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In this paper, we will show the Carleson measure characterizations for the boundedness and compactness of the Cesàro-type operator
\begin{equation*}\mathcal{C}_{\mu}(f)(z)=\sum^{\infty}_{n=0}\left( \int_{[0,1)}t^nd\mu(t)\right) \left(\sum^{n}_{k=0}a_k \right)z^n, \quad z\in \mathbb{D},\end{equation*}
acting on a number of important analytic function spaces on 
$\mathbb{D}$, where µ is a positive finite Borel measure. The function spaces are some newly appeared analytic function spaces (e.g., Bergman–Morrey spaces 
$A^{p,\lambda}$ and Dirichlet–Morrey spaces 
$\mathcal{D}_p^{\lambda}$) . This work continues the lines of the previous characterizations by Blasco and Galanopoulos et al. for classical Hardy spaces and weighted Bergman spaces and so forth.
Using the technique of Gabor analysis, we characterize the boundedness of 
$e^{i\Delta }: W^{p_1,q_1}_m\rightarrow W^{p_2,q_2}$ with modulation and translation operators, where 
and m is a v-moderate weight. The sharp exponents for the boundedness are also characterized in the case of power weight.
This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant\[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \]in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$
with smooth boundary $\partial \Omega$
. It is shown that if $m>\max \{1,\,\frac {3N-2}{2N+2}\}$
, for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$
in an appropriate sense as $t\rightarrow \infty$
, where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$
. This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys. 58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys. 66 (2015), 3159–3179) to $m>\frac {3N-2}{2N}$
in the case $N\geq 3$
, but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 669–686) to $m>\frac {3N}{2N+2}$
when $N\geq 2$
.
For a real number 
$0<\epsilon <1/3$, we show that the anti-canonical volume of an 
$\epsilon $-klt Fano 
$3$-fold is at most 
$3,200/\epsilon ^4$, and the order 
$O(1/\epsilon ^4)$ is sharp.
In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, 
$$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$
$\mu $ is a positive Radon measure, K is a 
$\mu $-measurable function on the open unit disk 
$\mathbb {D}$, and 
$\sigma _w(z)$ is the classical Möbius transform of 
$\mathbb {D}$.
This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system
in a smoothly bounded domain Ω ⊂ ℝ2, with parameters Dw > 0, Dz > 0, β > 0 and ρ ⩾ 0.
\[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\]
Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if β < 1, whereas whenever β > 1 and 
$({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$, infinite-time blow-up occurs at least in the particular case when ρ = 0.
In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β > 0, at each prescribed level γ ∈ (0, 1/(β − 1)+) one can identify an L∞-neighbourhood of the homogeneous distribution (u, v, w, z) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium (u∞, 0, 0, 0) with some u∞ > 0.
This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as
It is shown that whenever ρ ∈ ℝ, μ > 0 and
$$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta v-v + u,\quad x\in \Omega ,\;t > 0.} $$then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium
$$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.
$$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$
