Dynamics for wave equations connected in parallel with nonlinear localized damping
-
Yunlong Gao
und
Kaibin Zhang
Abstract
This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.
1 Introduction
The well-posedness, stability, blowup of solutions, and existence of attractors for evolution wave systems have attracted the attention of many scholars (e.g., [1,2,5–13,19–27,29–42,46–50]). In this work, our main aim is to investigate the long-time dynamical behavior of the following system of coupled wave equations in parallel with nonlinear localized damping:
subject to initial-boundary conditions
where
where
where
There have been some main results regarding the study of linear and nonlinear parallel coupled wave equations. For example, Li and Wang [36] studied the conditions for the existence of partially synchronized solutions in a coupled dissipationless system of one-dimensional wave equations with different wave speeds within the framework of classical solutions. If the principal operator of System (1.1) in three-dimensional space is the Lamé operator, then the Lamé system is equivalent to three coupled wave equations. Yang et al. [50] studied the stability of solutions and the existence of global and exponential attractors for weakly damped Lamé systems with nonlinear time-varying delay. By considering different boundary conditions and wave propagation speeds, Najafi and Wang [39] proved the exponential stability of the linear system (1.1) which contained the linear full damping and without nonlinear source. Rajaram and Najafi [42] further proved that such system achieved an exact controllability result with Dirichlet boundary controls, without any restrictions on the coupling parameters. Moreover, the control time was independent of the coupling constants and was determined by geometric optics. Wu et al. [48] also investigated the exact controllability of the linear system (1.1) in the case where the principal part has variable coefficients. For the nonlinear system (1.1) and (1.2) with nonlinear full damping and source terms, Freitas et al. [26] had studied the singular limit dynamics of equations (1.1) and (1.2). To the best of our knowledge, Problems (1.1) and (1.2) with local damping have not been studied yet. For more details on the parallel connection wave equation, please refer to [40,41].
In recent years, scientists have shown a growing interest in studying the behavior of solutions to nonlinear models. Several of these models, such as Timoshenko system [7,8,21], Bresse system [11,38], porous elastic system [19,27], Lamé system [50], among others, are constituted by coupled nonlinear damped wave equations. These dissipative wave systems can be divided into fully (locally) dissipative systems (i.e., all the damping coefficients in the study area are strictly greater than zero) and partially (nonlocally) dissipative systems based on the damping coefficients. Generally, the study of fully dissipative systems is relatively simple, as we do not need to overly consider how the geometric characteristics of the study area affect dissipation. There are many achievements in this area, such as research on the stability, existence of attractors, and precise controllability of fully damped wave systems, we can refer to [21,26,27,29–31,39–42,47–50] and other related studies.
In terms of physics, partially dissipative wave systems reflect the mechanism of localized energy loss in the medium. The energy of the wave will gradually decrease with the propagation distance but not completely disappear. This localized energy loss can reflect more better the energy attenuation characteristics of the medium in real systems. In terms of control and optimization, partially dissipative wave systems often have more applications. By designing control strategies, it is possible to adjust the region or frequency range of localized dissipation, thereby achieving goal control. Compared to fully dissipative wave systems, the study of partially dissipative systems is more challenging due to the potential requirement for geometric control conditions (GCCs) (e.g., [1,3,5]), and the scarcity of relevant literature, particularly for coupled systems. For example, Zuazua [22], Feireisl and Zuazua [23] investigated the exponential decay and existence of global attractors for semilinear single wave equation with locally distributed damping and nonlinear source term, and required to consider the case where the damping term grows linearly at infinity. For single wave equations with localized damping satisfying sublinear, or linear, or superlinear at infinity, their uniform decay rates had been studied in [9,16,17,35]. Note that these works all relied on the unique continuation property (UCP) of single wave equation proved in [33], which dictated whether wave equations that vanished in a subdomain must be identically null. For coupled wave systems with localized linear damping, the UCP can be derived from Holmgren’s uniqueness theorem, which is valid for equations with analytic coefficients [8,11,16]. However, the UCP for coupled wave systems with nonanalytic coefficients had been an open problem for a long time. Therefore, many previous studies on coupled wave equations with localized nonlinear damping and source terms treated the UCP as an a priori (without proof) condition (e.g., [7,20,24]). Recently, Cavalcanti et al. [10] proved the UCP of solutions to the general coupled wave equation system in hyper two-dimensional space in 2020. After that, Ma et al. [38] also obtained a new UCP for a system of one-dimensional wave equations through Carleman-type estimates. Combining the UCP with GCCs [1,3,5], the uniform stability and existence of attractors for various types of coupled hyperbolic equations (e.g., Klein-Gordon equation, Timoshenko system, Bress system, etc.) with localized damping had been studied in [2,6,7,10,25,38].
