Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
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Jiaxing Liu
Abstract
This study proposed and analyzed a vector-borne reaction–diffusion–advection model with vector-bias mechanism and heterogeneous parameters in one-dimensional habitat. The basic reproduction number
1 Introduction
Some well-known vector-borne diseases include mosquito-borne diseases (such as dengue fever, chikungunya, West Nile virus, and malaria) [23,31,32,36] and fly-borne diseases, which have become a major concern for national and social development due to environmental change and urbanization. It seems thus imperative to conduct vaccination, health measures, epidemic surveillance, and isolation of infected hosts to control the spread of disease. In recent years, more and more studies indicate that many factors can cause spatial heterogeneity. Specifically, heterogeneity of habitat plays a crucial role in the spread of infectious diseases [1,4,9–11,15,16,18,19,21,26,27,33]. Among the aforementioned works, Fitzgibbon et al. [9,10] utilized mathematical models that formulated a reaction–diffusion system on non-coincident domains to investigate the circulation of diseases between two host populations. Moreover, in [11], a diffusive vector-borne disease model was formulated to explore the Zika outbreak in Rio De Janeiro. Magal et al. [21,22] further revisited the model in [11], established the threshold dynamics of the model, and studied a deeper characterization of the basic reproduction number (BRN). Let
where
Recent studies [2,29,30] revealed that mosquitoes are more likely to select infected hosts than those who are uninfected to bite (called vector-bias mechanism). Denote by
where
The aforementioned works adopt the general reaction–diffusion model to study the spread of vector-borne diseases. But, as is known to all, due to the influence of water flow or wind, hosts and vectors may take passive movement in specific directions under some circumstances, which will bring a useful insight for studying the spread of disease. This phenomenon can be described as advection terms into the model. Based on the model of Allen et al. [1], Cui et al. [6,7], and Cui and Lou [8] considered a reaction–diffusion–advection susceptible-infective-susceptible (SIS) model in a spatially heterogeneous environment, and the asymptotic profiles of the BRN and the aggregation phenomenon of endemic equilibrium (EE) were addressed. It should be pointed out that only one associated eigenvalue problem consisting of one equation was used to determine the stability of the disease-free equilibrium. Furthermore, Zhao et al. [31] formulated a reaction–diffusion–advection model and explored the aggregation phenomenon of EE. Specifically, the asymptotic behaviors and monotonicity of the BRN with respect to (w.r.t.) the advection and diffusion rates were studied.
Inspired by the aforementioned works, it thus imperative to understand the possible impacts of the advection and diffusion rates on the BRN and aggregation phenomenon of EE of the model, although incorporating advection effects can make the analysis more difficult and complex. This study is a continuation of [30], and the main objective of this study is to carry out a thorough analysis on the effect of spatial heterogeneity, diffusion rates, and advection rates in a reaction–diffusion–advection system and to explore the asymptotic profiles and monotonicity of BRN and the level set classification of BRN. Moreover, consider the effects of vector-bias on disease transmission. Consider the model of [30] from another perspective, which can help us to understand the influence of different factors on disease transmission, such as diffusion coefficients, advection coefficients, and vector-bias; through the in-depth analysis and research in this article the conclusions in [30] are supplemented. Compared with the existing results, it should be pointed out that the main innovation of this study is the level set of BRN w.r.t. diffusion and advection rates is classified, the aggregation phenomenon is investigated, and the monotonicity of BRN w.r.t.
The remainder of this article is organized as follows: in Section 2, we formulate our model with advection terms and vector-bias and present the the well-posedness, threshold dynamics, asymptotic profiles, classification of level set of the BRN, and aggregation phenomenon of EE. The proof of the main results is given in Section 3. We concluded this article in Section 4.
2 Mathematical model and main results
We revisit the model in [30] in an advective heterogeneous environment. To make things simple, we assume that the diffusion rates of hosts and vectors are positive constants, which are written as
For
where
and
Biologically, at location
For any
the high-risk (HR) sites and
the low-risk (LR) sites, respectively. Upon these settings, we then consider (2.1) under two scenarios:
(H1) For any
(H2) For any
Schematic representation of
Remark 2.1
Biologically, (H1) (resp. (H2)) indicate that the upstream and downstream end belong to a high-risk (resp. low-risk) site and a low-risk (resp. high-risk) site, respectively.
For mathematically tractable, we further assume that:
(A1)
(A2) For
Remark 2.2
Mathematically, (A1) is a technical condition. From a biological perspective, this seems reasonable since the hosts and vectors may employ the proportional mobility strategy. (A2) measures that we can find only one low-risk area and one high-risk area in
2.1 Well-posedness
To proceed, we set
Theorem 2.1
For every
for
2.2 Threshold dynamics
The total vectors
To derive the BRN of (2.1), we define the operators
where
By [28], the BRN of (2.1) can be defined by
the spectral radius of
Lemma 2.1
Assume that (A1) holds. Let
with a positive eigenfunction. Then,
Lemma 2.2
Assume that (A1) holds. We then have
With Lemmas 2.1 and 2.2, the global dynamical results of (2.1) read as follows.
