Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
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Qi Zhou
Abstract
How individual dispersal patterns and human intervention behaviours affect the spread of infectious diseases constitutes a central problem in epidemiological research. By incorporating these two critical factors, this article proposes an impulsive non-local faecal-oral model with free boundaries. In this framework, non-local diffusion captures the long-range movement of individuals, impulsive effects model periodic disinfection measures, and free boundaries delineate the advancing fronts of infection. We first establish the existence and uniqueness of a non-negative global classical solution to the system. Subsequently, by employing the theory of resolvent positive operators and their perturbations, we analyze the principal eigenvalue, which depends on the infected domain, impulse intensity, and non-local diffusion kernel functions. On the basis of this eigenvalue, we derive sharp threshold criteria distinguishing between disease extinction and propagation. The incorporation of non-local diffusion and impulsive interventions introduces significant mathematical complexities, which are addressed through novel analytical techniques. Finally, numerical simulations are conducted to validate the theoretical results and explore the impact of pulsed interventions. Our findings demonstrate that periodic disinfection effectively mitigates disease transmission, whereas the influence of non-local diffusion varies with the selection of kernel functions.
1 Introduction
Human beings have always been more seriously threatened by diseases than by wars, poverty, disasters, etc. It is estimated that there are about 1.3–4 million cases of cholera globally each year, causing about 21,000 to 143,000 deaths [2]. Typhoid disease is estimated to cause between 9.9 and 14.7 million infection cases and between 75,000 and 208,000 deaths annually [27]. Global adult mortality rates significantly increased during the COVID-19 pandemic in 2020 and 2021, reversing past decreasing trends [18]. A common feature of the diseases mentioned earlier is that they can be spread by the faecal-oral route [7,11,19]. In addition, poliomyelitis, infectious hepatitis, hand-foot-mouth diseases, and enteric viruses like norovirus, rotavirus, and astrovirus can also be transmitted by the faecal-oral route [7,11,17].
Mathematical models have been widely used to predict the trends, design the control measures, and understand the underlying mechanisms of infectious diseases [39]. The first faecal-oral model has the following form:
with suitable initial conditions [8]. Model (1.1), formulated by Capasso and Paveri-Fontana, is used to understand how cholera was transmitted in the Mediterranean regions of Europe in 1973. In (1.1),
Taking into account the spatial movement of individuals, Capasso and Maddalena further developed the corresponding reaction–diffusion model in 1981 [7]. Subsequently, there have been a series of reaction–diffusion faecal-oral models based on the work [7]. We refer to [6] for an integro-differential model, [31] for an integro-differential model with a time delay, [36] for an model with two time delays, and [20] for a model with general growing function.
However, the infected region of infectious disease always moves gradually outwards, but the models mentioned earlier do not capture this. In 2010, Du and Lin introduced the free boundary to study the invasion of a single species [12]. Motivated by this work, Ahn et al. developed a diffusive faecal-oral model with free boundaries, where the free boundaries successfully capture the movement of the infected region [1]. The model in [1] does not take into account the movement of the infected humans. Considering the movements of the infected humans and diseases, Wang and Du [32] developed the following model:
with suitable initial functions
List of notations and their meanings in model (1.2)
| Notation | Biological meaning |
|---|---|
|
|
Spatial density of the infectious agents at location
|
|
|
Spatial density of the infectious persons at location
|
|
|
Left boundary of the infected region at time
|
|
|
Right boundary of the infected region at time
|
|
|
Diffusion coefficient of the infectious agents |
|
|
Diffusion coefficient of the infectious persons |
|
|
Expansion capacity of the infectious agents |
|
|
Expansion capacity of the infectious persons |
|
|
Length of the right boundary of the initial infection region |
Note that Fickian diffusion applies to a diffusion process that corresponds to the random walk only when the step size and time size are small compared with the spatial variable and time, respectively [42]. Therefore, the Laplace operator used in model (1.2) only describes the spatial properties of the infectious persons and the infectious agents locally. In 2003, Murray emphasised the importance and intuitively necessity of the long range effects in the biological areas [28, Chapter 17]. An extensively used non-local diffusion operator to replace the local diffusion term is given by
where the kernel function
Through appropriately selecting
where the kernel functions
Human intervention can prevent and control the large-scale outbreaks of infectious diseases. For the diseases transmitted by the faecal-oral route, the infectious agents in the environment have a significant impact on the evolution of the diseases. The spraying of disinfectant liquids is effective to dramatically reduce the number of infectious agents rapidly. Classical reaction diffusion models do not capture this instantaneous and abrupt phenomenon. However, this phenomenon can be described well by impulse differential equation models.