On the other hand, the coupled fully damped wave equations had shown some results in terms of stability under the influence of damping terms with different growth conditions at infinity (e.g., [19,27,29,30]). On the contrary, for the case of localized damping, the current results only consider the behavior of solutions with linear growth at infinity, and there are no stability results for other growth cases (i.e., sublinear, superlinear), [2,6–8,10,11,24,25,38,41]. Moreover, it is worth noting that Ferreira [24] also raised an open question about the mixed growth conditions, i.e., what is the stability of a coupled hyperbolic system with localized damping satisfying different growth conditions at infinity?
In this study, we will consider the uniform stability of solutions to Systems (1.1) and (1.2) under nine different combinations of growth conditions for nonlinear damping at infinity (Theorem 3.1). In addition, in contrast to the study of fully damped systems [26], we consider the existence of global attractors and determining functionals for the coupled system by imposing a minimal amount of support for the damping (Theorems 4.2 and 5.1).
In terms of stability analysis of Systems (1.1) and (1.2), we mainly face three challenges or innovations. First, we consider damping with different growth conditions at infinity, i.e.,
| Combination | (L, L) | (Sub, Sub) | (Super, Super) |
|
|
|
|
|
| Combination | (L, Sub) | (L, Super) | (Sub, Super) |
|
|
|
|
|
| Combination | (Sub, L) | (Super, L) | (Super, Sub) |
|
|
|
|
|
The manuscript is organized as follows: In Section 2, we present the notations needed, we list the standing assumptions on the nonlinear terms and give well-posedness results. In Section 3, we prove the uniform decay rates of energy under the effect of two nonlinear localized frictional viscous mechanisms. In Section 4, we study the existence of global attractors. In Section 5, we give the construction of determining functionals. In Section 6, we state microlocal analysis background and abstract results of dynamical systems throughout the manuscript.
2 Preliminaries and well-posedness
In this section, we will present the notations, assumptions, and well-posedness of solutions for Problems (1.1) and (1.2).
2.1 Preliminaries
The relation
and as usual
Now, let us consider the Hilbert space
where inner product and norm on
and
for
Finally, we consider the following some assumptions on
Assumption 2.1
For any
- (2.3)
and there exist constants
(2.4) - (2.5)
There exists a function
(2.6)and there exists
(2.7)and
(2.8)
Remark 2.1
By assumption (A1), if the order
Remark 2.2
There is a large class of nonlinear force satisfying (A2) and (A3). For instance,
2.2 Local and global well-posedness
Under the above assumptions and notations, we are able to state well-posedness of the Problems (1.1) and (1.2). To this end, denoting
where the nonlinear unbounded operator
Here the domain of
The autonomous forcing terms are represented by a nonlinear function
Next, we give the definition of weak and strong solutions for Problems (1.1) and (1.2).
Definition 2.1
A function
for all
then it is called a strong solution.
The well-posedness of Problems (1.1) and (1.2) is given in the following theorem.
Theorem 2.1
(Local and global in time) Suppose that (A1) and (A2) hold in Assumption 2.1.
(Energy identity) The following energy identity holds true:
(2.14)where
(Continuous dependence) The weak solution
(2.15)In particular, solution is unique.
Further, if the assumption (A3) holds, then
Proof
By employing the standard arguments of nonlinear semigroup theory, as outlined in [43] and Theorem 7.2 from [13], we can prove (i)–(iii) proceed with our proof following the methodology presented in the proofs of Theorems 2.4 and 2.5 in [27] (also referring to [9,16,30]). In order to prove (iv), we can use contradiction arguments. Indeed, let us suppose that
in which
that is,
Therefore, it follows that
Remark 2.3
If the inequalities (2.7) and (2.8) in (A3) are replaced by
and
then we can establish a potential well framework proposed by Sattinger [44] to prove that (iv) in Theorem 2.1 also holds with small initial energy. So under these conditions, we can also study the uniform stability of the third part [27,29–31,47].