Theorem 2.2
Let
If
If
In what follows, we will use “system (2.1) is UP-EE” as short form for “system (2.1) is uniformly persistent and admits at least one EE.”
2.3 Asymptotic profiles of
R
0
Now, we focus on the characterization of
2.3.1 For the case
c
1
=
c
2
=
0
When
Theorem 2.3
If
Fix
(2.5)Moreover, if
Fix
(2.6)Moreover, if
As
As
As
As
Remark 2.3
From Theorem 2.3
if
as
Theorem 2.4
If
If
If
Theorem 2.5
If
Remark 2.4
Theorems 2.4 and 2.5 indicate that under certain conditions, rapid movements of infected vectors or hosts can reduce the risk of disease spread and play a positive effects in disease control. The reason for this may be that it reduces exposure to uninfected hosts.
Theorem 2.6
If
If
If
Remark 2.5
Theorem 2.6 illustrates that when there are no advection effects, if
2.3.2 For the case
c
1
,
c
2
>
0
The asymptotic properties of
Theorem 2.7
Assume that (A1) holds. We then have
For
For
For
For
Remark 2.6
Theorem 2.7 (i) illustrates that when advection rates
2.4 Classification of level set of
R
0
With the diffusion rates of hosts and vectors (
Let
and its positive eigenfunction is denoted by
Lemma 2.3
Assume that (A1)–(A2) hold. If
Under the case of (H1),
Under the case of (H2),
2.4.1 Classification of dynamics for
R
0
a
>
1
Theorem 2.8
Under the conditions of
Lemma 2.3, assume that (A1)–(A2) hold. For the case that (H1), if
and
such that, system (2.1) is UP-EE for any
where
Remark 2.7
From assumption (A1), we know that
and
In order to describe Theorem 2.8 more clearly, we use Figure 2 to depict it. Specifically, we can see that when the diffusion rates
Classification of dynamics in Theorem 2.8. Here,
2.4.2 Classification of dynamics for
R
0
a
<
1
In this section, we explore the classification of
Theorem 2.9
Under the conditions of
Lemma 2.3, assume that (A1)–(A2) hold. If
In the case of (H1), then
As
and
such that system (2.1)is UP-EE for any
As
In the case of (H2), then
As
As
and
such that when
and
where
Classification of dynamics in (i) of Theorem 2.9. Here,
Remark 2.8
To illustrate the dynamical classification in Theorem 2.9(i), we define the regions
and
where
In Figure 3, our findings are as follows: if
Classification of dynamics in (ii) of Theorem 2.9. Here,
Remark 2.9
To illustrate the dynamical classification in Theorem 2.9(ii), we define the regions
and
where
In Figure 4, our findings are as follows: the disease will break out as
2.5 Aggregation phenomenon of EE
Theorem 2.10
Assume that
When
(2.7)When
(2.8)
Theorem 2.11
Under the conditions of Theorem 2.10, we then have
When
(2.9)When
(2.10)
Remark 2.10
Biologically, both Theorems 2.10 and 2.11 demonstrate that if
3 Proof of the main results
3.1 Well-posedness: Proof of Theorem 2.1
Taking a transformation
and plugging it into (2.1) obtains
for
where
Denote
According to [5], system (3.3) admits a unique global classical solution
Furthermore, for all
Based on equation (3.3), we obtain a constant
Consider the following system:
Using the arguments in [35, Lemma 2.1] and [12, Theorem 2.2], it is readily seen that system (3.6) possesses a unique, global attractive and positive steady state
Then, we can obtain a constant
Therefore, from (3.5) and (3.7), the solution of (3.2) exists globally and lies in the invariant region
Based on the transformation (3.1) and [13], the assertions stated in Theorem 2.1 hold. This completes the proof of Theorem 2.1.
It is easy to see from (3.3) that
3.2 Threshold dynamics: Proof of Lemmas 2.1 and 2.2 and Theorem 2.2
Proof of Lemma 2.1
Assume that
Like in [28], it is essential to show the uniqueness of
Multiplying the
Subtracting the aforementioned two equations gives
Likewise, we multiply the
Then, combining (3.10) and (3.11), we obtain
By the positivity of
Proof of Lemma 2.2
Letting
Let
Consider the following problem:
It is thus observed from the Krein-Rutman theorem that problem (3.13) has a unique principal eigenvalue
Let
Multiplying the
Subtracting the aforementioned two equations yields that
Likewise, relying on the
Combining (3.15) and (3.16) gives
Since
Next, we proceed to prove Theorem 2.2.
