy is negative. Based on the results by Messaoudi [36], it has been shown that for a particular type of semilinear thermoelastic equations, weak solutions collapse in finite time. Almost at the same time, Kirane and Tatar [20] examined the Cauchy problem for a nonlinear system arising in thermoelasticity and proved that its solution develops singularities in finite time, depending on the size and regularity of the initial data. Their work is distinguished from previous studies by relaxing the requirements on the initial data, allowing for a slightly more general and nonautonomous forcing term, and permitting the insertion of gradient terms in both equations of the system. This result improves upon Messaoudi’s work by accommodating a more general set of initial data. To provide threshold critical exponents depending on the space dimension
The study of the long-time dynamic behavior of nonlinear thermoelastic systems is of great significance. In classical models, the hyperbolic elastic system and the parabolic heat conduction model are usually studied separately. However, in the fields of mechanics, physics, and engineering, many string vibration problems are often accompanied by thermal effects, which in turn influence the vibrational behavior, thereby forming an interesting coupled hyperbolic–parabolic system. This coupled system is not only significant in physical reality but also highly valuable in its mathematical properties. As a coupled hyperbolic-parabolic system, it possesses both the typical characteristics of hyperbolic systems and the features of parabolic systems, which makes its study highly attractive. One key question is: In such a coupled system, which characteristic, the hyperbolic or the parabolic, dominates? It turns out that the answer to this question depends on various factors, such as the spatial dimension, initial conditions, nonlinear indices, and the specific type of problem being studied [11,13,30,39]. When studying the relationship between initial conditions and the dynamic behavior of solutions in thermoelastic problems, there are certain particularities. The dissipative effect in thermoelastic problems can be caused not only by the system’s inherent damping but also by the mechanism of heat conduction. Therefore, a natural question arises: Is the dissipative effect caused by heat conduction strong enough to prevent the explosive growth of solutions under large data or at least under small data conditions? To answer this question, it is necessary to deeply explore the dependence of the long-time dynamic behavior of solutions of nonlinear thermoelastic systems on the initial data. Many previous studies have focused particularly on the exponential decay of solutions to thermoelastic systems, that is, the asymptotic behavior of solutions decaying to the equilibrium point (exponentially or polynomially) as time tends to infinity. These studies are undoubtedly of great significance. However, the focus of this article is different from that of previous literature. We are concerned with the dependence of the solutions of thermoelastic systems on the initial data. Specifically, we will discuss the different dynamic behaviors of solutions caused by changes in the magnitude of the initial data and systematically characterize the features of the initial data.
The goal of this article is to establish the result for the global existence of a weak solution and the blow-up phenomena in a finite time frame for the nonlinear system (1.1) in the context of initial energy states, as well as the long-time behavior of global solutions. Specifically, inspired by [6], we consider a class of generalized nonlinear sources that depend on the characteristics of the domain, i.e.,
These nonlinear sources are particularly important in the study of
where
We end this section with a discussion of the organization of the article. Section 2 introduces some preparatory lemmas, defines the family of stable and unstable sets, and provides a local well-posedness result. In Section 3, we discuss the finite time blowup of solutions under negative initial energy. Section 4 is devoted to presenting a sufficient condition for the global existence and finite time blowup of weak solutions under subcritical initial energy, including the decay of global solutions and the lower and upper bound estimates for the blowup time. In Section 5, we extend the results obtained for subcritical initial energy to the critical initial energy case.
2 Preliminaries
Throughout this article, we denote the standard Lebesgue space
Definition 2.1
Let
for a.e.
Furthermore, the total energy of system (1.1) is denoted as follows:
Lemma 2.2
Let
Proof
Multiplying the equation (1.1)
and
Differentiating and then integrating the energy functional
By substituting (2.6) into (2.5), then (2.7) turns into
which yields
For the sake of completeness, we state here a local existence and uniqueness theorem for the nonlinear thermoelastic system (1.1). Its proof can be obtained by following a similar argument to the method mentioned in studies by Chen [3], Messaoudi [36], and Dafermos [8].
Proposition 2.3
(Local existence and uniqueness [3,36]) Let
Now, let’s introduce the potential energy functional
and
where
where the Nehari manifold
In addition, the depth of potential well
Moreover, the Nehari manifold
Lemma 2.4
Suppose that
Proof
We can use the basic variational method to prove (i). The details are as follows, according to the proof of Lemma 2.1 in Payne-Sattinger’s paper [40], we known that a critical point of functional
is a solution of Euler-Lagrange equation (
where the existence and uniqueness of the solution can be found in Chapter 2 of [28]. Given that the functional
Next, we calculus the second variation of
and
Then the second variation of
Choosing the test function
Due to
(ii) By assumption
Fixed
With the help of
which means that
(iii) From assumption
then we obtain
where
Lemma 2.5
Suppose that
Then the function
Proof
Taking the derivative of
Using (i) in Lemma 2.4, we know that
The property
Furthermore, by using the fact
where
If the nonlinearity source
Lemma 2.6
Suppose that
There exists a unique
(2.11)
Proof
(i) According to
we obtain the result immediately.
