We incorporate local ventilation effects into a spatially dependent generalisation of the Wells–Riley model for airborne disease transmission and consider a room in which an air purifier supplements the ventilation provided by a poorly functioning air-conditioning (AC) unit. Aerosol production and removal through ventilation, biological deactivation and gravitational settling as well as transport around a recirculating AC flow and global turbulent mixing are modelled using an advection–diffusion–reaction equation. This local ventilation model is compared with a global ventilation model, where all ventilation is treated as a global sink. We undertake this comparison for a weak purifier (clean air delivery rate of 140 m$^3$h$^{-1}$) and a strong purifier (clean air delivery rate of 1,000 m$^3$h$^{-1}$). For each purifier, we determine a total effective air exchange rate and compare against the global model with equivalent ventilation. We find that, as expected, the purifier removes fewer aerosols when the distance between the infectious person and the purifier is increased, resulting in a greater average aerosol concentration in the room. Moreover, the concentration is generally lowest when the infectious person is upstream of the purifier, located in regions where the airflow streamlines are directed into the purifier. For these infectious source locations, the global ventilation model significantly overestimates the concentration throughout the room. For infectious sources outside of these regions, there is generally good agreement between the models, particularly for the weak purifier. We also studied, for a fixed distance from the purifier, how the infection risk to a susceptible person varies as the infectious person changes location. A susceptible person faces the highest infection risk when they are directly downstream of the infectious person, where the aerosol concentration is the highest. There is better agreement between local and global ventilation models for the weak purifier than for the strong purifier since the weak purifier has less impact on the airflow in the room.

This paper studies a class of degenerate parabolic partial differential equation models that describe the dynamics of single-species populations with cognitive functions in toxic environments. The core innovation lies in introducing cognitive functions to simulate the ability of species to perceive toxins and adaptively adjust their behaviours, which in turn affects population density. This mechanism is characterized by a nonlinear degenerate Pratial Differential Equation. The degradation of the model is mainly manifested in the diffusion coefficient and response term approaching zero when the population density is zero or the cognitive state reaches the boundary. For this model, we have achieved the following theoretical breakthroughs: First, by using regularization techniques, we constructed a non-degenerate approximate system. Combining prior energy estimates with the fixed point theorem, we proved the existence of local classical solutions for this regularized system; subsequently, we constructed an appropriate Lyapunov function. Based on strict energy dissipation analysis and uniform prior estimates, we proved that under conditions such as bounded toxins and regular initial values, the classical solutions of the regularized system exist globally and remain bounded on the interval $[0, \infty )$ ultimately, based on this, through compactness theory and limit processes, the existence of the global weak solution for the original degenerate system is further derived. Numerical simulations verify the rationality of the theoretical results and visually demonstrate the dynamic regulatory effect of cognitive function on population tolerance.

In this study, we develop epidemic reaction-diffusion models by incorporating the dependency of the diffusion rate of susceptible individuals on new infection cases, employing both Fickian and Fokker–Planck-type diffusion laws. As the first part of a two-part series, we focus on epidemics driven by frequency-dependent incidence. We explore linear, exponential and algebraic relationships between diffusion rate of the susceptible population and new infection cases to provide deeper biological insights. Our analysis establishes the global existence of solutions and characterizes the threshold dynamics using basic reproduction numbers. We find that in quasilinear parabolic systems, the Fokker–Planck-type diffusion law tends to induce spatial segregation of susceptible and infected individuals, while the Fickian law favours spatial homogenization of susceptible individuals. Additionally, the Fokker–Planck-type model, where the diffusion rate of infected individuals depends on new recovery cases, more accurately captures the cognitive diffusion behaviour of individuals.

This paper concerns the isentropic compressible Navier–Stokes equations in a three-dimensional (3D) bounded domain with slip boundary conditions and vacuum. It is shown that the classical solutions to the initial-boundary-value problem of this system with large initial energy and vacuum exist globally in time and have an exponential decay rate, which is decreasing with respect to the adiabatic exponent $\gamma \gt 1$ provided that the fluid is nearly isothermal (namely, the adiabatic exponent is close enough to 1). This constitutes an extension of the celebrated result for the one-dimensional Cauchy problem of the isentropic Euler equations that has been established in 1973 by Nishida and Smoller (Comm. Pure Appl. Math. 26 (1973), 183–200). In addition, it is also shown that the gradient of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). In contrast to previous related works, where either small initial energy is required or boundary effects are absent, this establishes the first result on the global existence and exponential growth of large-energy solutions with vacuum to the 3D isentropic compressible Navier–Stokes equations with slip boundary conditions.

We study the long-time behaviour of solutions to a free boundary model in river networks with small water flow speeds. First, we establish the well-posedness of our model. Under suitable assumptions, we then establish a spreading–vanishing dichotomy result for solutions to our model. By introducing a parameter $\sigma $ in the initial data, we derive sharp threshold behaviours.