Proof of (i) of Theorem 2.2
Linearizing system (2.1) at
Let
Denote by
where
Proof of (ii) of Theorem 2.2
From (3.19) and (3.4), we know that
either (3.13). As
admits a unique solution
where
Due to the fact that (3.20) admits a unique solution
Proof of (iii) of Theorem 2.2
Similar to the arguments in [30, Lemma 4.2 and Theorem 4.2], following a contradictory argument and the persistence theory (see [20, Theorem 3.7]), we can prove (iii) of Theorem 2.2.□
3.3 Asymptotic behaviors of
R
0
with no advection effect: Proof of Theorems 2.3–2.6
Before going into details, we first prove the boundedness of
Let us integrate the first two equations of (3.21) by parts over
Thus,
Hence, we have the following result on the boundedness of
Lemma 3.1
For
Remark 3.1
By the definition of
3.3.1 Proof of Theorem 2.3
Proof of (i) of Theorem 2.3
When
For the case that
Let
and
Moreover, it follows from (2.5) and (3.24) that
and
Thus,
Multiplying the
Hence,
In a similar manner, we can obtain that
This, together with (3.27) and (3.28), indicates that
i.e.,
Since
Next, we consider the case that
Using the elliptic regularity estimate to the
Since
This proves (i) of Theorem 2.3.□
Proof of (ii) of Theorem 2.3
Similar to the proof of (i) and so is omitted.□
Proof of (iii) of Theorem 2.3
With the aid of Lemma 3.1, passing to a sequence if necessary,
Let the operators
where
By virtue of [17, Proposition 3.4], we know that
Furthermore, from in [17, Theorem 1.4], we obtain
where
Thus,
where
Proof of (iv) of Theorem 2.3
With the similar procedures as the proof of (ii), letting
for any
Proof of (v) of Theorem 2.3
Similar to (iv), one can obtain
Proof of (vi) of Theorem 2.3
By applying the assertions in (i), (v), and (iv), (vi) is obvious. This completes the proof of Theorem 2.3.□
3.3.2 Proof of Theorem 2.4
Proof of (i) of Theorem 2.4
When
Similar to the arguments in [14, Lemma 15.1] and [5, Proposition 2.20], we know that
where
Multiplying the
where
From the
i.e.,
Thus,
Proof of (ii) of Theorem 2.4 is similar to (i).
3.3.3 Proof of Theorem 2.5
Proof of (i) of Theorem 2.5
When
where
Similarly,
Adding (3.37) and (3.38) and substituting
Thus,
Proof of (ii) of Theorem 2.5
(ii) can be proved by the arguments similar to (i), which will not be repeated here.□
3.3.4 Proof of Theorem 2.6
Proof of (i) of Theorem 2.6
Theorem 2.3 tells us that
Proof of (ii) of Theorem 2.6
Combined with the fact that
3.4 Asymptotic behaviors of
R
0
with advection effect: Proof of Theorem 2.7
We first prove the following lemmas.
Lemma 3.2
If (A1) holds, then
Proof
Let
For
Let
If we make
which is a contradiction. Hence,
Furthermore, we have
Similarly, if
For
Let
Due to the continuity of
This combined with Lemma 3.1 implies that
which is a contradiction. Hence,
Furthermore, we have
Similarly, let
for some
This together with (3.41), (3.44), (3.42), and (3.45) implies that
and
Denote
and
Let us integrate (2.4) into
Accordingly,
Since
Consequently,
and
In virtue of the Lebesgue-dominant convergence theorem and (3.48)–(3.49), we can obtain that
Lemma 3.3
If (A1) holds. For any fixed
Proof
With the aid of Lemma 3.1, passing to a sequence if necessary,
Direct calculation gives
Proof of Theorem 2.7
(i) is obvious. (ii)–(iii) and (iv) directly follow from Lemmas 3.2 and 3.3. (v) holds from Lemma 3.2, since
3.5 Classification of level set of
R
0
: Proof of Lemma 2.3 and Theorems 2.8 and 2.9
3.5.1 Proof of Lemma 2.3
By virtue of (A2) and (H1), we can find a
By a
Since
Accordingly,
and
which implies that
The proof of (ii) resembles that of (i), so we omit the specifics. This proves Lemma 2.3.
3.5.2 Proof of Theorem 2.8
Fixed
where
Subtracting the aforementioned two equations yields
Similarly, we deal with the
As a result,
Adding (3.52) and (3.53) and according to the assumption (A1), and
By Theorem 2.7, (A1), and (H1), we directly have
Thanks to Theorems 2.6 and 2.7, we obtain
As a consequence, we can find at least a
Due to
Arguing by contradiction, if there exist
which contradicts with
Similarly, if
Integrating (3.55) over
With the help of Theorem 2.7 and the elliptic regularity estimate, there exist
Since
Recall that
3.5.3 Proof of Theorem 2.9
Theorem 2.9 will be achieved through proving the following two lemmas.