3 Asymptotic stability
In dissipative dynamical systems, the uniform stability of energy plays a significant role in control and optimization, analyzing long-term behavior, model validation, and more, especially in cases of polynomial and exponential decay. Therefore, in this section, we will investigate the uniform decay rate of energy for equations (1.1) and (1.2) by establishing an observability inequality.
3.1 Observability inequality
Before proving the observability inequality, we still need to make some assumptions. Due to the presence of localized damping and nonconstant coefficients
Assumption 3.1
Remark 3.1
When we do not have any control on the geodesics of the metric
For all
For all geodesic
One of the ingredients for obtaining stability of wave equations with localized damping is a UCP. So we make the following lemma:
Lemma 3.1
Suppose that Assumptions (1.3) and (1.4) hold and
of problem
where
Proof
Its proof can be obtained from Theorem 3.2 in [38].□
Remark 3.2
The lemma is called UCP. In the case of a single equation, these kinds of results were studied by Koch and Tataru [33], but for coupled systems, it remained an open problem before the year 2020. Recently, inspired by Dos Santos Ferreira [18] and Koch and Tataru [33], Cavalcanti et al. ([6], Proposition 2.3 and [10], Proposition 3.6) and Faria and Souza Franco ([20], Proposition 1.2) gave an example where the UCP holds at least locally, i.e., for the system
where
Now, we state and prove the observability inequality lemma for Problems (1.1) and (1.2).
Lemma 3.2
Under the assumptions of Assumptions 2.1, 3.1, and Lemma 3.1, let
Proof
If
If we multiply (3.4)
This implies that
Next our proof relies on contradiction arguments. If Lemma 3.2 does not hold, then there exists
By (2.16), it implies that
which is equivalent to
Once
Due to the lengthy proof process, we will prove the lemma in the following four steps:
Step 1. Assert
On the one hand, since
and from Aubin-Lions theorem ([45], Corollary 4), we obtain
which yields that, as
Indeed, by (2.5), (3.9), Hölder’s inequality, and the embedding
where
At this point in the proof we will divide it into two cases: Case (a):
passing to the limit in (3.12) and taking (3.6)–(3.11) into account, we obtain
Thus, for
which implies that
Multiplying (3.15) by
Thus, we obtain that
Case (a):
For each
Taking the derivatives of (3.17)
If we consider the following functions:
by using Assumptions (2.5) and (3.7), one can easily obtain
and then
which implies
Finally, multiplying (3.19)
Again using Assumptions (2.7) and (2.8), we infer that
so
Case (b):
From (3.7), (3.8), and (3.9), now we have that
Step 2. Assert
Now, we define
Using (3.5) and the definition of
On the other hand, for each
The energy functional associated with (3.27) is given by
Taking in the same way as (2.16), we have
Moreover, it is not difficult to check that
In order to achieve a contradiction, our main goal is to prove that
This will complete the proof (3.3) as desired. Indeed, since
Since
Next let us prove that
Case (A):
Passing to the limit as
Hence, taking derivatives over
where
If we denote the following functions:
we can rewrite (3.36) as
Moreover, by (2.5) and (3.32), it is easy to obtain
Case (B):
From (3.24),
Taking (3.26), (3.32), (3.33), (3.38), (3.39), and (3.40) into account, passing to the limit in (3.27) arrive at
For
Denoting
Then, in both cases
Step 3. Assert
Let us denote
Taking into account (3.26), (3.34), (3.43), and (3.44), we deduce that as
and
strongly in
Let us also denote by
From Theorem 6.2 (Appendix), the support of the measures
Indeed, from Theorem 6.4 and Proposition 6.1, it follows that
where
From the convergence
From Remark 6.1, it follows that
Proceeding analogously as above, we also deduce
Step 4. Derive contradictory point
Next it is easy to show that
Therefore, taking the limit as
which implies that
Moreover, we have that
then, from (2.8) we conclude
Combining the above convergence we obtain that
Then, by nonincreasing property of the energy function, we obtain
Combining the energy identity
along with (3.26), we can obtain that
Therefore, we have that for
which is a contradiction with (3.30) and it allows us to conclude that (3.3) holds.□
Remark 3.3
The combination of the Microlocal analysis method and GCC condition is mainly used to prove that
3.2 Stability result
In order to characterize decay rates for the energy, we need to introduce several special functions, which in turn will depend on the growth of
Proceed as in [27,34], since
We define
In this way,
Assumption 3.2
(Regularity assumptions)
If
for some
If
for some
If
for some
If
for some
where
In this setting, the following technical lemma due to Lasiecka and Tataru [34] is given.
