We study the timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition. Given a global solution u to the scalar wave equation with sufficiently small $C_c^\infty $ initial data, we derive an asymptotic formula for this global solution inside the light cone (i.e. for $|x|<t$). It involves the scattering data obtained in the author’s asymptotic completeness result in [75]. Using this asymptotic formula, we prove that u must vanish under some decaying assumptions on u or its scattering data, provided that the wave equation violates the null condition.

The Bray–Liebhafsky reaction is one of many intricate chemical systems that is known to exhibit periodic behaviour. Although the underlying chemistry is somewhat complicated and involves at least ten chemical species, in a recent work we suggested a reduced two-component model of the reaction involving the concentrations of iodine and iodous acid. Although it is drastically simplified, this reduced system retains enough structure so as to exhibit many of the oscillatory characteristics seen in experimental analyses. Here, we consider the possibility of spatial patterning in a nonuniformly mixed solution. Since many practical demonstrations of chemical oscillations are undertaken using circular containers such as beakers or Petri dishes, we develop both linearized and nonlinear pattern solutions in terms of cylindrical coordinates. These results are complemented by an analysis of the patterning that might be possible within a rectangular domain. The simulations give compelling evidence that spatial patterning may well be feasible in the Bray–Liebhafsky process.

This article is concerned with the following chemotaxis-growth system

\begin{equation*}\begin{cases}u_t = \nabla\cdot \left(\nabla u - u\bf{w} \right) + \rho u - \mu u^\alpha ,&x \in \Omega ,\ t \gt 0,\\v_{1,t} = \Delta v_1 - v_1 + u,&x \in \Omega ,\ t \gt 0,\\v_{2,t} = \Delta v_2 - v_2 + v_1,&x \in \Omega ,\ t \gt 0,\\\ldots \\v_{k,t} = \Delta v_k - v_k + v_{k - 1},&x \in \Omega ,\ t \gt 0,\\{\bf{w}}_t = \Delta {\bf{w}} - {\bf{w}} + \nabla v_k,&x \in \Omega ,\ t \gt 0,\end{cases}\end{equation*}

under the homogeneous Neumann boundary condition for u, vi and the homogeneous Dirichlet boundary condition for $\bf{w}$ in a smooth bounded domain $\Omega \subset {\mathbb{R}^n}\left( {n \geqslant 1} \right),$ where ρ > 0, µ > 0, α > 1 and $i=1,\ldots,k$. We reveal that when the index α, the spatial variable n, and the number of equations k satisfy certain relationships, the global solution of the system exists and converges to the constant equilibrium state in the form of exponential convergence.

In the second part of this series of papers, we address the same evolution problem that was considered in part 1 (see [16]), namely the nonlocal Fisher-KPP equation in one spatial dimension,

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

$\phi *u$

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

$[0,a]$

For multi-scale differential equations (or fast–slow equations), one often encounters problems in which a key system parameter slowly passes through a bifurcation. In this article, we show that a pair of prototypical reaction–diffusion equations in two space dimensions can exhibit delayed Hopf bifurcations. Solutions that approach attracting/stable states before the instantaneous Hopf point stay near these states for long, spatially dependent times after these states have become repelling/unstable. We use the complex Ginzburg–Landau equation and the Brusselator models as prototypes. We show that there exist two-dimensional spatio-temporal buffer surfaces and memory surfaces in the three-dimensional space-time. We derive asymptotic formulas for them for the complex Ginzburg–Landau equation and show numerically that they exist also for the Brusselator model. At each point in the domain, these surfaces determine how long the delay in the loss of stability lasts, that is, to leading order when the spatially dependent onset of the post-Hopf oscillations occurs. Also, the onset of the oscillations in these partial differential equations is a hard onset.

In this article, we deal with non-existence results, i.e., Liouville type results, for positive radial solutions of quasilinear elliptic equations with weights both in the entire $\mathbb R^N$ and in a ball, in the latter case under Dirichlet boundary conditions. The presence of weights, possibly singular or degenerate, makes the study fairly delicate. The proofs use a Pohozaev type identity combined with an accurate qualitative analysis of solutions. In the last part of the article, a non-existence theorem is proved for a Dirichlet problem with a convection term.

Asymptotic homogenisation is considered for problems with integral constraints imposed on a slowly varying microstructure; an insulator with an array of perfectly dielectric inclusions of slowly varying size serves as a paradigm. Although it is well-known how to handle each of these effects (integral constraints, slowly varying microstructure) independently within multiple scales analysis, additional care is needed when they are combined. Using the flux transport theorem, the multiple scales form of an integral constraint on a slowly varying domain is identified. The proposed form is applied to obtain a homogenised model for the electric potential in a dielectric composite, where the microstructure slowly varies and the integral constraint arises due to a statement of charge conservation. A comparison with multiple scales analysis of the problem with established approaches provides validation that the proposed form results in the correct homogenised model.