(ii) By a simple calculations, we can easily obtain
(iii) Let
Since
(iv) Suppose that there two roots of equation
and
Removing the term
This expression can be rewritten by the following substitution
One can easily verify that equation (2.12) does not hold. Thus, we obtain the uniqueness of critical point.
Finally, according to Lemma 2.5, we deduce the case (v)–(vi), and
To introduce the family of potential wells
and
Lemma 2.7
(Sobolev inequality) Let
Lemma 2.8
Suppose that
where
if
if
if
Proof
According to Lemma 2.7, we obtain from
which means that
Since
which yields
According to
which means that
Lemma 2.9
(Theorem 1.1, [19]) There exists
where
In addition, one can recall that
Next lemma describes the properties of
Lemma 2.10
Suppose that
Proof
If
where
tells us that for
(2.14)Thus, for
From Lemma 2.6, one can deduce that
and
Obviously, we deduce the existence of
Indeed, we only need to demonstrate that
where
where one can noticed the fact that
For
Remark 2.11
According to Lemma 2.10, it is easily to verify that the depth of potential well is a positive constant, i.e.,
where
Lemma 2.12
Suppose
if
In particular, for
if
In particular, for
if
Proof
The case (i) follows from
The proof of cases (ii) and (iii) follows from a similar argument.□
Then a family of potential wells can be defined for
In the case
From the definition of
Lemma 2.13
If
Lemma 2.14
Suppose
Proof
From
Therefore, the assumption does not hold, and the original proposition is valid.□
3 Finite time blow up of the solution for system (1.1) with negative initial energy
E
(
0
)
<
0
In this section, we discuss the finite time blow up of the solution to the thermoelastic system (1.1) with
Lemma 3.1
[24] Suppose that
where
then we have
and
Theorem 3.2
Let
where the maximal existence time
Proof
Assume that
with the constant
By employing the condition (iii), Lemma 2.9 and
Therefore,
Then by taking
Using the assumption condition that
thus we deduce
Taking
Hence, by Lemma 3.1, we see that the expression (3.4) for
That completes the proof.□
4 Long-time behavior of the solution for system (1.1) with subcritical initial energy
E
(
0
)
<
d
In this section, we will discuss the initial data partition problem under the subcritical initial energy level. We will prove that when the initial energy is below the depth of the potential well
4.1 Invariant sets of solutions
Theorem 4.1
Let
All solutions of problem (1) with
All solutions of problem (1) with
Proof
For case (i), we assume that
According to (4.1) and the definition of
In virtue of (4.1), it easily verifies that
For case (ii), we again employ that
As in the previous case,
The proof of
for some
Hence, by the definition of
Now we construct a global weak solution for system (1.1) by the Galerkin approximation method. We now begin with the definition of the Krasnoselskii genus [21]. We denote by
Definition 4.2
Let
Lemma 4.3
([50], Theorem 3.1.1) Let
is compact.
Lemma 4.4
([50], Lemma 3.1.7 and Remark 3.1.4) Let
Then
Lemma 4.5
Let
then
4.2 Global existence of solutions
Theorem 4.6
(Global existence for
where c is a positive constant.
Proof
We divide this proof into two steps. Step 1: Global existence of solutions.
For
where
are the eigenvalues of the
is a hemicontinuous, coercive, and monotone operator on
Moreover, it is easy to verify that there is a basis
Hence, we can choose
which satisfy, for
with initial conditions
According to Carathéodory’s theorem, we find that (4.5) lead to a system of ordinary differential equations in the variable
Next, multiplying (4.5) by
Substituting (4.10) into (4.9), we obtain
where
and
From
holds for sufficiently large
Since (4.12), we know that
we establish
for sufficiently large
and
where
Having in mind the aforementioned energy estimates, one can deduce that the sequence
Moreover, according to (4.14)–(4.18) for each
By (4.21), (4.25), and Lemma 4.3, we obtain
Then
which, combining with (4.14) and (4.19), yields
Next, we integrate the two equations of (4.5) over
By (4.21) and (4.23), we obtain that, for each
which tells us that, for a.e.
Therefore, making
Since
hold for any
Now, for sufficiently small
then
Since
Note
By multiplying the both sides of (4.35) by
Therefore, employing (4.14)–(4.18), and Hölder’s inequality, it is easy to obtain
Since
Since
i.e.,
Below, for sufficiently small
hence,
Since
Note
By multiplying the both sides of (4.40) by
Therefore, from (4.16), (4.17), and Hölder’s inequality, it follows that
where
Due to the embedding
By using the fact that
i.e.,
In view of (4.38), (4.41), (4.21), (4.23), and (4.25), by using Lemma 4.4, we have
hold for a.e.
To the end, we claim that (2.4) holds for a.e.
Note (1.3), (4.24), and arbitrariness of
Therefore, by (4.19), (4.42), (4.44), and Hölder’s inequality, one can obtain, for a.e.
as
Step 2: Exponential asymptotic behavior of solutions.