Lemma 3.3
[34, Lemma 3.3] Let p be a positive, increasing function such that
Then,
Moreover, if
We are now in a position to state and briefly prove the main result of our work. It reads as follows.
Theorem 3.1
Let Assumption 3.2 hold and
with
where
the constant
Proof
Proceed as in Daoulatli et al.’s work [16] (Lemma 3.1 of the referred paper) adapted to our context. We define
The proof will require the following inequalities:
Case (1): Damping near the origin. Let
Analogously, we can conclude that
Since each
where
Case (2):
Analogously, we can conclude that
Thus, by using (3.69) and (3.70), we obtain
Case (3a):
On the other hand, let
and choosing
that is,
Therefore, from (2.4) we have
Combining (3.72) and (3.74), we conclude that
Similar to previous calculations, we also conclude that
Summing (3.75) and (3.76), we obtain
Case (3b):
Next let
and choosing
that is,
From (2.4), we know
Now, using (3.78) and (3.80), we have
Analogously, we can conclude that
Then, combining (3.87) and (3.88), we obtain
Case (3c):
Case (3d):
Case (3e):
Case (3f):
Case (3g):
Case (3h):
Finally, combining the above estimates, the observability inequalities (3.3) and (3.62), we deduce that
where the constant
To finish the proof, we replace
Thus, using Lemma 3.3 with
Finally, observing that for every
where we have used the fact that the solution
Remark 3.4
If
3.3 Some examples
While Theorem 3.1 provides an abstract estimate of uniform stability, the specific decay behavior is determined by the growth properties of the nonlinear damping function
where
which satisfies Inequality (3.60). However, if the dampings are bounded by sublinear functions near the origin, namely, for all
where
which also satisfies Inequality (3.60). Therefore, by (3.93) and (3.95), there exists constant
where
Next, we rely on a corollary provided by Lasiecka and Toundykov [35] or Cavalcanti et al. [9] for calculations and estimations.
Corollary 3.1
Suppose that the assumptions of Theorem 3.1 are satisfied and that the function h in (3.62) can be expressed as
where
with
Finally, we will provide some specific examples of energy stability based on Corollary 3.1.
Example 3.1
If
Therefore, according to (3.64) and (3.65), the function
Example 3.2
If at least one of
where
which satisfies ODE
where
Then, it is known from (3.97) that the decay rate of
4 Global attractors
In this section, our main task is to study the existence of global attractors for Problems (1.1) and (1.2). In order to obtain a nontrivial global attractor, we need to modify Assumptions (2.4), (2.7), and (2.8) as follows:
Assumption 4.1
Suppose that
(4.1)Moreover, we assume that
(4.2)There exists constants
(4.3)(4.4)
Remark 4.1
Assumption (4.1) can be replaced with a weaker assumption
Assumption (G2) guarantees existence of nontrivial global attractors. Moreover, Theorem 2.1 also holds under the assumption (G2). Then, Problems (1.1) and (1.2) generate a dynamical system in the space
where
4.1 Gradient system
In this subsection, we prove that the dynamical system
Lemma 4.1
Suppose that (A1), (A2) of Assumptions 2.1 and 4.1 hold. Then, the dynamical system
Proof
Let
From (2.3) and (2.16), we have that
Thus, the function
Next, we suppose that
Then, using (4.1), we can deduce for all
and
which shows that for all
and
This means that
By density arguments, we can assume
If we apply to (2.5) and set the following functions:
we can rewrite (4.9) as
Using Lemma 3.1, we conclude that
Now, it follows from (4.3) and Poincaré inequality that
and therefore, (4.11) together with
On the other hand, using the embedding
So, we obtain
Combining (4.12) and (4.13), we have
and it implies that (4.5) holds. The proof is complete.□
Lemma 4.2
Suppose that (A1), (A2) of Assumption 2.1 and Assumption 4.1 hold. Then, the set
Proof
Stationary points
