We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order $O(\delta _1^{4N+2}+\delta _2^{4N+2})$ in the sense of consistency, where $\delta _1$ and $\delta _2$ are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and $N$ is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.

For $s\in [\tfrac {1}{2},\, 1)$, let $u$ solve $(\partial _t - \Delta )^s u = Vu$ in $\mathbb {R}^{n} \times [-T,\, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} < \infty$. We show that if for some $0<\mathfrak {K} < T$ and $\epsilon >0$

\[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \]

$u \equiv 0$

$\mathbb {R}^{n} \times [-T,\, 0]$

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption

\[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \]

$(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$

$m>1$

$q\in (0,\,1)$

$\sigma =\sigma _c:=2(1-q)/ (m-1)$

\[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \]

$\beta ^*\in (0,\,\infty )$

$(u_A)_{A>0}$

$(f_A)_{A>0}$

$f_A(0)=A$

$f_A'(0)=0$

$\sigma \in (0,\,\sigma _c)$

In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem:

\[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \]

$b,\, \omega >0$

$p>2$

In this article, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced an estimate of the Morse index of such hypersurfaces.

The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

This paper is concerned with stochastic Schrödinger delay lattice systems with both locally Lipschitz drift and diffusion terms. Based on the uniform estimates and the equicontinuity of the segment of the solution in probability, we show the tightness of a family of probability distributions of the solution and its segment process, and hence the existence of invariant measures on $l^2\times L^2((-\rho,\,0);l^2)$ with $\rho >0$. We also establish a large deviation principle for the solutions with small noise by the weak convergence method.

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$. For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$, where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$. In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$. Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$, then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$. For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media

\[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \]

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

$\mathcal {C}^{1,1}$

$\nu$

$\lambda <0$

$2^{\ast }_{\alpha }=3+\alpha$

$0<\alpha <3$

$W^{\alpha,2^{\ast }_{\alpha }}_{0}(\mathrm {curl};\Omega )$

The paper is concerned with positive solutions to problems of the type

\[ -\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \]

$\mathbb {B}^N$

$1< p<2^*-1:=\frac {N+2}{N-2}$

$\;\lambda < \frac {(N-1)^2}{4}$

$f \in H^{-1}(\mathbb {B}^{N})$

$f \not \equiv 0$

$a\in L^\infty (\mathbb {B}^N)$

$\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$

$d(x,\, 0)$

$a(x) \leq 1$

$a(x) \geq 1$

$\mu ( \{ x : a(x) \neq 1\}) > 0.$

$a(x) \equiv 1$

We generalize the known collision results for a solid in a 3D compressible Newtonian fluid to compressible non-Newtonian ones, and to Newtonian fluids with temperature-depending viscosities.

In this note, we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi _n$ and $\phi _n$ associated with a sequence $\{nP\}_{n\in \mathbb {N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.

In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.

In this paper, we consider the following non-linear system involving the fractional Laplacian0.1

\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation}

$0< s<1$

$x_n$

We study minimizers of the Allen–Cahn system. We consider the $\varepsilon$-energy functional with Dirichlet values and we establish the $\Gamma$-limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the $\varepsilon$-energy functional as $\varepsilon \rightarrow 0$.

In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magnetic field $(\alpha,0 )^\top$ on $\mathbb {T}\times \mathbb {R}$. More precisely, for sufficiently large $\alpha$, we show that when the initial datum of the shear flow satisfies $\left \| U(y)-y\right \|_{H^{N+6}}\ll 1$, with $N>1$, and the initial perturbations ${u}_{\mathrm {in}}$ and ${b}_{\mathrm {in}}$ satisfy $\left \| ( {u}_{\mathrm {in}},{b}_{\mathrm {in}}) \right \| _{H^{N+1}}=\epsilon \ll \nu ^{\frac 56+\tilde \delta }$ for any fixed $\tilde \delta >0$, then the solution of the 2D MHD equations remains $\nu ^{-(\frac {1}{3}+\frac {\tilde \delta }{2})}\epsilon$-close to $( e^{\nu t \partial _{yy}}U(y),0)^\top$ for all $t>0$.

Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.

In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the $p$-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.