The impulse differential models have the characteristics of both continuous and discrete models. In the last three decades, the theoretical research on impulsive differential equations has yielded numerous results. In the field of biomathematics, Lewis and Li proposed an impulsive reaction–diffusion equation model to study persistence and spread of species with a reproductive stage and a dispersal stage in bounded and unbounded domains [22]. Subsequently, some impulsive partial differential equation models rooted in biomathematical background were developed and analyzed. We refer to [14] for higher dimensional models extended from [22], [45] for an impulse epidemic model in a periodically evolving environment, [23] for a pest growth model with multiple pulse perturbations, [15] for a hybrid impulsive reaction-advection-diffusion model, [37] for a birth pulse population model with non-local dispersal, [40] for a birth pulse population model with shifting environments, and [38] for an impulsive hybrid population model in a heterogeneous landscape.
In this article, we develop a pulsed non-local faecal-oral model with free boundaries by introducing the impulsive intervention into model (1.4), to examine how impulsive intervention and non-local diffusion together affect the spread of the diseases. The developed model can be reduced to model (1.4) by choosing the impulse intervention function properly. The introduction of the impulsive intervention results in the appearance of a periodic solution and reduces the regularity of the solution. As a result, it may make analysis hard, and the results become complex and different and will naturally raise some new problems.
With the introduction of the impulse intervention, it is natural to ask what kind of regularity does the solution possess? Whether the spreading-vanishing dichotomy is still true? What are the new criteria for determining spreading and vanishing? Whether the impulsive intervention and the non-local diffusion influence the movement speed of the infected area? Whether the combination of the impulsive intervention and the non-local diffusion affects or even alters the dynamical behaviour? These are the research motivations. The main contributions of this research are listed as follows:
Introduce pulsed intervention into a faecal-oral model with free boundaries and non-local diffusion.
Investigate the principal eigenvalue, which depends on the impulse intensity, the infected region, and the kernel functions.
Prove a vanishing-spreading dichotomy, and provide the sufficient conditions for determining vanishing and spreading.
The article is arranged as follows. Section 2 develops an impulsive faecal-oral model with free boundaries and non-local diffusion and presents some preliminaries including necessary notations, a frequently used comparison principle, a prior estimate, and the well-posedness of the solution. The related impulse problem with fixed boundaries and the corresponding eigenvalue problem with impulse are treated in Section 3. Here, the existence of the principal eigenvalue is obtained by applying the theory of resolvent positive operators with their perturbations. Section 4 proves a vanishing-spreading dichotomy and obtains the sufficient conditions for determining vanishing and spreading based on the results obtained in Section 3. In Section 5, numerical simulation is used to present the effect of the impulsive intervention and to corroborate the validity of the theoretical findings. Finally, a brief discussion and conclusion are given in Section 6.
2 Model formulation and preliminaries
In this section, we formulate a pulsed non-local faecal-oral model in a moving infected region, the global existence of its unique non-negative classical solution is then given, and finally a frequently used comparison principle is provided.
2.1 Model formulation
As stated in Section 1, the transmission of faecal-oral diseases is usually suppressed by human control measures. As an example, periodic disinfection of the infected area plays an important role in cutting off the transmission route and protecting the susceptible population. This kind of measure generally results in a dramatic decrease in the spatial density of the infectious agents within a very small time span, and there is no direct impact on the infected individuals.