Multiplying the first equation in equation (4.15) by
By Poincare inequality, (4.3), and (4.4), we have
Then, using
The proof is complete.□
4.2 Asymptotic smoothness
In this subsection, we will show that the dynamical system
Lemma 4.3
Under the hypotheses Lemma 4.1 given a bounded subset
where
Proof
For
Then,
with the Dirichlet boundary condition and initial condition
Let us consider
with
Now, multiplying the equations in (4.19) by
Then, observing that
we deduce that
Step 1. Now, let us estimate the left-hand side of (4.23). First, employing the embedding
Using Hölder’s and Young’s inequalities yields that
To estimate the damping terms, we use the assumption (G1), which implies that for any
Thus, by (4.2), (4.26), and Young’s inequality, we obtain for every
Similarly, we obtain
This allows us to conclude the following estimate:
Next we estimate the source terms. Using the assumption (2.5), Hölder’s inequality with exponents
and analogously, we also have
The above implies that
Now, we combine (4.28), (4.29), (4.32) with (4.23). For
Step 2. Next we estimate the fourth integral on the right in (4.33). We consider a cutoff function
Thus, multiplying the first and second equations on equation (4.19) by
We shall estimate the right-hand side of (4.35). Analogous to (4.25), we also have
Again applying the estimate (4.26), we can conclude that
Now, integrating by parts and using the embedding
Again using (4.2), (4.26), and Hölder’s and Young’s inequalities, we deduce that
Analogous to (4.30), we see that
Substituting the estimates (4.36)–(4.40) into (4.35) and choosing
Inserting (4.41) into (4.33), and choosing
Step 3. Next we estimate damping terms in (4.42). Multiplying the first and second equation of (4.19) by
Analogous to (4.32), we can conclude that
Consequently, combining (4.44) (
We return to (4.42) and using (4.45), with
Step 4. Integrating the energy equality (4.43) with respect to
Similarly (4.44), we find
By (4.26), and substituting (4.48) into (4.47) yields
Step 5. Finally, by the estimates (4.46) and (4.49), we can find
Therefore, we see that
Let
Theorem 4.1
Under the assumptions of Lemma 4.3. Then, the system
Proof
In order to prove Theorem 4.1, we will use the abstract result Theorem 6.5. Let
where
Let
Hence,
Thus, by Alaoglu’s theorem, there exists a subsequence, reindexed again by
By Simon’s compactness theorem [45], we conclude that
Therefore,
That is,
The asymptotic smoothness follows by Theorem 6.5. This completes the proof.□
Remark 4.2
All the constants in the proofs of Lemma 4.3 and Theorem 4.1 are independent of the variable
4.3 Existence of global attractors
In this section, we prove the existence of global attractors for the dynamical systems
Theorem 4.2
Under the assumptions of Lemma 4.3. Then, the system
where
Proof
Using Lemma 4.1, Lemma 4.2, and Theorem 4.1, we know all the assumptions of [15, Corollary 7.5.7] are fulfilled and consequently the system
Remark 4.3
Suppose
As in [26], if the proof process has been slightly modified, we can prove finiteness of fractal dimension and smoothness of global attractors by Quasi-stability ([14], Theorem 3.4.18). We can also establish that the singular limit of two wave equations connected in parallel is a single wave equation when
5 Determining functionals
Determining functionals can be interpreted as some kinds of measurements of the state of the system. Based on the fourth section, we now present several results on the determining functionals of the dynamical system
5.1 The concept and a related lemma of determining functionals
In order to state the main conclusion, we first provide the definitions of completeness defect and the determining functionals for the dynamical system
Definition 5.1
Let
which implies that
Definition 5.2
Let
Next, we prove the following key lemma.