Lemma 3.4
Under the conditions of
Lemma 2.3, assume that (A1)–(A2) hold. If
In the case of (H1), then
When
When
In the case of (H2), then
When
When
Proof
We only prove (i), since (ii) can be similarly achieved. Based on the proof of Theorem 2.8 and (ii) of Theorem 2.6, if
Obviously, we cannot find
Remark 3.2
In the forthcoming discussions, we will demonstrate that
Lemma 3.5
Under the conditions of
Lemma 3.4, assume that (A1)–(A2) hold. If
In the case of (H1), there exists function
In the case of (H2), there exists function
where
Proof
Based on Lemma 3.4, the existence of
which is a contradiction, since
To deal with
As
Let us multiply
Similarly, relying on the
From (3.59) and (3.60), we then have
owing to
To proceed to prove (ii), we claim that
Applying the
Similarly, relying on the
Subtracting the two equations of (3.62) and (3.63), we obtain
According to the assumption (A1) and
By (ii) of Lemma 2.3, we obtain
By differentiating the equation
Recall that
Thus,
Since
Like in the proof of (i) and Theorem 2.8, let
This completes the proof.□
Theorem 2.9 is a direct consequence of aforementioned lemmas.
3.6 Aggregation phenomenon of EE: Proof of Theorems 2.10 and 2.11
To prove Theorems 2.10 and 2.11, we shall appeal to the following lemmas.
Lemma 3.6
For
Proof
Let
which is cooperative. With the aid of strong maximum principle (see, e.g., [25, Lemma 2.4] and [34, Lemma 2.1.2]), we know that
The following result gives the sub- and super-solutions of system (2.1).
Lemma 3.7
Suppose that (A1) holds, and
with
and
where
Proof
By simple calculations, we obtain
and
Setting
and
Based on Lemma 3.6, we see that for any given
Lemma 3.8
For
and
satisfy
Proof
Direct calculation gives
where
With Lemmas 3.6–3.8, we proceed to prove Theorems 2.10 and 2.11.
3.6.1 Proof of Theorem 2.10
By Theorem 2.2 and Lemma 3.2, (2.1) admits at least one EE. For the case that
Relying on
According to Lemma 3.8, letting
and
Hence, for
Substituting it into (3.66) yields
Consequently, we have
Similarly, we can cope with (2.8) for the case that
3.6.2 Proof of Theorem 2.11
If
Furthermore, if
It then follows that
Due to
we obtain
By applying the same arguments, the other scenarios can also be validated. This proves Theorem 2.11.
4 Discussion
We formulated and analyzed a vector-borne reaction–diffusion–advection model with vector-bias mechanism in a spatially heterogeneous environment. One of the significant features in our model is that we can consider the effects of advection and diffusion rate on the disease transmission, which poses new challenges to theoretical analysis. We obtained the threshold dynamics in terms of
We proved the boundedness of solution of system (2.1) by utilizing the comparison principle, [31, Theorem 2.1] and [21], addressing the well-posedness of the model (2.1) (see Theorem 2.1). In addition, the DFE
The level set of
As shown in Theorem 2.8, there exist unique surfaces
As indicated in (i) of Theorem 2.9, there exist two constants
As indicated in (ii) of Theorem 2.9, there are two vital constants
Finally, we analyzed the aggregation phenomenon of EE (Theorems 2.10 and 2.11). Theorems 2.10 and 2.11 indicated that under certain circumstances, if the advection rates are large relative to dispersal rates, the infected individuals will aggregate at downstream end.
Our findings complement the results of vector-borne disease in non-advective environments and may provide several new clues for the investigation and control of the disease. It should be pointed out that due to the complexity of model (2.1), some assumptions are harsh, just for mathematical purposes only. There are quite a few ways to improve our model. For example, we can consider the vector-borne disease model in the case that hosts can random walk without advective effects but vectors with, or consider the influence of periodic delay of vectors on the disease dynamics. We leave these questions for further investigations.
Acknowledgments
The authors would like to thank the referees and editor for many helpful comments. J. Wang was supported by the National Natural Science Foundation of China (Number 12071115 and 12471460), the Heilongjiang Natural Science Funds for Distinguished Young Scholar (Number JQ2023A005), and the Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, P. R. China.
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Author contributions: Jiaxing Liu: Writing-original draft, Methodology. Jinliang Wang: Writing-review & editing, Supervision, Methodology, Funding acquisition, Conceptualization.
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Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this article.
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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”
