We study the long time dynamic properties of the nonlocal Kuramoto–Sivashinsky (KS) equation with multiplicative white noise. First, we consider the dynamic properties of the stochastic nonlocal KS equation via a transformation into the associated conjugated random differential equation. Next, we prove the existence and uniqueness of solution for the conjugated random differential equation in the theory of random dynamical systems. We also establish the existence and uniqueness of a random attractor for the stochastic nonlocal equation.

In this paper, we study the existence of solutions to the following Hartree equation

\begin{align*}\begin{cases}-\Delta u+\lambda V(x) u+\mu u=\left(\int_{\mathbb{R}^N}\frac{|u|^p}{|x-y|^{N-\alpha}}\right)|u|^{p-2}u,\ \text{in}\ \mathbb{R}^N,\\\int_{\mathbb{R}^N}|u|^2=\omega,\end{cases}\end{align*}

Where $N\geq 3$, $\omega,\lambda \gt 0$, $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$ and µ will appear as a Lagrange multiplier. We assume that $0\leq V\in L^{\infty}_{loc}(\mathbb{R}^N)$ has a bottom $int V^{-1}(0)$ composed of $\ell_0$ $(\ell_{0}\geq1)$ connected components $\{\Omega_i\}_{i=1}^{\ell_0}$, where $int V^{-1}(0)$ is the interior of the zero set $V^{-1}(0)=\{x\in\mathbb{R}^N| V(x)=0\}$ of V. It is worth pointing out that the penalization technique is no longer applicable to the local sublinear case $p\in \left(\frac{N+\alpha}{N},2\right)$. Therefore, we develop a new variational method in which the two deformation flows are established that reflect the properties of the potential. Moreover, we find a critical point without introducing a penalization term and give the existence result for $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$. When ω is fixed and satisfies $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}$ sufficiently small, we construct a $\ell$-bump $(1\leq\ell\leq \ell_{0})$ positive normalization solution, which concentrates at $\ell$ prescribed components $\{\Omega_i\}^{\ell}_{i=1}$ for large λ. We also consider the asymptotic profile of the solutions as $\lambda\rightarrow\infty$ and $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}\rightarrow 0$.

We study existence and uniqueness of spherically symmetric solutions of

\begin{equation*}S_k(D^2v)+\beta \xi\cdot\nabla v+\alpha v+\left\vert v\right\vert^{q-1}v=0\;\; \mbox{in}\;\; \mathbb{R}^n,\end{equation*}

where $\alpha,\beta$ are real parameters, $n \gt 2,\, q \gt k\geqslant 1$ and $S_k(D^2v)$ stands for the k-Hessian operator of v. Our results are based mainly on the analysis of an associated dynamical system and energy methods. We derive some properties of the solutions of the above equation for different ranges of the parameters α and β. In particular, we describe with precision its asymptotic behaviour at infinity. Further, according to the position of q with respect to the first critical exponent $\frac{(n+2)k}{n}$ and the Tso critical exponent $\frac{(n+2)k}{n-2k}$ we study the existence of three classes of solutions: crossing, slow decay or fast decay solutions. In particular, if k > 1 all the fast decay solutions have a compact support in $\mathbb{R}^n$. The results also apply to construct self-similar solutions of type I to a related nonlinear evolution equation. These are self-similar functions of the form $u(t,x)=t^{-\alpha}v(xt^{-\beta})$ with suitable α and β.

We consider radially symmetric solutions of the degenerate Keller–Segel system

\begin{align*}\begin{cases}\partial_t u=\nabla\cdot (u^{m-1}\nabla u - u\nabla v),\\0=\Delta v -\mu +u,\quad\mu =\frac{1}{|\Omega|}\int_\Omega u,\end{cases}\end{align*}

in balls $\Omega\subset\mathbb R^n$, $n\ge 1$, where m > 1 is arbitrary. Our main result states that the initial evolution of the positivity set of u is essentially determined by the shape of the (nonnegative, radially symmetric, Hölder continuous) initial data u0 near the boundary of its support $\overline{B_{r_1}(0)}\subsetneq\Omega$: It shrinks for sufficiently flat and expands for sufficiently steep u0. More precisely, there exists an explicit constant $A_{\mathrm{crit}} \in (0, \infty)$ (depending only on $m, n, R, r_1$ and $\int_\Omega u_0$) such that if $u_0(x)\le A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0\in(0,r_1)$ and $A \lt A_{\mathrm{crit}}$ then there are T > 0 and ζ > 0 such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\le r_1 -\zeta t$ for all $t\in(0, T)$, while if $u_0(x)\ge A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0 \in (0, r_1)$ and $A \gt A_{\mathrm{crit}}$ then we can find T > 0 and ζ > 0 such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +\zeta t$ for all $t\in(0, T)$.

The aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in the deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity – an imaginary gauge potential and on-site gain and loss – we prove that all but two eigenmodes of the system pass through exceptional points and decouple. By tuning the gain-to-loss ratio, the system changes from a phase with unbroken parity-time symmetry to a phase with broken parity-time symmetry. At the macroscopic level, this is observed as a transition from symmetrical eigenmodes to condensated eigenmodes at one edge of the structure. Mathematically, it arises from a topological state change. The results of this paper open the door to the justification of a variety of phenomena arising from the interplay between non-Hermitian reciprocal and nonreciprocal mechanisms not only in subwavelength wave physics but also in quantum mechanics, where the tight-binding model coupled with the nearest neighbour approximation can be analysed with the same tools as those developed here.

This paper develops a geometric and analytical framework for studying the existence and stability of pinned pulse solutions in a class of non-autonomous reaction–diffusion equations. The analysis relies on geometric singular perturbation theory, matched asymptotic method and nonlocal eigenvalue problem method. First, we derive the general criteria on the existence and spectral (in)stability of pinned pulses in slowly varying heterogeneous media. Then, as a specific example, we apply our theory to a heterogeneous Gierer–Meinhardt (GM) equation, where the nonlinearity varies slowly in space. We identify the conditions on parameters under which the pulse solutions are spectrally stable or unstable. It is found that when the heterogeneity vanishes, the results for the heterogeneous GM system reduce directly to the known results on the homogeneous GM system. This demonstrates the validity of our approach and highlights how the spatial heterogeneity gives rise to richer pulse dynamics compared to the homogeneous case.

In this article, we consider a fully nonlinear equation associated with the Christoffel–Minkowski problem in hyperbolic space. By using the full rank theorem, we establish the existence of h-convex solutions when the prescribed functions on the right-hand side are under some appropriate assumption.

This article studies the optimal boundary regularity of harmonic maps between a class of asymptotically hyperbolic spaces. To be precise, given any smooth boundary map with nowhere vanishing energy density, this article provides an asymptotic expansion formula for harmonic maps under the assumption of $C^1$ up to the boundary.

In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional Nonlinear Schrödinger Equation (NLS),

\begin{equation*}\mathrm{i} \, \partial_t u=(-\Delta)^s u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^N,\end{equation*}

where $N \geq 2$, $1/2 \lt s \lt 1$, and $0 \lt \sigma \lt 2s/(N-2s)$. In the mass critical and supercritical cases, we establish a criterion for blow-up of solutions to the problem for cylindrically symmetric data. The results extend the known ones with respect to blow-up of solutions to the problem for radially symmetric data.

It was proved in [11, J. Funct. Anal., 2020] that the Cauchy problem for some Oldroyd-B model is well-posed in $\dot{B}^{d/p-1}_{p,1}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,1}(\mathbb{R}^d)$ with $1\leq p \lt 2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\dot{B}^{d/p-1}_{p,r}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,r}(\mathbb{R}^d)$ with $1\leq p\leq \infty$ and $1 \lt r\leq\infty$ due to the lack of continuous dependence of the solution.

We are concerned with the following coupled Schrödinger system with a Hardy potential in the critical case

\begin{align*}\begin{cases}-\Delta u_{i}-\frac{\lambda_{i}}{|x|^2}u_{i}=|u_i|^{2^*-2}u_i+\sum_{j\neq i}^{3}\beta_{ij}|u_{j}|^{\frac{2^*}{2}}|u_i|^{\frac{2^*}{2}-2}u_i, ~x\in \mathbb{R}^N,\\u_i\in D^{1,2}(\mathbb{R}^N),\,\, N\geq 3,\,\, i=1,2,3,\end{cases}\end{align*}

$2^*=\frac{2N}{N-2}$

$\lambda_i\in (0,\Lambda_N), \Lambda_N:= \frac{(N-2)^2}{4}$

$\beta_{ij}=\beta_{ji}$

$\beta_{ij} \gt 0$

$\beta_{i_{1}j_{1}} \gt 0$

$\beta_{i_{2}j_{2}} \lt 0$

$(i_{1}, j_{1})$

$(i_{2}, j_{2})$

$\beta_{ij}\ge0$

$N\ge5$

$\beta_{ij} \gt 0$

$N=3,4$

$N=3, 4$

$N\geq 5$

$N\geq 5$

$N\ge5$

We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non-linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $\Lambda^3(\mathcal{T})$, where ${\mathcal{T}}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms.

Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular, we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing–Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterization of algebraically special Riemannian metrics of type D in dimensions higher than four.