Lemma 5.1
Let
Proof
Let
By the definition in (5.3), we have that
for some
5.2 Construction of determining functionals
According to Theorem 4.2, if there is a global attractor, then the dynamic system
Theorem 5.1
Take the hypotheses from Theorem 4.2 and (4.57), and consider
Proof
Let
which implies that
Indeed, suppose that the assumption in (5.6) holds and note that this is equivalent to
Based on (4.50), Assumption (4.57),
for some
By using interpolation and the embedding
Then, from [14, (3.3.9), p. 123] with
It follows from (5.10) and (5.11) that we obtain
With the Lipschitz estimate on
Then, from (5.9) and (5.13), we conclude that
Choosing
Thus, again using the semigroup property, we can iterate on intervals of size
From here, taking
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions that help to improve and clarify the article greatly.
-
Funding information: This work was supported by the National Natural Science Foundation of China (Grant No. 12271227).
-
Author contributions: Yunlong Gao: writing – original draft preparation, Chunyou Sun: writing - reviewing and editing, and Kaibin Zhang: writing – original draft preparation.
-
Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this article.
-
Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Appendix A Part I: Microlocal analysis
For the readers comprehension, we will recall some results which can be found in Burq and Gérard [4] and in Gérard [28] and were used in the proof of the stabilization.
Theorem A.1
[4, Theorem 5.14] Let
Definition A.1
Under the circumstances of Theorem A.1,
Remark A.1
Theorem A.1 assures that for all bounded sequence
so that
The second important result reads as follows.
Theorem A.2
[4, Theorem 5.18] Let P be a differential operator of order m on
Theorem A.3
[4, Theorem 5.19] Let P be a differential operator of order m on
We complete this section by examining the case of the wave equation in an inhomogeneous medium:
whose principal symbol is given by
where
for
Proposition A.1
Unless a change in variables, the bicharacteristics of (A4) are curves of the form
where
The main result as follows:
Theorem A.4
[4, Theorem A.1] Let P be a self-adjoint differential operator of order m on
B Part II: Abstract results of dynamical systems
For the reader’s convenience, we present here some definitions and theorems related to dynamics of dissipative systems that can be found in classical works (e.g., [14,15]).
Definition B.2
Let
where
Definition B.3
Let
where
Theorem B.5
[14, Theorem 2.2.17] Let
where
for every sequence
Theorem B.6
[14, Theorem 2.3.5] Let
for every bounded absorbing set
whereas above
Definition B.4
Let
Then, the unstable manifold emanating from
Definition B.5
A dynamical system
Theorem B.7
[15, Corollary 7.5.7] Let
and that the set of stationary points
References
[1] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065. 10.1137/0330055Suche in Google Scholar
[2] C. Buriol, L. G. Delatorre, V. H. Gonzalez Martinez, D. C. Soares, and E. H. G. Tavares, Asymptotic stability for a generalized nonlinear Klein-Gordon system, J. Differ. Equ. 280 (2021), 517–545. 10.1016/j.jde.2021.01.011Suche in Google Scholar
[3] N. Burq and P. Gérard, A necessary and sufficient condition for the exact controllability of the wave equation, C. R. Acad. Sci. Paris Seeeer. I Math. 325 (1997), no. 7, 749–752. 10.1016/S0764-4442(97)80053-5Suche in Google Scholar
[4] N. Burq and P. Gérard, Contróle Optimal des équations aux dérivés partielles, 2001, http://www.math.u-psud.fr/ burq/articles/coursX.pdf. Suche in Google Scholar
[5] M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result, Arch. Ration. Mech. Anal. 197 (2010), 925–964. 10.1007/s00205-009-0284-zSuche in Google Scholar
[6] M. M. Cavalcanti, V. N. D. Domingos Cavalcanti, S. Mansouri, V. H. Gonzalez Martinez, Z. Hajjej, M. R. Astudillo Rojas, Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping, J. Differ. Equ. 268 (2020), 447–489. 10.1016/j.jde.2019.08.011Suche in Google Scholar
[7] M. M. Cavalcanti, W. J. Corrêa, V. N. Domingos Cavalcanti, M. A. Jorge Silva, and J. P. Zanchetta, Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping, Z. Angew. Math. Phys. 72 (2021), 191. 10.1007/s00033-021-01622-7Suche in Google Scholar
