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*}

In this paper, we are mainly devoted to the limiting behaviour of smooth inertial manifolds for a class of random retarded differential equations with a singular parameter $\delta$ and their Galerkin approximations, which have not been considered before. Under appropriate conditions, we show not only that the inertial manifolds for this class of random retarded equations converge pointwise to those of the corresponding stochastic equations driven by white noise as $\delta\rightarrow 0$, but also that the inertial manifolds of their Galerkin approximations converge pointwise to those of stochastic equations driven by white noise described above under the simultaneous limits $\delta\rightarrow 0$ and $M\rightarrow +\infty$, where $M$ denotes the dimension index of the orthogonal projection operator $P_M$ in the Galerkin scheme.

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 provide a first-order homogenization result for quadratic functionals. In particular, we identify the scaling of the energy and the explicit form of the limiting functional in terms of the first-order correctors. The main novelty of the paper is the use of the dual correspondence between quadratic functionals and PDEs, combined with a refinement of the classical Riemann–Lebesgue lemma.

In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form

\begin{equation*}A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u \, d\mu(s),\end{equation*}

$\mu$

$[0,1]$

$p$

$p$

$p$

$p$

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.

We investigate axisymmetric surfaces in Euclidean space that are stationary for the energy $E_\alpha=\int_\Sigma |p|^\alpha\, d\Sigma$. Using a phase plane analysis, we classify these surfaces under the assumption that they intersect the rotation axis orthogonally. We also provide applications of the maximum principle, characterizing closed stationary surfaces and compact stationary surfaces with circular boundary in the case $\alpha=-2$. Finally, we prove that helicoidal stationary surfaces must in fact be rotational surfaces.

We study the decay properties of non-negative solutions to the one-dimensional defocusing damped wave equation in the Fujita subcritical case under a specific initial condition. Specifically, we assume that the initial data are positive, satisfy a condition ensuring the positiveness of solutions, and exhibit polynomial decay at infinity. To show the decay properties of the solution, we construct suitable supersolutions composed of an explicit function satisfying an ordinary differential inequality and the solution of the linear damped wave equation. Our estimates correspond to the optimal ones inferred from the analysis of the heat equation.

We prove that for bounded, divergence-free vector fields $\boldsymbol{b}$ in $L^1_{loc}((0,1];BV(\mathbb{T}^d;\mathbb{R}^d))$, there exists a unique incompressible measure on integral curves of $\boldsymbol{b}$. We recall the vector field constructed by Depauw in [8], which lies in the above class, and prove that for this vector field, the unique incompressible measure on integral curves exhibits stochasticity.

In this paper, we are interested in the existence and concentration of normalized solutions for the following logarithmic Schrödinger–Bopp–Podolsky type system involving the $p$-Laplacian in $\mathbb{R}^3$:

\begin{equation*}\left\{\begin{array}{ll}\displaystyle -\varepsilon^p\Delta_p u+Z(x)|u|^{p-2}u-\kappa\phi u=\lambda |u|^{p-2}u+|u|^{p-2}u\log|u|^p& \text{in} \ \mathbb{R}^3, \\\displaystyle -\varepsilon^2\Delta\phi+a^2\varepsilon^4\Delta^2\phi=4\pi u^2& \text{in} \ \mathbb{R}^3, \\\displaystyle \int_{\mathbb{R}^3}|u|^pdx=d^p\varepsilon^3,\end{array}\right.\end{equation*}

where $\Delta_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator, $Z$ is a given external potential, $\kappa \gt 0$ a constant, $a \gt 0$ is the Bopp–Podolsky constant and $\varepsilon \gt 0$ is a small parameter. The unknowns are $u,\phi:\mathbb{R}^{3}\to \mathbb{R}$ and the Lagrange multiplier $\lambda\in\mathbb{R}$. If $p\in[2,\frac{12}{5})$, we obtain, via the variational method, that the number of positive solutions depends on the profile of $Z$ and the solutions concentrate around the global minimum points of $Z$ in the semiclassical limit as $\varepsilon\to 0^{+}$.

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.

We consider the following problem

\begin{equation*}\varepsilon^{2 s} P_{\hat{h}}^s u+u=u^p, \quad u \gt 0 \quad \text {on } \,(M, \hat{h}),\end{equation*}

$(X, g^{+})$

$(M,[\hat{h}])$

$s\in (0,1)$

$P_{\hat{h}}^s$

$1 \lt p \lt \frac{n+2s}{n-2s}$

$\varepsilon$

$C^1$

$H$

$0 \lt s \lt {1}/{2}$

$C^1$

${1}/{2}\le s \lt 1$

In this paper, we study the existence and stability of solitary wave solutions for the generalized Benjamin equation in both the $L^2$-critical and $L^2$-supercritical cases by applying the variational methods and the non-homogeneous Gagliardo–Nirenberg inequality. Our main results generalize and complement the existing results in the literature.

In this paper, we are concerned with the following Dirichlet problems for nonlinear equations involving the fractional $p$-Laplacian:

\begin{equation*}\begin{cases}(-\Delta)_p^\alpha u=f(x,u,\nabla u),\ \ u \gt 0,\ \ \text{in}\ \ E,\\\ \ \ \ \ \ u\equiv0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \ \mathbb{R}^{n}\setminus E,\end{cases}\end{equation*}

where $E \subseteq \mathbb{R}^{n}$ is a coercive epigraph, i.e., there exists a continuous function $\phi: \, \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ satisfying

\begin{equation*}\lim_{|x'|\rightarrow+\infty}\phi(x')=+\infty,\end{equation*}

such that $E:=\{x=(x',x_{n}) \in \mathbb{R}^{n}|\,x_{n} \gt \phi(x')\}$, where $x':= (x_{1},...,x_{n-1}) \in \mathbb{R}^{n-1}$. Under some mild assumptions on the nonlinearity $f(x,u,\nabla u)$, we prove strict monotonicity of positive solutions to the above Dirichlet problems involving fractional $p$-Laplacian in coercive epigraph $E$.

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions $7$ and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov’s Lemma for ODE that is to our knowledge new.

Given $p\in[1,\infty)$ and a bounded open set $\Omega\subset\mathbb{R}^d$ with Lipschitz boundary, we study the $\Gamma$-convergence of the weighted fractional seminorm

\begin{equation*}[u]_{s,p,f}^p=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|\tilde u(x)- \tilde u(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y,\end{equation*}

as $s\to1^-$ for $u\in L^p(\Omega)$, where $\tilde u=u$ on $\Omega$ and $\tilde u=0$ on $\mathbb{R}^d\setminus\Omega$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\mathbb{R}^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\mathbb{R}^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\mathbb{R}^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}^p$ $\Gamma$-converges to the Dirichlet $p$-energy weighted by $f^2$. In the case $p=2$, we also prove the convergence of the corresponding gradient flows.