We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space ${\mathbb H}^n$. The free energy consists of two competing terms: an entropy, corresponding to slow nonlinear diffusion, that favours spreading, and an attractive interaction potential energy that favours aggregation. We establish necessary and sufficient conditions on the interaction potential for ground states to exist on the hyperbolic space ${\mathbb H}^n$. To prove our results, we derived several Hardy–Littlewood–Sobolev (HLS)-type inequalities on general Cartan–Hadamard manifolds of bounded curvature, which have an interest in their own.

We consider the drifting p-Laplace operator

\begin{equation*}\Delta_{p,v}u=e^{-v} \text{div}\,(e^v |\nabla u|^{p-2}\nabla u)\end{equation*}

and discuss generalized weighted Hardy-type inequalities associated with the measure $d\mu=e^{v(x)}dx$. As an application, we obtain several Liouville-type results for positive solutions of the non-linear elliptic problem with singular lower order term

\begin{equation*}-\Delta_{p,v} u\ge c(x) u^{p-1}+B \frac{|\nabla u|^p}{u} \quad \text{in}\ \Omega,\end{equation*}

where Ω is a bounded or an unbounded exterior domain in ${\mathbb{R}}^N$, $N \gt p \gt 1$, $B+p-1 \gt 0$, as well as of the non-autonomous quasilinear elliptic problem

\begin{equation*}-\Delta_{p,v} u+b(x)|\nabla u|^{p-1} \ge c(x) u^{p-1}\quad \text{in}\ \Omega,\end{equation*}

with general weights $b\ge0$ and c > 0. Liouville-type results are also discussed for a class of higher order differential equations.

We investigate the pullback measure attractors for non-autonomous stochastic p-Laplacian equations driven by nonlinear noise on thin domains. The concept of complete orbits for such systems is presented to establish the structures of pullback measure attractors. We first present some essential uniform estimates, as well as the existence and uniqueness of pullback measure attractors. A novel technical proof method is shown to overcome the difficulty of the estimates of the solutions in $W^{1,p}$ on thin domains. Then, we prove the upper semicontinuity of these measure attractors as the $(n + 1)$-dimensional thin domains collapse onto the lower n-dimensional space.

This paper develops methods for simplifying systems of partial differential equations (PDEs) that have families of conservation laws which depend on arbitrary functions of the independent or dependent variables. Cases are identified in which such methods can be combined with reduction using families of symmetries to give a multiple reduction; this is analogous to the double reduction of order for ordinary differential equations (ODE) with variational symmetries. Applications are given, including a widely used class of pseudoparabolic equations and several mean curvature equations.

This overview discusses the inverse scattering theory for the Kadomtsev–Petviashvili II equation, focusing on the inverse problem for perturbed multi-line solitons. Despite the introduction of new techniques to handle singularities, the theory remains consistent across various backgrounds, including the vacuum, 1-line and multi-line solitons.

In this paper, we study the Cauchy problem for pseudo-parabolic equations with a logarithmic nonlinearity. After establishing the existence and uniqueness of weak solutions within a suitable functional framework, we investigate several qualitative properties, including the asymptotic behaviour and blow-up of solutions as $t\to +\infty$. Moreover, when the initial data are close to a Gaussian function, we prove that these weak solutions exhibit either super-exponential growth or super-exponential decay.

Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. In this paper, we investigate the global (higher-order) Sobolev regularity of weak solutions to the fractional Dirichlet problem

\begin{equation*}\begin{cases}(-\Delta)^su=f \ \ & \text{in}\ \ \Omega,\\u=0 \ \ & \text{in}\ \ \mathbb{R}^n\setminus\Omega.\end{cases}\end{equation*}

Precisely, we prove that there exists a positive constant $\varepsilon\in(0,s]$ depending on n, s, and the Lipschitz constant of Ω such that, for any $t\in[\varepsilon,\min\{1+\varepsilon,2s\})$, when $f\in L^q(\Omega)$ with some $q\in(\frac{n}{2s-t},\infty]$, the weak solution u satisfies

\begin{equation*}\|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for all $p\in[1,\frac{1}{t-\varepsilon})$. In particular, when Ω is a bounded C1 domain or a bounded Lipschitz domain satisfying the uniform exterior ball condition, the aforementioned global regularity estimates hold with $\varepsilon=s$ and they are sharp in this case. Moreover, if Ω is a bounded $C^{1,\kappa}$ domain with $\kappa\in(0,s)$ or a bounded Lipschitz domain satisfying the uniform exterior ball condition, we further show the global BMO-Sobolev regularity estimate

\begin{equation*}\left\|(-\Delta)^{\frac{s}{2}}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}+\left\|\nabla^{s}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for some $q\in(\frac{n}{s},\infty]$, which is sharp in the sense that the BMO norm can not be improved to the $L^\infty$ norm.

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analysing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker–Planck equation can be approximated using the Keller–Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker–Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller–Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker–Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.