Motivated by this biological phenomenon, we consider here a situation with two stages: a human intervention stage and a natural developmental stage. In a human intervention stage, the spatial density of the infectious agents decreases impulsively through a discrete-time map. We use
Let
In (2.1), it is always assumed that
The impulsive function
It should be mentioned that (H) is natural assumptions for the impulsive function. Some frequently used function forms satisfying assumption (H) are the linear function
This subsection formulates a pulsed non-local epidemic model in a moving infected environment. In the next subsection, the global existence and uniqueness of the solution to model (2.1) will be investigated.
2.2 Preliminaries
We give some notations first. For any given
For any given
where
where
where
We now prove the following estimate that has an important function in the proof of the global solution of (2.1).
Lemma 2.1
Assume that
admits a unique solution
Proof
Because the initial function
which means that
The hypothesis of
which implies that the conclusion holds for
Since
it can be viewed as a new initial function for
Since
With the help of (2.4), it follows that
This ends the proof.□
On the basis of the preceding lemma, we can now establish the following theorem.
Theorem 2.1
For any given
Proof
Because the initial function
which means that
Since
it can be viewed as a new initial function for
(1) When
(2) When
Finally, we present the following comparison principle, which is used frequently in later analysis.
Lemma 2.2
Assume that
Then,
Proof
When
3 The associated problem with fixed boundaries
In this section, we investigate the fixed boundary problem corresponding to model (2.1), that is, model (2.1) with
which is represented by the following equations:
It turns out that the methods required here are quite different from previous works where the impulsive function
3.1 The corresponding eigenvalue problem
There are some works focusing on the existence of principal eigenvalues of the time-periodic cooperative problems with long-range diffusion, see, for instance, [3,16,24]. Motivated by these works, the existence of the principal eigenvalue of the linearised system corresponding to (3.1) will be established next. For the specific definitions of some terms mentioned in this subsection, we refer to [29, Section 3.1] for positive perturbation, [30, Definition 4.5] for essentially compact perturbation, [16, Section 2.1] for positive operator, strongly positive operator, resolvent positive operator, spectral bound, real spectral bound, spectral radius, and principal eigenvalue.
It is a well-known fact that the dynamical behaviours of model (3.1) are controlled by the principal eigenvalue of its linearised eigenvalue problem at
To overcome the difficulties presented by the impulse intervention, we consider the following eigenvalue problem:
where
For any given
with norm
Let
Now, define two operators
and
for
Let
and
With the help of [16, Lemma 2.8] (or [3, Lemma 2.1]), we have the following lemma.
Lemma 3.1
The eigenvalue problem
and for any
Based on the aforementioned preparations, some properties of the operator
Proposition 3.1
The resolvent operator
Proof
To begin with, the fundamental theorem of ordinary differential equations gives that for every
To prove that
With the help of the evolution family
Proposition 3.2
There exists
Proof
We consider the following equation:
With the help of the variation of constant formula, it can be immediately obtained that
Let
In virtue of (3.4)–(3.6), it follows that if
Therefore,
A simple calculation yields that the matrix
has a real eigenvalue
which has a positive eigenvector
In addition, assumption (J) yields that
In virtue of [29, Lemma 5.8], one can obtain that
where
Therefore, we have that
On the other hand, the definition of the operator
Now we present the main theorem of this subsection.
