In this article, we study the followingsemilinear wave equation\begin{equation*}u_{tt}-\Delta u+f ( u ) =0\end{equation*}which has the following nonlinear boundary condition\begin{equation*}\frac{\partial u}{\partial \nu }+g ( u_{t} ( x,t ) )= | u | ^{p ( x ) -2}u.\end{equation*}We also consider the uniform energy decay rate for solutions of thesemilinear wave equation, and our main results focus on the interactionbetween highly nonlinear sources and damping about the system.

Hyperbolic problems can at times be solved employing symbolicarguments. This is especially true for the construction of forward(and backward) fundamental solutions. We formulate a correspondingabstract scheme and illustrate its practicality by a number ofinstructive examples.

The Benjamin-Bona-Mahony (BBM) equation has proven to be a good approximation for the unidirectional propagation of small amplitude long wavesin a channel where the crosswise variation can be safely ignored. The Benjamin-Bona-Mahony-Kadomtsev-Petviashvili (BBM-KP) equation is the regularized version of the Kadomtsev-Petvia-shvili equation which arises in various modeling scenarios corresponding to nonlinear dispersive waves that propagateprincipally along the $x$-axis with weak dispersive effects undergone in the direction parallel to the$y$-axis and normal to the primary direction of propagation. There is much literature on mathematical studies regarding these well known equations, however the relationship between the solutions of their underlying pure initial value problems is not fully understood. In this work, it is shown that the solution of the Cauchy problem for the BBM-KP equation convergesto the solution of the Cauchy problem for the BBM equation in a suitable function space, provided thatthe initial data for both equations are close as the transverse variable $y \rightarrow \pm \infty$.

We study the Allen-Cahn equation with respect to a transport term in a bounded domain. We prove that the limit interface is the mean curvature flow with the transport term, given the condition that the energy is uniformly bounded with respect to time. Using this result, we show the existence of the mean curvature flow with a gradient vector field as the transport term.

The aim of this paper is to establish existence and uniqueness of bounded viscosity solutions of fully nonlinear second-order elliptic equations in domains like slabs and cylinders. A global Hölder regularity is also proved. The issue is motivated by different applications based on the First Passage Time (FPT). In particular, the equations modelize the expected value of the FPT for a time-homogeneous stochastic process. Explicit solutions have been computed for the Wiener process, and the influence of a drift term on the mean first hitting time has been also discussed.

In this paper, we study the $p$-Poisson equation with Robin boundary conditions, where the Robin parameter is a function.By means of some weighted isoperimetric inequalities, we provide various sharp bounds for the solutions to the problems under consideration. We also derive a Faber-Krahn-type inequality.