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.

In recent years, there has been significant interest in the effect of different types of adversarial perturbations in data classification problems. Many of these models incorporate the adversarial power, which is an important parameter with an associated trade-off between accuracy and robustness. This work considers a general framework for adversarially perturbed classification problems, in a large data or population-level limit. In such a regime, we demonstrate that as adversarial strength goes to zero that optimal classifiers converge to the Bayes classifier in the Hausdorff distance. This significantly strengthens previous results, which generally focus on $L^1$-type convergence. The main argument relies upon direct geometric comparisons and is inspired by techniques from geometric measure theory.

This paper investigates sharp stability estimates for the fractional Hardy–Sobolev inequality:

\begin{align*}\ \ \ \ \ \ \ \ \ \mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,\mathrm{d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-\Delta)^{\frac{s}{2}} u \right|^2 \,\mathrm{d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right)\end{align*}

where $s \in (0,1)$, $0 \lt t \lt 2s$, $N \gt 2s$ is an integer, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $\mu_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality. The primary focus is on the quantitative stability results of the above inequality and the corresponding Euler–Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler–Lagrange equation, offering a thorough characterization of the Palais–Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell [J. Funct. Anal. 1991] as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi [Int. Math. Res. Not. 2017] in the framework of fractional calculus.

We consider the existence, nonexistence and multiplicity of normalized solutions to the $p$-Laplacian equation on a bounded domain(0.1)

\begin{equation}\left\{\begin{array}{ll}-\Delta_p u+\lambda |u|^{p-2}u=g(u),&\text{in }\Omega,\\\int_\Omega |u|^p=\rho, u=0&\text{on }\partial \Omega,\end{array}\right.\end{equation}

where $\Omega$ is a bounded domain, $p\geq 2$. Firstly, under suitable assumptions on $\rho$, if $g$ is at most mass-critical at infinity, we prove the existence of infinitely many solutions. Secondly, for $\rho$ large, if $g$ is mass-supercritical, we perform a blow-up analysis to show the nonexistence of finite Morse index solutions. At last, for $\rho$ suitably small, combining with the monotonicity argument, we obtain a multiplicity result. In particular, when $p=2$, we obtain the existence of infinitely many normalized solutions.

We study the existence and multiplicity of positive bounded solutions for a class of nonlocal, non-variational elliptic problems governed by a nonhomogeneous operator with unbalanced growth, specifically the double phase operator. To tackle these challenges, we employ a combination of analytical techniques, including the sub-super solution method, variational and truncation approaches, and set-valued analysis. Furthermore, we examine a one-dimensional fixed-point problem.To the best of our knowledge, this is the first workaddressing nonlocal double phase problems using these methods.

This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.

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.