Let $$H_{p(\cdot ),q}$$ be the variable Lorentz–Hardy martingale spaces. In this paper, we give a new atomic decomposition for these spaces via simple $$L_r$$ -atoms $$(1<r \le \infty )$$ . Using decomposition, consider dual of Lorentz-Hardy case $$0<p(\cdot )\le 1$$ , $$0<q\le and )<2$$ $$1<q<\infty $$ respectively, prove that they are equivalent to BMO with exponent. Furthermore, also obtain ...

This paper deals with a flux-limited chemotaxis-haptotaxis system nonlinear diffusion \begin{document}$ \begin{eqnarray*} \left\{ \begin{split}{} &amp;u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u|\nabla v|^{p-2}\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &amp;v_t \Delta v-v+u, &amp;w_t -vw, \ \end{split} \right. \end{eqnarray*} $\end{document} in $ \Omega\times(0, \infty) $, where...

We extend two constructions of Alan Koch, exhibiting methods to construct brace blocks, that is, families group operations on a set $G$ such any them induce skew structure $G$. these by using bilinear maps and liftings endomorphisms quotient groups with respect central subgroup. provide several examples the construction, showing there are blocks which consist distinct given cardinality. One we ...

We consider structured optimisation problems defined in terms of the sum a smooth and convex function, proper, l.s.c., (typically non-smooth) one reflexive variable exponent Lebesgue spaces $L_{p(\cdot)}(\Omega)$. Due to their intrinsic space-variant properties, such can be naturally used as solution space combined with functionals for ill-posed inverse problems. For this purpose, we propose an...

In this note, we consider the large time behavior of following chemotaxis-Navier-Stokes system$ \begin{equation*} \label{AH4} \left\{ \begin{split} n_t + {\bf u}\cdot\nabla n &amp; = \Delta n^m - \nabla \cdot\big(n\nabla c\big), \qquad x\in\Omega ,\, t&gt;0,\\ c_t c nc, &amp;\qquad u}_t u} \Delta{\bf +\nabla P n\nabla \phi, \cdot{\bf 0, t&gt;0 \end{split} \right. \end{equation*} $with $ m&gt;1 ...

We establish a continuous embedding $W^{s(\cdot),2}(\Omega)\hookrightarrow L^{\alpha(\cdot)}(\Omega)$, where the variable exponent $\alpha(x)$ can be close to critical $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\bar{\Omega}$. Subsequently, this is used prove multiplicity of solutions nonlocal degenerate Kirchhoff problems singular exponent. Moreover, we also ob...

in this paper we give some general results on the non-splitextension group $overline{g}_{n} = 2^{2n}{^{cdot}}sp(2n,2), ngeq2.$ we then focus on the group $overline{g}_{4} =2^{8}{^{cdot}}sp(8,2).$ we construct $overline{g}_{4}$ as apermutation group acting on 512 points. the conjugacy classes aredetermined using the coset analysis technique. then we determine theinertia factor groups and fischer...

Let $$p(\cdot ):\ {\mathbb {R}^n}\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Hölder continuous condition and A general expansive matrix on $${\mathbb {R}^n}$$ . In this article, authors introduce anisotropic Campanato-type spaces give some applications. Especially, using known atomic finite characterizations of Hardy space $$H_A^{p(\cdot )}(\mathbb {R}^...