[8] M. M. Cavalcanti, V. N. Domingos Cavalcanti, F. A. F. Nascimento, I. Lasiecka, and J. H. Rodrigues, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys. 65 (2014), 1189–1206. 10.1007/s00033-013-0380-7Suche in Google Scholar
[9] M. M. Cavalcanti, V. N. Domingos Cavalcanti, V. H. Gonzalez Martinez, and T. Özsari, Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation, Appl. Math. Optim. 87 (2023), 2. 10.1007/s00245-022-09918-4Suche in Google Scholar
[10] M. M. Cavalcanti, V. N. Domingos Cavalcanti, V. H. Gonzalez Martinez, V. A. Peralta, and A. Vicente, Stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping, J. Differ. Equ. 269 (2020), no. 10, 8212–8268. 10.1016/j.jde.2020.06.013Suche in Google Scholar
[11] W. Charles, J. A. Soriano, F. A. Falcao Nascimento, and J. H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, J. Differ. Equ. 255 (2013), no. 8, 2267–2290. 10.1016/j.jde.2013.06.014Suche in Google Scholar
[12] Y. X. Chen and R. Z. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal. 192 (2020), Paper no. 111664, 39 pp. 10.1016/j.na.2019.111664Suche in Google Scholar
[13] I. Chueshov, M. Eller, and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. Part. Differ. Equ. 27 (2002), 1901–1951. 10.1081/PDE-120016132Suche in Google Scholar
[14] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer International Publishing, Switzerland, 2015. 10.1007/978-3-319-22903-4Suche in Google Scholar
[15] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. 10.1007/978-0-387-87712-9Suche in Google Scholar
[16] M. Daoulatli, I. Lasiecka, and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 1, 67–94. 10.3934/dcdss.2009.2.67Suche in Google Scholar
[17] M. Daoulatli, Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping, Nonlinear Anal. 73 (2010), 987–1003. 10.1016/j.na.2010.04.026Suche in Google Scholar
[18] D. Dos Santos Ferreira, Sharp Lp carleman estimates and unique continuation, Journées équations aux Dérivés Partielles, vol. VI, 2003, p. 12. Suche in Google Scholar
[19] M. J. Dos Santos, B. Feng, D. S. Almeida, and M. L. Santos, Global and exponential attractors for a nonlinear porous elastic system with delay term, Discrete Contin. Dyn. Syst. Ser. B 22 (2020), no. 11, 1–24. Suche in Google Scholar
[20] J. C. O. Faria and A. Y. Souza Franco, Well-posedness and exponential stability for a Klein-Gordon system with locally distributed viscoelastic dampings in a past-history framework, J. Differ. Equ. 346 (2023), 108–144. 10.1016/j.jde.2022.11.022Suche in Google Scholar
[21] L. H. Fatori, M. A. Jorge Silva, and V. Narciso, Quasi-stability property and attractors for a semilinear Timoshenko system, Discrete Contin. Dyn. Syst. 36 (2016), no. 11, 6117–6132. 10.3934/dcds.2016067Suche in Google Scholar
[22] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Commun. Partial Differ. Equ. 15 (1990), 205–235. 10.1080/03605309908820684Suche in Google Scholar
[23] E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Commun. Partial Differ. Equ. 18 (1993), 1539–1555. 10.1080/03605309308820985Suche in Google Scholar
[24] J. S. Ferreira, Exponential decay for a nonlinear system of hyperbolic equations with locally distributed dampings, Nonlinear Anal. 18 (1992), no. 11, 1015–1032. 10.1016/0362-546X(92)90193-ISuche in Google Scholar
[25] M. M. Freitas, R. Q. Caljaro, and A. J. A. Ramos, Long-time dynamics of ternary mixtures with localized dissipation, J. Math. Phys. 63 (2022), 121508. 10.1063/5.0098498Suche in Google Scholar
[26] M. M. Freitas, R. Q. Caljaro, M. L. Santos, and A. J. A. Ramos, Singular limit dynamics and attractors for wave equations connected in parallel, Appl. Math. Optim. 85 (2022), 9. 10.1007/s00245-022-09849-0Suche in Google Scholar
[27] M. M. Freitas, M. L. Santos, and L. A. Langa, Porous elastic system with nonlinear damping and sources terms, J. Differ. Equ. 264 (2018), 2970–3051. 10.1016/j.jde.2017.11.006Suche in Google Scholar