In this paper, we use a degenerate reaction-diffusion system with free boundaries to model the spatial spread of rabies among foxes, whose population is divided into three sub-populations: susceptible ($S$), infected ($E$), and rabid ($I$). Based on established biological observations, susceptible and infected foxes are assumed to be territorial (random diffusion is ignored for $E$ and $S$ in the model), whereas rabid foxes disperse randomly (random diffusion is assumed for $I$), causing the spread of the disease. While $S$ evolves over the entire real line $\mathbb{R}$, $E$ and $I$ are found only in the infected region represented by an interval $[g(t), h(t)]$, which expands with moving fronts $x = g(t)$ and $x = h(t)$ as time $t$ increases. We show that this system admits a unique global solution and then analyse its dynamics and establish a spreading-vanishing dichotomy in certain natural parameter regimes. Moreover, we supply some simple sufficient conditions for the vanishing and spreading of the rabies, respectively. For example, we show that if the corresponding ODE system has basic reproduction number $\mathcal{R}_0 \gt 1$, then a spreading-vanishing dichotomy holds, and the outcome depends on the initial size of the infected region, while if a certain quantity $\mathcal R_0^* \in (\mathcal R_0,\infty)$ is no bigger than 1, then the rabies will always vanish. The degenerate nature of the model, combined with the evolving infected region, causes considerable difficulties in the mathematical treatment, both in proving the well-posedness and in understanding the long-time dynamics. This paper appears to be the first to treat a free boundary model where one reaction-diffusion equation is coupled with two ordinary differential equations.

We investigate the qualitative properties of positive solutions to mixed local–nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$-direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator ${(-\Delta )}^s - \Delta $ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local–nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems.

This work investigates the dynamics of positive classical solutions to a diffusive susceptible-exposed-infected-recovered-susceptible epidemic model with a mass-action incidence mechanism in spatially heterogeneous environments. Under minimal assumptions on the initial data, the global existence of classical solutions is established. Moreover, the eventual boundedness of these solutions is proved when either the spatial domain has dimension five or lower or the susceptible and exposed subpopulations share the same diffusion rate. Next, we define the basic reproduction number, $\mathcal{R}_0$, and demonstrate that the disease-free equilibrium is globally stable when $\mathcal{R}_0$ is sufficiently small. However, due to the complex interaction between population movement and spatial variation in transmission rates, we find that the disease may persist even when $\mathcal{R}_0$ is slightly less than one. In such cases, we show that the system admits at least two endemic equilibrium (EE) solutions, an outcome not observed under the frequency-dependent incidence mechanism. These results highlight the significant influence of the transmission mechanism on disease dynamics. Furthermore, we examine the spatial profiles of the EE solutions when diffusion rates are small. Our analysis suggests that limiting the movement of the susceptible population can significantly reduce disease prevalence, provided that the total population remains below a specific threshold. In contrast, restricting the movement of the infected, exposed, or recovered populations alone may not eradicate the disease. Overall, our findings provide important insights into the spatial dynamics of infectious diseases and may offer guidance for developing and implementing effective containment strategies.

The doubly degenerate nutrient taxis system(0.1)

\begin{equation} \left \{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha }v\nabla v)+\ell uv,&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v-uv,&x\in \Omega ,\, t\gt 0,\\ \end{aligned} \right . \end{equation}

$\Omega \subset \mathbb{R}^3$

$\alpha \gt 0,\chi \gt 0$

$\ell \gt 0$

$\alpha \in (\frac {3}{2},\frac {19}{12})$

$(u_\infty , 0)$

$t\rightarrow \infty$

$u_{\infty }$

$v_0$

$u_0$

Recent research has demonstrated the importance of spatial diffusion and environmental heterogeneity in influencing the transmission dynamics of infectious diseases. At the same time, human mobility patterns have been shown to exhibit scale-free, nonlocal dynamics characterized by an anomalous Lévy process diffusion, which is mathematically represented by nonlocal equations involving fractional Laplacian operators. To investigate the effects of environmental heterogeneity and long-range geographical disease transmission, we propose a time-periodic susceptible-infectious-susceptible (SIS) epidemic model that incorporates anomalous diffusion and spatial heterogeneity. The key issues of this paper include the existence and stability of both disease-free and endemic periodic equilibria, as well as the impact of diffusion rates and fractional powers on the spatial distribution of these periodic states. Our analytical findings indicate that spatio-temporal heterogeneity promotes disease persistence and that the fractional power can modulate the transmission threshold.

