We investigate a regularity for weak solutions of the following generalized Leray equations \begin{equation*} (-\Delta)^{\alpha}V- \frac{2\alpha-1}{2\alpha}V+V\cdot\nabla V-\frac{1}{2\alpha}x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from study self-similar to Naiver-Stokes in $\mathbb R^3$. Firstly, by making use vanishing viscosity and developing non-local effects fractional diff...

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, Jensen’s chain rule on calculus and leveraging the properties convex submultiplicative functions. Nabla time scales provides a unified framework that unifies continuous discrete calculus, making it powerful tool for studying various mathematical problems scales. By utilizing approach,...

We consider a smooth, complete and non-compact Riemannian manifold $$(\mathcal {M},g)$$ of dimension $$d \ge 3$$ , we look for solutions to the semilinear elliptic equation $$\begin{aligned} -\varDelta _g w + V(\sigma ) = \alpha (\sigma f(w) \lambda \quad \hbox {in }\mathcal {M}. \end{aligned}$$ The potential $$V :\mathcal {M} \rightarrow \mathbb {R}$$ is continuous function which coercive in s...

Through the paper, we present several inequalities involving C-monotonic functions with C≥1, on nabla calculus via time scales. It is known that dynamic generate many different in calculus. The main results will be proved by applying chain rule formula As a special case for our results, when T=R, obtain continuous analouges of had previously been literature. When T=N, to best authors’ knowledge...

This paper deals with the classical solution of following chemotaxis system generalized logistic growth and indirect signal production \begin{eqnarray} \left\{ \begin{array}{llll} & u_t=\epsilon\Delta u-\nabla\cdot(u\nabla v)+ru-\mu u^\theta,\\ 0=d_1\Delta v-\beta v+\alpha w,\\ 0=d_2\Delta w-\delta w+\gamma u \end{array} \right. \qquad(0.1)\end{eqnarray} so-called strong $W^{1, q}(\Omega)$-solu...

The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential$ \begin{equation*} \frac{\partial \boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta|\boldsymbol{y}|^{r-1}\boldsymbol{y}+\nabla p+\Psi(\boldsymbol{y})\ni\boldsymbol{g},\ \nabla\cdot\boldsymbol{y} = 0, \end{equatio...

We consider the drift-diffusion equation $$\begin{aligned} u_t-\varepsilon \varDelta u+\nabla \cdot (u\ \nabla K*u)=0 \end{aligned}$$ in whole space with global-in-time solutions bounded all Sobolev spaces; for simplicity, we restrict ourselves to model case $$K(x)=-|x|$$ . quantify mass concentration phenomenon, a genuinely nonlinear effect, radially symmetric of this small diffusivity $$\vare...

This article concerns the existence of solutions to Schrodinger-Poisson system $$\displaylines{ -\Delta_p u+|u|^{p-2}u+\lambda\phi u=|u|^{q-2}u+h(x) \quad \hbox{in }\mathbb{R}^3,\\ -\Delta \phi=u^2 }\mathbb{R}^3, }$$ where \( 4/3 < p 12/5 \), q p^{*}=3p/(3-p) \(\Delta_p u =\hbox{div}(|\nabla u|^{p-2}\nabla u)\), \(\lambda >0\), and \(h \not= 0\). The multiplicity results are obtained by u...

This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=\Delta v+\alpha u-\beta v, w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) smooth boundary $\partial\Ome...