This paper is concerned with the existence of solutions for the following weighted fractional $p$-Laplace problem involving singular nonlinearity: $$ \begin{cases} \displaystyle 2\int_{\mathbb{R}^{N}}\frac{|x|^{\alpha p} |y|^{\alpha p}|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy\\ \qquad \qquad \qquad \qquad \qquad \qquad = a(x)u^{-q}+\lambda |x|^{\beta}u^{r-1}\,\, \ & {\rm in}\ \Omega,\\ u=0\ \ \ & {\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} $$ where $s\in(0,1)$, $\alpha \in\mathbb{R}$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $1 < p < N/s$, $0 < q < 1, p < r < p^{\ast}_s$, $\lambda>0$, $p_s^{\ast}=Np/(N-sp)$ is the critical Sobolev exponent. By the Nehari manifold method and the fractional Caffarelli--Kohn--Nirenberg inequalities, we show that there exists $\lambda^* > 0$ such that the problem admits at least two positive solutions as $\lambda\in(0,\lambda^*)$.

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: $$ \begin{cases} \Delta_p u= v^m| \nabla u |^\alpha & \text{in}\quad \Omega\\ \Delta_p v= v^\beta| \nabla u |^q & \text{in}\quad \Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^N\ (N\geq 1)$ is either equal to $ \mathbb{R}^N $ or equal to a ball $B_R$ centered at the origin and having radius $R>0$, $1 < p < \infty$, $m,q > 0$, $\alpha\geq 0$, $0\leq \beta\leq m$ and $\delta:=(p-1-\alpha)(p-1-\beta)-qm \neq 0$. Our aim is to establish the asymptotics of the blowing-up radial solutions to the above system. Precisely, we provide the accurate asymptotic behavior at the boundary for such blowing-up radial solutions. For that, we prove a strong maximal principle for the problem of independent interest and study an auxiliary asymptotically autonomous system in $\mathbb{R}^3$.

We consider the convergence rate in periodic homogenization of higher order parabolic systems with time-dependent coefficients. The sharp $ O(\varepsilon) $-order scaling invariant convergence rate in the space $L^{2}(0,T;W^{m-1,p_{0}}(\Omega))$, $p_{0}=\frac{2d}{d-1}$, is derived by the duality argument. This largely improves the corresponding result by Niu and Xu in Discrete Contin. Dynam. Systems. Series A 38(8): 4203--4229 (2018).

New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation $$ \dot{x}(t) + \sum_{k=1}^m a_k(t)x(h_k(t)) + \int_{g(t)}^t K(t,s) x(s)ds=0, $$ where $h_k(t)\leq t$, $g(t)\leq t$, $a_k(\cdot)$ and the kernel $K(\cdot,\cdot)$ are oscillatory and, generally, discontinuous functions. The proofs are based on establishing boundedness of solutions and later using the exponential dichotomy for linear equations stating that either the homogeneous equation is exponentially stable or a non-homogeneous equation has an unbounded solution for some bounded right-hand side. Explicit tests are applied to models of population dynamics, such as controlled Hutchinson and Mackey-Glass equations. The results are illustrated with numerical examples, and connection to known tests is discussed.

Linear differential equations with variable coefficients and Prabhakar-type operators featuring Mittag-Leffler kernels are solved. In each case, the unique solution is constructed explicitly as a convergent infinite series involving compositions of Prabhakar fractional integrals. We also extend these results to Prabhakar operators with respect to functions. As an important illustrative example, we consider the case of constant coefficients, and give the solutions in a more closed form by using multivariate Mittag-Leffler functions.

This paper is concerned with an initial-boundary value problem for the two-dimensional axisymmetric rotating shallow water equations. The Dirichlet boundary conditions are imposed only on the radial velocity, while no boundary condition is imposed on the height of the fluid or the angular velocity. A series of a priori estimates for the solution of the approximate linear problem are derived and the strong convergence of the approximate solution sequences is verified. Consequently, we establish the local well-posedness in time of strong solutions for the initial-boundary value problem of the model.

In this work, we introduce a diffusive stochastic size-struc\-tured population model. After formulating the problem in an abstract framework, we derive mild solution to the given model. We show the existence and uniqueness of the mild solution to the given model in a Hilbert space. We use the Burkholder-Davis-Gundy's lemma to prove the uniqueness of the mild solution. We also show that the system is quasi invariant in the second moment and also mean square dissipative.