We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in $\mathbb{R}^3$:

\begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*}

$\alpha \gt \frac {3}{4}$

$\tilde {\textrm {a}}$

$\alpha \gt \frac {1}{2}$

$\alpha \geq \frac {5}{4}$

$\frac {3}{4}\lt \alpha \lt \frac {5}{4}$

$L^2$

Bioreactor scaffolds must be designed to facilitate adequate nutrient delivery to the growing tissue they support. For perfusion bioreactors, the dominant transport process is determined by the scale of fluid velocity relative to diffusion and the geometry of the scaffold. In this paper, models of nutrient transport in a fibrous bioreactor scaffold are developed using homogenisation via multiscale asymptotics. The scaffold is modelled as an ensemble of aligned strings surrounded by viscous, slowly flowing fluid. Multiple scales analysis is carried out for various parameter regimes which give rise to macroscale transport models that incorporate the effects of advection, reaction and diffusion. Multiple scales in both space and time are employed when macroscale advection balances macroscale diffusion. The microscale model is solved to obtain the effective diffusion coefficient and simple solutions to the macroscale problem are presented for each regime.

This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.

In this article, we study the effect of the Hardy potential on existence, uniqueness, and optimal summability of solutions of the mixed local–nonlocal elliptic problem

\begin{equation*}-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \,\text{in } \Omega, \ u=0 \,\text{in } {\mathbb R}^n \setminus \Omega,\end{equation*}

${\mathbb R}^n$

In this paper, we establish Newton–Maclaurin-type inequalities for functions arising from linear combinations of primitively symmetric polynomials. This generalization extends the classical Newton–Maclaurin inequality to a broader class of functions.

We consider the Cauchy problem of the non-linear Schrödinger equation with the modulated dispersion and power type non-linearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk–Gubinelli [7] and multilinear estimates which are based on divisor counting and show the local well-posedness. This generalizes the result by Chouk–Gubinelli [7] in terms of the dimension and the order of the non-linearity.

In this article, we investigate the following non-linear Schrödinger (NLS) equation with Neumann boundary conditions:

\begin{equation*}\begin{cases}-\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\\displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega\end{cases}\end{equation*}

coupled with a constraint condition:

\begin{equation*}\int_{\Omega}|u|^2 dx=c,\end{equation*}

where $\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, ν represents the unit outer normal vector to $\partial \Omega$, c is a positive constant, and λ acts as a Lagrange multiplier. When the non-linearity f exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various non-linearities, including cases such as $f(u)=|u|^{p-2}u$, $|u|^{q-2}u+ |u|^{p-2}u$, and $-|u|^{q-2}u+|u|^{p-2}u$, where $2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$. The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.

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.

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $\Omega \subset \mathbb{R}^n$ in order to have the following Hardy inequality:

\begin{equation*}\int_{\Omega} W(x) |\nabla u|_A^2 dx \geq \int_{\Omega} |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(\Omega),\end{equation*}

where d(x) is a suitable quasi-norm (gauge), $|\xi|^2_A = \langle A(x)\xi, \xi \rangle$ for $\xi \in \mathbb{R}^n$ and A(x) is an n × n symmetric, uniformly positive definite matrix defined on a bounded domain $\Omega \subset \mathbb{R}^n$. We also give its Lp analogue. As a consequence, we present examples for a standard Laplacian on $\mathbb{R}^n$, Baouendi–Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam [16].

Let $n\ge 2$ and $\mathcal {L}=-\mathrm {div}(A\nabla \cdot )$ be an elliptic operator on $\mathbb {R}^n$. Given an exterior Lipschitz domain $\Omega $, let $\mathcal {L}_D$ be the elliptic operator $\mathcal {L}$ on $\Omega $ subject to the Dirichlet boundary condition. Previously, it was known that the Riesz transform $\nabla \mathcal {L}_D^{-1/2}$ is not bounded for $p>2$ and $p\ge n$, even if $\mathcal {L}=\Delta $ is the Laplace operator and $\Omega $ is a domain outside a ball. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and $\partial \Omega $ is $C^1$. We prove that for $p>2$ and $p\in [n,\infty )$, it holds that

$$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$

for $f\in \mathring {W}^{1,p}(\Omega )$. Here, $\mathcal {K}_p(\mathcal {L}_D^{1/2})$ is the kernel of $\mathcal {L}_D^{1/2}$ in $\mathring {W}^{1,p}(\Omega )$, which coincides with $\tilde {\mathcal {A}}^p_0(\Omega ):=\{f\in \mathring {W}^{1,p}(\Omega ):\ \mathcal {L}_Df=0\}$ and is a one-dimensional subspace. As an application, we provide a substitution of $L^p$-boundedness of $\sqrt {t}\nabla e^{-t\mathcal {L}_D}$ which is uniform in t for $p\ge n$ and $p>2$.

This article is dedicated to investigating limit behaviours of invariant measures with respect to delay and system parameters of 3D Navier–Stokes–Voigt equations. Firstly, the well-posedness of such a system is obtained on arbitrary open sets that satisfy the Poincaré inequality, and then a unique minimal pullback attractor is attained by using the energy equation method and asymptotic compactness property. Furthermore, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors. Specifically, when the external forcing terms are periodic in time, the periodic invariant measure can be obtained. Finally, as the delay approaches zero and system parameters tend to some numbers, the limit of the invariant measure sequences for this class of equations must be the invariant measure of the corresponding limit equations.

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source

\begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*}

$p\gt 1$

$\sigma \ge -2$

$t\to \infty$

$\sigma =-2$