Theorem 3.1
The eigenvalue problem (3.2) has a unique eigenvalue
Proof
In virtue of Proposition 3.1, one can obtain that
By standard calculation, the eigenvalue problem
has a principal eigenpair
Case (iii) in [29, Theorem 3.6] tells us that there exist
Let
Next, we prove uniqueness. Let
The strong positivity of the eigenfunction
Since the operator
Now (3.8) together with (3.9) yields that
This implies that
where
Remark 3.1
By the proof of Theorem 3.1, it follows that the unique eigenvalue
The aforementioned theorem exhibits that the eigenvalue problem (3.2) corresponding to (3.1) has a unique principal eigenvalue
Lemma 3.2
If there exists a non-negative vector-valued function
then
Proof
We first prove the first assertion. Let
where
By multiplying the first equations of (3.11) and (3.12) by
Then, integrating the two sides of (3.13) on
Recalling that the assumption
In addition, the impulse and period conditions of (3.11) and (3.12) yield that
Inserting (3.15) and (3.16) into (3.14) yields that
A similar process results in
Adding (3.17) and (3.18) gives that
Noting that
The proof of the remaining conclusion is shown next. When the equalities of (3.11) all hold, the conclusion
Lemma 3.3
If there exists a non-negative vector-valued function
then
Proof
This proof is similar to Lemma 3.2, and we omit it here.□
Subsequently, we will investigate the monotonicity and continuity of the eigenvalue
Lemma 3.4
Assume that
Proof
(1) First, we show that
Then, Lemma 3.3 yields that
Subsequently, we show that
while if
Let
and
Then, the functions
where
which implies that (3.21) holds. When
which implies that (3.22) also holds. We have now proved the desired continuity.
(2) This proof is similar to [33, Lemma 3.8], and we omit it here.□
Consider the following eigenvalue problem:
By the standard method (see, e.g., the proof of [44, Lemma 3.3]), it follows that problem (3.24) has a unique principal eigenvalue
Lemma 3.5
Assume that
Proof
For any given
and
Lemma 3.3 gives that
Subsequently, we show that
It follows from the proof of [33, Lemma 3.9(2)] that for any sufficiently small
For any
and
With the help of Lemma 3.2, it follows that
3.2 The asymptotic profile in a fixed domain
In this subsection, we still use some notations from Section 2.2 with the free boundaries
and
where
Before discussing the asymptotic profile of model (3.1), we first give the definitions of the ordered upper and lower solutions of problems (3.1) and (3.25) and then prove that the comparison principles of problems (3.1) and (3.25) are also valid. They form the basis of this subsection.
Definition 3.1
For any given
are called the ordered upper and lower solutions of (3.1), respectively, if
and
respectively.
Remark 3.2
Denote the ordered upper and lower solutions of problem (3.25) by the pairs
Lemma 3.6
Assume that r, s,
then,
Proof
By the similarity to the proof of Lemma 2.2, the desired conclusion can be obtained. Since it is actually much simpler, we omit the details here.□
Remark 3.3
Note that the fixed boundary problem (3.1) and the corresponding steady-state problem (3.25) have a very similar structure. Therefore, when the initial conditions
Now, we present the major results of the subsection.
Theorem 3.2
Let
Proof
(1) We first prove the first assertion. Clearly,
Substituting (3.26) into (3.25) yields that
In virtue of Lemma 3.2, we have from (3.27) that
The second assertion is proved next. Its proof is divided into the two situations below.
Case 1:
It follows from Remark 3.1 that the unique principal eigenvalue
where
This combined with the condition
Case 2:
By a standard calculation, the function pair
where
Then, the initial conditions satisfy
The impulse conditions satisfy
The equations satisfy
Based on these, Lemma 3.6 yields that
Then, it follows from (3.29) that
This together with
Recalling that
for all
where
for all
for all
for all
which implies that
(2) We first prove the first assertion. To begin with, we construct the following functions:
where
and
where
and
A standard calculation shows that the constructed
The two sequences of iteration
where
where
It remains to show that the positive periodic solution of problem (3.25) is unique. Arguing indirectly, suppose that
is well defined and finite. In addition, we have that
To simplify notation, let
If
This implies that the aforementioned inequality can hold only if
which is impossible. Hence,
Next, we prove the second assertion. A standard calculation yields that
for
for
Noting that
This proof is finished.□
Theorem 3.2 gives the asymptotical profiles of problem (3.1) when the infection environment
Lemma 3.7
Assume that
where
Proof
It is well known that the asymptotic profiles of problem (3.37) are determined by the eigenvalue
Now, we claim that if
and
Then, Lemma 3.3 yields that
By using a similar proof method as in the first assertion of Theorem 3.2(2), it follows that when
4 The asymptotic profile in a moving environment
This section first presents some basic knowledge. Based on this, the vanishing-spreading dichotomy is proved and sufficient conditions used for determining spreading or vanishing are provided. Throughout this section, we always denote the solution of (2.1) by
4.1 Some introductory results
In here, the definition of the order upper and lower solutions of model (2.1), the comparison principle applicable to model (2.1), and the definition of the spreading and vanishing of model (2.1) will be presented in turn.