By using Theorem 4.1, we have
we obtain that
Next, we estimate the constant
On the other hand, by the nonincreasing property of energy functional
Together with (4.47) and (4.48), it follows from Theorem 4.1 and
Thus, in virtue of (4.46) and (4.49), we claim that
Moreover, as a consequence, it follows from the definition of
To prove the asymptotic behavior of solutions, we now denote the following Lyapunov functional:
where
With the help of the Hölder and
for any
According to (4.51), we deduce from (4.55) that
Next, we plan to give the estimate for
In virtue of the Poincaré inequality
Recalling
Thus, taking
In virtue of (4.15), it follows that
and
Therefore, (4.60) turns into
According to the definition of
and
Thus, it follows from (4.61), (4.62), and (4.63) that
where
Now, we show the equivalent of
where
Then, we obtain by taking
Finally, it follows from (4.64) that there exist a positive constant
which yields (4.3) immediately.□
4.3 Finite time blow up of solutions
Lemma 4.7
Suppose
Proof
Denote
Since
In addition, it easily verify that
Below, we give (4.66). Taking
Hence, by Lemma 2.8-(ii), it follows that
Finally, we obtain that
which yields (4.66).□
Theorem 4.8
Let
Furthermore, the upper bound of blowup time T is estimated by
And the lower bound of blowup time T is estimated in the following two cases:
if
(4.70)where
if
(4.71)where
Proof
We divide this proof into two steps.
Step 1: Blow up in finite time and upper bound estimate of the blowup time.
Let
By the definition of
Then by Hölder’s inequality, the definition of
Take
Step 2: Lower bound estimate of the blow-up time.
First, by Step 1, we know that the weak solutions of problem (1.1) will blow up in a finite time, i.e.,
In the following, we divide the remainder proof into two cases.
Case 1:
We define the auxiliary function
Since
where
where
Integrating (4.77) from 0 to
which, combining with
Case 2:
We know the embedding
Since
Since
where
Combining (4.74), (4.80), and (4.81), we obtain
In virtue of the interpolation inequality, we arrive at
where
Using the Young inequality and (4.74), it follows that
where
This implies that
Let
where
By integrating (4.85) from 0 to
which, together with
5 Long-time behavior of the solution for system (1.1) with critical initial energy
E
(
0
)
=
d
5.1 Global existence of solutions
In this subsection, we will employ the scaling method to extend the conclusions obtained under low initial energy levels to the critical initial energy level. This primarily includes the global existence of solutions, their asymptotic behavior, and the dependence of finite-time blow-up and blow-up time estimates on the initial data.
Theorem 5.1
(Global existence for
where c is a positive constant.
Proof
We divide this proof into two steps. Step 1: Global existence of solutions.
For the case
According to
Hence, it follows that
which tells us that
for any
In addition, with the help of (2.4), we know
which, combining with the same argument as we did in Theorem 4.1, we deduce that
For the case
Let
Obviously, it follows that
In virtue of (5.6) and Theorem 4.6, we conclude that there exists a global weak solution
Step 2: Exponential asymptotic behavior of solutions.
Recall that the existence of a global weak solution of problem (1.1) is given in Theorem 5.1 with
Assume that
then by using the fact that
and
which yields (5.1) immediately.
Assume that there exists
However, it is easy to verify from
which means that
which tells us that
Therefore,
5.2 Finite time blowup of solutions
Theorem 5.2
Let
Furthermore, the upper bound of blowup time T is estimated by
And the lower bound of blowup time T is estimated in the following two cases:
if
(5.8)where
if
(5.9)where
Proof
Let
holds. Then we know from Theorem 4.1-(ii) that
Obviously, the proof for the remaining part is the same as the process we argue in Theorem 4.8, we hence omit it and obtain the results in the case
Acknowledgment
The author would like to thank the anonymous referees for their valuable comments and suggestions that help to improve and clarify the article greatly.
-
Funding information: The research was supported by the Fundamental Research Funds in Heilongjiang Natural Science Funds for Distinguished Young Scholar (JQ2023A005).
-
Author contributions: The sole author is responsible for the entire work and has read and approved the final manuscript.
-
Conflict of interest: The author states no conflict of interest.
-
Data availability statement: No data were used for the research described in the article.
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- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
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Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
- Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
- Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
- Shape of extremal functions for weighted Sobolev-type inequalities
- One-dimensional boundary blow up problem with a nonlocal term
- Doubling measure and regularity to K-quasiminimizers of double-phase energy
- General solutions of weakly delayed discrete systems in 3D
- Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
- Optimal large time behavior of the 3D rate type viscoelastic fluids
- Local well-posedness for the two-component Benjamin-Ono equation
- Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
- Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
- Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
- On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
- Normal forms of piecewise-smooth monodromic systems
- Fractional Dirichlet problems with singular and non-locally convective reaction
- Sharp forced waves of degenerate diffusion equations in shifting environments
- Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
- Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
- Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
- Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
- Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
- Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
- Generalized quasi-linear fractional Wentzell problems
- Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
- Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
- Boundedness and global stability in a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics
- Review Article
- Existence and stability of contact discontinuities to piston problems