[28] P. Gérard, Microlocal defect measures, Commun. Partial Differ. Equ. 16 (1991), 1761–1794. 10.1080/03605309108820822Suche in Google Scholar
[29] Y. Q. Guo and M. A. Rammaha, Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z. Angew. Math. Phys. 64 (2013), 621–658. 10.1007/s00033-012-0252-6Suche in Google Scholar
[30] Y. Q. Guo and M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc. 366 (2014), 2265–2325. 10.1090/S0002-9947-2014-05772-3Suche in Google Scholar
[31] J. B. Han, R. Z. Xu, and C. Yang, Continuous dependence on initial data and high energy blowup time estimate for porous elastic system, Commun. Anal. Mech. 15 (2023), no. 2, 214–244. 10.3934/cam.2023012Suche in Google Scholar
[32] L. F. Ho, Exact controllability of the one dimensional wave equation with locally distributed control, SIAM J. Control Optim. 28 (1990), no. 3, 733–748. 10.1137/0328043Suche in Google Scholar
[33] H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math. 58 (2005), no. 2, 217–284. 10.1002/cpa.20067Suche in Google Scholar
[34] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation, Differ. Integral Equ. 6 (1993), 507–533. 10.57262/die/1370378427Suche in Google Scholar
[35] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal. 64 (2006), no. 8, 1757–1797. 10.1016/j.na.2005.07.024Suche in Google Scholar
[36] Y. C. Li and C. M. Wang, On a partially synchronizable system for a coupled system of wave equations in one dimension, Commun. Anal. Mech. 15 (2023), no. 3, 470–493. 10.3934/cam.2023023Suche in Google Scholar
[37] W. Lian and R. Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal. 9 (2020), no. 1, 613–632. 10.1515/anona-2020-0016Suche in Google Scholar
[38] T. F. Ma, R. N. Monteiro, and P. N. Seminario-Huertas, Attractors for locally damped Bresse systems and a unique continuation property, 2021, arXiv:2102.12025. Suche in Google Scholar
[39] M. Najafi and H. Wang, Exponential stability of wave equations coupled in parallel with viscous damping, In: Proceedings of 32nd IEEE Conference on Decision and Control, December 1993, pp. 15–17. Suche in Google Scholar
[40] M. Najafi, G. R. Sarhangi, and H. Wang, Stabilizability of coupled wave equations in parallel under various boundary conditions, IEEE Trans. Autom. Control 42 (1997), no. 9, 1308–1312. 10.1109/9.623099Suche in Google Scholar
[41] M. Najafi, Energy decay estimate for the boundary stabilization of the coupled wave equations, Math. Comput. Model. 48 (2008), 1796–1805. 10.1016/j.mcm.2008.06.002Suche in Google Scholar
[42] R. Rajaram and M. Najafi, Exact controllability of wave equations in Rn coupled in parallel, J. Math. Anal. Appl. 356 (2009), 7–12. 10.1016/j.jmaa.2009.02.034Suche in Google Scholar
[43] R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. Suche in Google Scholar
[44] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30 (1968), 148–172. 10.1007/BF00250942Suche in Google Scholar
[45] J. Simon, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl. 146 (1987), no. 4, 65–96. 10.1007/BF01762360Suche in Google Scholar
[46] J. T. Webster, Attractors and determining functionals for a flutter model: finite dimensionality out of thin air, Pure Appl. Funct. Anal. 5 (2020), 85–119. Suche in Google Scholar
[47] X. C. Wang, Y. X. Chen, Y. B. Yang, J. H. Li, and R. Z. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal. 188 (2019), 475–499. 10.1016/j.na.2019.06.019Suche in Google Scholar
[48] J. Q. Wu, S. J. Li, and S. G. Chai, Exact controllability of wave equations with variable coefficients coupled in parallel, Asian J. Control 12 (2010), no. 5, 650–655. 10.1002/asjc.179Suche in Google Scholar
[49] C. Yang, V. D. Rădulescu, R. Z. Xu, and M. Y. Zhang, Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models, Adv. Nonlinear Stud. 22 (2022), no. 1, 436–468. 10.1515/ans-2022-0024Suche in Google Scholar
[50] X. Y. Yang, S. B. Wang, and M. A. J. Silva, Lamé system with weak damping and nonlinear time-varying delay, Adv. Nonlinear Anal. 12 (2023), no. 1, 22. 10.1515/anona-2023-0115Suche in Google Scholar
© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Artikel in diesem Heft
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
