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We consider the existence and stability of constrained solitary wave solutions to the generalized Ostrovsky equationwhere the homogeneous nonlinearities 
\begin{align*}\partial_x\left(\partial_t u+ \alpha\partial_x u+\partial_x(f(u))+\beta \partial_x^3u\right)=\gamma u,\quad \|u\|_{L^2}^2=\lambda \gt 0,\end{align*}
$f(s)=\alpha_0|s|^p+\alpha_1|s|^{p-1}s$, with p > 1. If 
$\alpha_0,\alpha_1 \gt 0$, 
$\alpha\in\mathbb{R}$, and γ < 0 satisfying 
$\beta\gamma=-1$, we show that for 
$1 \lt p \lt 5$, there exists a constrained ground state traveling wave solution with travelling velocity 
$\omega \gt \alpha-2$. Furthermore, we obtain the exponential decay estimates and the weak non-degeneracy of the solution. Finally, we show that the solution is spectrally stable. This is a continuation of recent work [1] on existence and stability for a water wave model with non-homogeneous nonlinearities.
We study the existence and strong instability of standing wave solutions for the fractional nonlinear Schrödinger equation
\begin{equation*}\begin{cases}\mathrm{i} \psi_{t}=\left(-\Delta\right)^s \psi-\left(|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi\right), & \quad(t,x)\in(0,\infty)\times\mathbb{R}^N,\\\psi(0,x)=\psi_0(x),&\quad x\in\mathbb{R}^N,\end{cases}\end{equation*}
where 
$N\geq2$, 
$0 \lt s \lt 1$, 
$2 \lt q \lt p \lt 2_s^*=2N/(N-2s)$, and 
$\mu\in\mathbb{R}$. The primary challenge lies in the inhomogeneity of the nonlinearity.We deal with the following three cases: (i) for 
$2 \lt q \lt p \lt 2+4s/N$ and µ < 0, there exists a threshold mass a0 for the existence of the least energy normalized solution; (ii) for 
$2+4s/N \lt q \lt p \lt 2_s^*$ and µ > 0, we reveal the existence of the ground state solution, explore the strong instability of standing waves, and provide a blow-up criterion; (iii) for 
$2 \lt q\leq2+4s/N \lt p \lt 2_s^*$ and µ < 0, the strong instability of standing wave solutions is demonstrated. These findings are illuminated through variational characterizations, the profile decomposition, and the virial estimate.
In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schrödinger equation (NLS),
\begin{equation*}\text{i } \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^d,\end{equation*}
where 
$d \geq 1$, 
$\mu \in \mathbb{R}$ and 
$0 \lt \sigma \lt \infty$ if 
$1 \leq d \leq 4$ and 
$0 \lt \sigma \lt 4/(d-4)$ if 
$d \geq 5$. In the mass critical and supercritical cases, we establish the existence of blowup solutions to the problem for cylindrically symmetric data. The result extends the known ones with respect to blowup of solutions to the problem for radially symmetric data.
In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problemwhere 
\begin{equation*}-\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N,\end{equation*}
$q\ge p\ge2$, r > q, 
$0 \lt \mu \lt N$ and 
$w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative functions such that 
$w(x) \le C_1|x|^a$ and 
$f(x) \ge C_2|x|^b$ for all 
$|x| \gt R_0$, where 
$R_0,C_1,C_2 \gt 0$ and 
$a,b\in\mathbb{R}$. Under some appropriate assumptions on p, q, r, µ, a, b and N, we prove various Liouville-type theorems for weak solutions which are stable or stable outside a compact set of 
$\mathbb{R}^N$. First, we establish the standard integral estimates via stability property to derive the non-existence results for stable weak solutions. Then, by means of the Pohožaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.
We consider a parabolic-parabolic chemotaxis system with singular chemotactic sensitivity and source functions, which is originally introduced by Short et al to model the spatio-temporal behaviour of urban criminal activities with the particular value of the chemotactic sensitivity parameter 
$\chi =2$. The available analytical findings for this urban crime model including 
$\chi =2$ are restricted either to one-dimensional setting, or to initial data and source functions with appropriate smallness, or to initial data and source functions with some radial symmetry. In the present work, our first result asserts that for any 
$\chi \gt 0$ the initial-boundary value problem of this urban crime model possesses a global generalised solution in the two-dimensional setting, without imposing any small or radial conditions on initial data and source functions. Our second result presents the asymptotic behaviour of such solution, under some additional assumptions on source functions.
We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u)=w(\left \vert x \right \vert )g(u)$
on the Euclidean space $\mathbb {R}^n$
. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behaviour at infinity. All the estimates are given in terms of the spatial dimension $n$
, the values of $k$
and the behaviour at infinity of the growth rate function of $w$
.
This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$
with $\mathbb {T}=[0,\,1]$
being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$
. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$
as $t\to \infty$
. As a consequence, the solution converges to its horizontal average asymptotically.
We show that the energy norm of weak solutions to Vlasov equation coupled with a shear thickening fluid on the whole space has a decay rate the energy norm $E(t) \leq {C}/{(1+t)^{\alpha }}, \forall t \geq 0$
for $\alpha \in (0,3/2)$
.
We study the fractional parabolic Lichnerowicz equation
$$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$
where 
$p>0$
and
$ 0<s<1 $
. We establish a Liouville-type theorem for positive solutions in the case
$p>1$
and give a uniform lower bound of positive solutions when
$0<p\leq 1$
. In particular, when v is independent of the time variable, we obtain a similar result for the fractional elliptic Lichnerowicz equation
$$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$
with
$p>0$
and
$0<s<1$
. This extends the result of Brézis [‘Comments on two notes by L. Ma and X. Xu’, C. R. Math. Acad. Sci. Paris349(5–6) (2011), 269–271] to the fractional Laplacian.
We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension 
$d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order 
$2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in 
$C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.
We consider a Keller–Segel model that describes the cellular chemotactic movement away from repulsive chemical subject to logarithmic sensitivity function over a confined region in 
${{\mathbb{R}}^n},\,n \le 2$
. This sensitivity function describes the empirically tested Weber–Fecher’s law of living organism’s perception of a physical stimulus. We prove that, regardless of chemotaxis strength and initial data, this repulsive system is globally well-posed and the constant solution is the global and exponential in time attractor. Our results confirm the ‘folklore’ that chemorepulsion inhibits the formation of non-trivial steady states within the logarithmic chemotaxis model, hence preventing cellular aggregation therein.
This paper is concerned with the static Choquard equation where 
$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$
$N,p>2$ and 
$\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$. We prove that if 
$u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then 
$u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and 
$\unicode[STIX]{x1D6FC}\neq 2$.
