We present a novel multiscale mathematical model of espresso brewing. The model captures liquid infiltration and flow through a packed bed of ground coffee, as well as coffee solubles transport (both in the grains and in the liquid) and solubles dissolution. During infiltration, a sharp interface separates the dry and wet regions of the bed. A matched asymptotic analysis (based on fast dissolution rates) reveals that the bed can be described by four asymptotic regions: a dry region yet to be infiltrated by the liquid, a region in which the liquid is saturated with solubles and very little dissolution occurs, a slender region in which solubles are rapidly extracted from the smallest grains, and region in which slower extraction occurs from larger grains. The position and extent of each of these regions move with time (one being an intrinsic moving internal boundary layer) making the asymptotic analysis intriguing in its own right. The analysis yields a reduced model that elucidates the rate-limiting physical processes. Numerical solutions of the reduced model are compared to those to the full model, demonstrating that the reduced model is both accurate and significantly cheaper to solve.

We consider local and nonlocal Cahn–Hilliard equations with constant mobility and singular potentials including, e.g., the Flory–Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we examine the essential assumptions required for the reaction term to ensure the existence of a weak solution. Also, we explore the scenario involving the nonlocal Cahn–Hilliard equation and provide some illustrative examples that contextualize within our abstract framework.

We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data.

\begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+, &\ x\to +\infty, \\[3pt] 1/c_-, &\ x\to -\infty, \end{array}\right .} \end{align*}

$c_+$

$c_-$

$(y,t)$

$(\xi, c)$

$c=c_+/c_-$

$\{ (\xi, c)\,:\, \xi \in \mathbb{R}, \ c\gt 0, \ \xi =y/t\}$

$u(y,t)$

$\mathcal{O}(t^{-2})$

$\mathcal{O}(t^{-1})$

This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.

This article is concerned with the spreading speed and traveling waves of a lattice prey–predator system with non-local diffusion in a periodic habitat. With the help of an associated scalar lattice equation, we derive the invasion speed for the predator. More specifically, when the dispersal kernel of the predator is exponentially bounded, the invasion speed is finite and can be characterized in terms of principal eigenvalues; while the dispersal kernel is algebraically decaying, the invasion speed is infinite and the accelerated spreading rate is obtained. Furthermore, the existence and non-existence of traveling waves connecting the semi-equilibrium point to a uniformly persistent state are established.

In this paper, we consider the defocusing nonlinear wave equation $-\partial _t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb {R}\times \mathbb {R}^d$. Building on our companion work (Self-similar imploding solutions of the relativistic Euler equations, arXiv:2403.11471), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time.

We establish a weak local boundedness to Lane–Emden systems in two-dimensional domains involving general second-order elliptic operators in divergence form and arbitrary positive powers whose product equals 1. Our result is complete in the sense that it reduces to that of Trudinger for single equations. As a counterpart, we derive a new Harnack estimate for such systems and, as a by-product, for biharmonic equations.

We prove interior boundedness and Hölder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et al. in 2022 and 2023 for the nonlinear integro-differential problems in Heisenberg setting. Our proof of the a priori estimates bases on De Giorgi–Nash–Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev–Poincaré type inequality in local domain, which may be of independent interest and potential applications.

In this work, the Riemann–Hilbert (RH) problem is employed to study the multiple high-order pole solutions of the cubic Camassa–Holm (cCH) equation with the term characterizing the effect of linear dispersion under zero boundary conditions and nonzero boundary conditions. Under the reflectionless situation, we generalize the residue theorem and obtain the multiple high-order pole solutions of cCH equation by solving an algebraic system. During the process of establishing the solution of RH problem, to simplify the calculations involving the implicitly expressed of variables (x, t) in the solution, we introduce a new scale (y, t) to ensure the solution of RH problem is explicitly expressed with respect to it. Finally, the exact solutions are obtained for cases involving one high-order pole and N high-order poles.

In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of m species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in $L^1$. This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form $\partial _t v_i - \Delta v_i^{r_i} = R_i$, for an exponent $r_i < 2$. The case $r_i \geq 2$ remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.

In this paper, we study the existence of travelling wave solutions and the spreading speed for the solutions of an age-structured epidemic model with nonlocal diffusion. Our proofs make use of the comparison principles both to construct suitable sub/super-solutions and to prove the regularity of travelling wave solutions.

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$

$u=1$

$D$

$\Delta _1 \approx 0.00297$

$D \ll 1$

$O(1)$

$u=O(1)$

$u$

$D \to 0^+$

$D \geq \Delta _1$

$2 \sqrt{D}$

$0 \lt D \lt \Delta _1$

$u=0$

$D$

This article offers an advanced and novel investigation into the intricate propagation dynamics of the Belousov–Zhabotinsky system with non-local delayed interaction, which exhibits dynamical transition structure from bistable to monostable. We first solved the enduring open problem concerning the existence, uniqueness and the speed sign of the bistable travelling waves. In the monostable case, we developed and derived new results for the minimal wave speed selection, which, as an application, further improved the existing investigations on pushed and pulled wavefronts. Our results can provide new estimate to the minimal speed as well as to the determinacy of the transition parameters. Moreover, these results can be directly applied to standard localised models and delayed reaction diffusion models by choosing appropriate kernel functions.

This work describes a hyperbolic model for cell-cell repulsion with population dynamics. We consider the pressure produced by a population of cells to describe their motion. We assume that cells try to avoid crowded areas and prefer locally empty spaces far away from the carrying capacity. Here, our main goal is to prove the existence of travelling waves with continuous profiles. This article complements our previous results about sharp travelling waves. We conclude the paper with numerical simulations of the PDE problem, illustrating such a result. An application to wound healing also illustrates the importance of travelling waves with a continuous and discontinuous profile.

This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumour. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study is we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and non-linear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.

We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems:

$$ \begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*} $$

where $\Omega \subset \mathbb {R}^{n}$ is a bounded domain of class $C^{1,1}$, $1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$ and $\mathcal {N}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above problem achieve an $L^{\infty }$-bound independent of $d.$ Using this together with suitable $L^{r}$-estimates on $u_d,$ we show that the least energy solution $u_d$ achieves a maximum on the boundary of $\Omega $ for d sufficiently small.

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant

\[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \]

$\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$

$\partial \Omega$

$m>\max \{1,\,\frac {3N-2}{2N+2}\}$

$(\bar {u}_0,\,0)$

$t\rightarrow \infty$

$\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$

$m>\frac {3N-2}{2N}$

$N\geq 3$

$m>\frac {3N}{2N+2}$

$N\geq 2$

Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.