Definition 4.1
If for any
then the quadruple
Remark 4.1
By reversing all the inequalities in (4.1), a lower solution of model (2.1) can also be defined. In this section, denote the upper and lower solutions of model (2.1) by the quadruples
Lemma 4.1
Denote the unique solution of model (2.1) by
Proof
Here, we only need to prove that the conclusion holds for the upper solution, due to the fact that a similar analysis yields that the conclusion also holds for the lower solution.
By using assumption (
By viewing
Theorem 2.1 and Lemma 2.2 show that
are always well-defined.
Definition 4.2
The model (2.1) is called as vanishing if
and spreading if
where
4.2 Spreading-vanishing dichotomy
This subsection will prove that when the time
Lemma 4.2
Let
Moreover, we have that
Proof
The first conclusion is proved first. Define, for
where
Finally, assumption (
and
for
Therefore, taking
This proves the first assertion.
Next, we prove the second assertion. The first assertion tells us that a constant
for
for
which implies that
Therefore, the second assertion is valid. The proof is complected.□
Lemma 4.2 shows that model (2.1) is vanishing when
Lemma 4.3
Assume that the region at infinity is particularly unfavourable for the survival of the infectious agents, that is,
Proof
Let
where
From the condition
This together with (4.3) yields that
This ends the proof.□
Next, we will discuss the dynamic behaviour of model (2.1) when
Lemma 4.4
If
and
Proof
Begin by showing the second assertion. Arguing indirectly, suppose that
The assumption that
Then, for given
Now, we consider the following problem:
where
Therefore, from Lemma 3.6, we have that
Noting that
where
for
For
which implies that
The first assertion is shown next. Let
With the help of Theorem 3.2(1), it follows that
and hence the first assertion holds. The proof is finished.□
Lemma 4.4 implies that if
Lemma 4.5
If
Proof
Without loss of generality, we suppose by contradiction that
Consider the following problem:
Then,
Now, we begin to analyze the dynamical behaviour of model (2.1) when
Lemma 4.6
If
where
Proof
In virtue of Lemma 4.4, it follows from
Next verify the first claim by showing
uniformly in
locally uniformly in
where
uniformly for
uniformly in
We then prove that (4.6) also holds. Since
Then, it follows from Lemma 3.6 that
Since
uniformly in
uniformly in
locally uniformly in
On the basis of Lemmas 4.2,4.3 and 4.6, we now present the main result of this subsection.
Theorem 4.1
The model (2.1) is either vanishing or spreading.
4.3 Spreading-vanishing criteria
In this subsection, we will look for criteria guaranteeing vanishing or spreading for (2.1). For convenience, let
Based on Lemmas 4.2 and 4.3, the following conclusion can be obtained directly.
