In this paper, the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the fractional and pseudo-differential operators is obtained.

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows

\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[   \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}

Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong  damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.

Consider (M,g) as a complete, simply connected Riemannian manifold. The aim of this paper is to provide various geometric estimates in different cases for the first eigenvalue of (p,q)-elliptic quasilinear system in both Dirichlet and Neumann conditions on Riemannian manifold. In some cases we add integral curvature condition and maybe we prove some theorems under other conditions.

The paper proves the nonexistence of the solution for the following Cauchy problem
\begin{align*}\begin{cases}u_{t} ={\rm div}\left(\left|\nabla u^{m} \right|^{p-2} \nabla u^{m} \right)-\lambda \; u^{q},&\qquad \left(x,t\right)\in S_{T} ={\mathbb{R}}^N \times \left(0,T\right), \\u\left(x,\; 0\right)=\delta \left(x\right), &\qquad x\in {\mathbb{R}}^,\end{cases}\end{align*}
under some conditions on \textit{m,p,q},$\lambda$, where $\delta $ is Dirac function.

This paper investigates a reaction-advection-diffusion equation with double free boundaries. The stationary solution of the system is studied by phase plane analysis. Then, the scale logarithm change sequence method is introduced to show the exact heteroclinic of the system with corresponding parameters. Moreover, a complete description of the types of traveling wave solutions is given with different advection term coefficients.