This study is concerned with nonnegative solutions of the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system with a logistic source, as given by

\begin{align*}\left\{\begin{array}{ll}u_t = \nabla\cdot (uv\nabla u) - \nabla\cdot (u^2 v\nabla v)+ u - u^2,\\[1mm]v_t = \Delta v -uv,\end{array} \right.\end{align*}

$\Omega$

$(u,v)|_{t=0}=(u_0,v_0)$

$v_0 \gt 0$

$\overline{\Omega}$

\begin{align*}\int_\Omega \ln u_0~ \gt ~-\infty,\end{align*}

$t\to\infty$

\begin{align*} u(\cdot,t) \rightharpoonup 1 \quad \text{in } L^1(\Omega) \qquad\text{and}\qquad v(\cdot,t)\to 0 \quad \text{in } L^\infty(\Omega). \end{align*}

This paper studies a time-switching advection-diffusion system modelling the competition between Aedes albopictus and Aedes aegypti mosquitoes in heterogeneous environments. The switching mechanism is induced by periodic releases of sterile Ae. albopictus mosquitoes, which are active only during their sexual lifespan within each release period. By defining a minimal release amount and four critical release period thresholds, we establish the periodic dynamics of the system, providing new insights into optimal control strategies of mosquitoes. Specifically, the trivial steady state is globally asymptotically stable if sterile releases are sufficiently frequent and abundant, which ensures the eradication of both Aedes species. For less frequent sterile releases, we prove the global asymptotic stability of the two semi-trivial periodic solutions and demonstrate the existence of a coexisting periodic solution, indicating cases where mosquito control fails. Numerical simulations are presented to validate our theoretical findings.

We study the well-posedness of solutions to the general nonlinear parabolic equations with merely integrable data in time-dependent Musielak–Orlicz spaces. With the help of a density argument, we establish the existence and uniqueness of both renormalized and entropy solutions. Moreover, we conclude that the entropy and renormalized solutions for this equation are equivalent. Our results cover a variety of problems, including those with Orlicz growth, variable exponents and double-phase growth.

We consider steady-state diffusion in a bounded planar domain with multiple small targets on a smooth boundary. Using the method of matched asymptotic expansions, we investigate the competition of these targets for a diffusing particle and the crucial role of surface reactions on the targets. We start from the classical problem of splitting probabilities for perfectly reactive targets with Dirichlet boundary conditions and improve some earlier results. We discuss how this approach can be generalised to partially reactive targets characterised by a Robin boundary condition. In particular, we show how partial reactivity reduces the effective size of the target. In addition, we consider more intricate surface reactions modelled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behaviour of the eigenvalues and eigenfunctions for these spectral problems in the small-target limit. Finally, we show how our asymptotic approach can be extended to interior targets in the bulk and to exterior problems where diffusion occurs in an unbounded planar domain outside a compact set. Direct applications of these results to diffusion-controlled reactions are discussed.

We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_{\phi}$. We distinguish between the cases $p \gt 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $\rho$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $\rho$.

This note establishes sharp time-asymptotic algebraic rate bounds for the classical evolution problem of Fujita, but with sublinear rather than superlinear exponent. A transitional stability exponent is identified, which has a simple reciprocity relation with the classical Fujita critical blow-up exponent.

In this paper, we consider a reaction-diffusion equation that models the time-almost periodic response to climate change within a straight, infinite cylindrical domain. The shifting edge of the habitat is characterised by a time-almost periodic function, reflecting the varying pace of environmental changes. Note that the principal spectral theory is an important role to study the dynamics of reaction-diffusion equations in time heterogeneous environment. Initially, for time-almost periodic parabolic equations in finite cylindrical domains, we develop the principal spectral theory of such equations with mixed Dirichlet–Neumann boundary conditions. Subsequently, we demonstrate that the approximate principal Lyapunov exponent serves as a definitive threshold for species persistence versus extinction. Then, the existence, exponential decay and stability of the forced wave solutions $U(t,x_{1},y)=V\left (t,x_{1}-\int ^{t}_{0}c(s)ds,y\right )$ are established. Additionally, we analyse how fluctuations in the shifting speed affect the approximate top Lyapunov exponent.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

We consider the propagation dynamics of a single species with a birth pulse and living in a shifting environment driven by climate change. We describe how birth pulse and environment shift jointly impact the propagation properties. We show that a moderate environment shifting speed promotes the spatial–temporal propagation represented by a stable forced KPP wave, and that the birth pulse shrinks the survival region.

We consider the asymptotics of long-time behavior of a solution u of the semilinear parabolic problem $\partial _tu=\Delta u-u+u|u|^{p-2}$ in ${\mathbb {R}^N}\times (0,\infty )$, $u(0)=u_0\in H^1({\mathbb {R}^N})\cap L^\infty ({\mathbb {R}^N})$. Since the spatial domain on which the problem is posed is noncompact, we cannot expect the relative compactness of the solution orbit, e.g., in $H^1({\mathbb {R}^N})$ in general. In this article, we prove that the compactness of the orbit holds up to the ground state energy level, namely, if $\lim _{t\to \infty }I(u(t))\leq d_\infty $, where I is the energy functional associated with (P) and $d_\infty $ its ground state energy, then the orbit of $u(t)$ is compact in $H^1({\mathbb {R}^N})$. Our result includes the previous results in [4, 5].