We study some hyperbolic partial integro-differential systems in domains with time dependent cracks. In particular, we give conditions on the cracks which imply the uniqueness of the solution with prescribed initial-boundary conditions, and its continuous dependence on the cracks.

In this paper, we are interested in a new Kirchhoff-Schrödinger-Poisson type system on the Heisenberg group of the form: \begin{equation*} \begin{cases} \displaystyle - \Big ( a-b\int_{\Omega}|\nabla_{H}u|^{2}dx \Big ) \Delta_{H}u +\mu\phi u=\lambda|u|^{q-2}u, & \mbox{in}\quad \Omega,\\ -\Delta_{H}\phi=u^2 & \mbox{in}\quad \Omega,\\ u=\phi=0 & \mbox{on}\quad \partial\Omega, \end{cases} \end{equation*} where $a, b > 0$, $\Delta_H$ is the Kohn-Laplacian on the first Heisenberg group $\mathbb{H}^1$, and $\Omega\subset \mathbb{H}^1$ is a smooth bounded domain, $\mu, \lambda > 0$ are some real parameters and $2 < q < 4$. According to the different ranges of these parameters, we discuss the existence and multiplicity of solutions for the above equation using variational methods under suitable conditions. Our result is new even in the Euclidean case

We consider the inhomogeneous Neumann boundary value problem for nonlinear Schrödinger equations $$ \begin{cases} i\partial _{t} u+\frac{1}{2}\Delta u=\lambda | u | ^{p-1}u, & t > 0,\ x= ( x^{\prime },x_{n} ) \in \mathbb{R}_{+}^{n}, \\ u(0,x)=u_{0}(x), & x\in \mathbb{R}_{+}^{n}, \\ \partial _{x_{n}}u ( t,x^{\prime },0 ) =h ( t,x^{\prime } ) , & t > 0,\text{ }x^{\prime }\in \mathbb{R}^{n-1}, \end{cases} $$ where $1+\frac{4}{n+2} < p < 1+\frac{4}{n-2}$. We present some results on asymptotic behavior in time of small solutions to the integral equation associated with the original problem.

In this paper, we consider the Cauchy problem for one-dimensional damped wave equations with variable coefficients $a(x)$ having power type nonlinearity of order $p$. We discuss the global existence of solutions for small initial data and investigate the relation between the range of $a(x)$ and the order $p$.

We investigate the best constants for the regional fractional $p$-Poincaré inequality and the fractional $p$-Poincaré inequality in cylindrical domains. For the special case $p=2$, the result was already known due to Chowdhury-Csató-Roy-Sk [Study of fractional Poincaré inequalities on unbounded domains, Discrete Contin. Dyn. Syst., 41(6), 2021]. We addressed the asymptotic behavior of the first eigenvalue of the nonlocal Dirichlet $p$-Laplacian eigenvalue problem when the domain is becoming unbounded in several directions.