Theorem 4.2
If
In addition, if
Next, we will consider the case of
Theorem 4.3
Assume that
Proof
Suppose by contradiction that
which shows a contradiction with the condition
Next, we will explore the case of
Lemma 4.7
Assume that
Proof
A suitable upper solution of model (2.1) is constructed to prove this conclusion. Since
Denote by
where the non-negative constants
We start by examining some conditions and determining
A standard calculation yields that, for
and
provided that
and
In addition, assumption (J) yields that
provided that
Similarly, we have that
For the initial value, if we choose
then
Now it is clear that the quadruple
which implies that
Remark 4.2
In the proof of Lemma 4.7, it follows from observing the definition of
Lemma 4.8
Assume that
Proof
Since
Lemma 4.9
Suppose that
Proof
Denote the unique solution of model (2.1) by
If not, suppose that for any
By Theorem 2.1, we have that
are well defined. By Lemma 4.8, it follows that
Then, for the fixed
By integrating both sides of the fourth equation in the model over
where
This implies that
which shows a contradiction with the assumption. Therefore, (4.14) holds. Finally, it follows from Lemma 4.6 that the conclusion holds. The proof is completed.□
Now we consider the case
Theorem 4.4
Assume that
Proof
We first prove the first assertion. In the proof of Lemma 4.7, we have from observing the definition of
5 Numerical simulation
In this section, we will present a numerical example to demonstrate the theoretical outcomes and further understand the effect of the impulse intervention on the spread of the faecal-oral diseases.
Since the relationship between the infectious agents and the infected individuals is cooperative, we only give the numerical simulation of the infectious agents to save space. Now we choose the dispersal kernel function as follows:
where
Dispersal kernel function
The initial infection region is chosen as
The other parameters of model (2.1) are given in the following example.
Example 5.1
Fix
When there is no the impulsive intervention, it can be seen from Figure 2(c) that
Then, it follows from Lemma 4.6 that
When
When the impulse function is taken as
Then, we have from Lemma 4.2 that the disease-free equilibrium point
When
It can be obtained from comparing Figures 2 and 3 that the implementation of the impulsive intervention can reduce the number of the infectious agents, slow down the expansion speed of the diseases, decrease the infected region of the diseases, and extinguish the diseases which are originally persistent. Therefore, the impulsive intervention is beneficial in combating the diseases transmitted by the faecal-oral route.
6 Conclusion and future work
In this article, we develop an impulsive non-local model to capture the effect of the combination of periodic spraying of disinfectant and non-local diffusion of individuals on the diseases transmitted by the faecal-oral route. Our model extends the model of Chang and Du [9], which does not take into account the pulse intervention. With the introduction of the pulse intervention, the model is not continuous at the pulse point and the corresponding steady-state is governed by a periodic parabolic problem, hence the techniques used here are rather different from those in [9].
Based on the work of Wang and Du [33], Theorem 2.1 shows that the model has a unique non-negative global classical solution. Then, the corresponding periodic eigenvalue problem (3.2) is presented to study the asymptotical profiles of the model. The existence and uniqueness of the principal eigenvalue of problem (3.2) is proved in Theorem 3.1 by using the theory of resolvent positive operators with their perturbations. With the help of this eigenvalue, Theorem 4.1 shows that model (2.1) is either vanishing or spreading. To provide the criteria for determining spreading and vanishing, the principal eigenvalues of the eigenvalue problem (3.2) with
If
If
If
If
there exist two positive constants
or if
Our theoretical results suggest that the implementation of the impulse intervention is beneficial in combating the diseases transmitted by the faecal-oral route. Numerical simulation is used to further understand this finding. In addition, the initial infected region also has a significant impact on the evolution of the diseases. Specifically, the smaller the initial infected area, the more advantageous for human beings to fight against the diseases. The effect of the non-local diffusion of individuals on the evolution of the diseases depends on the choice of the diffusion kernel functions.
When
Acknowledgments
The authors are grateful to the anonymous reviewers for careful reading and valuable comments that helped to improve the article.
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Funding information: The first author is supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX24_3711) and the third author acknowledges the support of the National Natural Science Foundation of China (No. 12271470).
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Author contributions: All the authors Qi Zhou, Michael Pedersen, and Zhigui Lin contribute equally.
-
Conflict of interest: The authors state no conflict of interest.
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Data availability statement: The authors confirm that the data supporting the findings of this study are available within the manuscript.
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- 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
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- 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
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- 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
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- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
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- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
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- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
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- Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
- Generalized quasi-linear fractional Wentzell problems
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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
