Let $\Omega$ be a bounded domain in ${ℝ}^n$ with a smooth boundary $\Gamma$. In this work we study the existence of solutions for the following boundary value problem: $$\frac{{\partial }^{2}y}{\partial t^2}-M\left({\int }_{\Omega }{\left|\nabla y\right|}^2\mathrm{d}x\right)\Delta y-\frac{\partial }{\partial t}\Delta y=f\left(y\right)\text{in}Q=\Omega \times (0,\infty ),.1y=0\text{in}{\Sigma }_1={\Gamma }_1\times (0,\infty ),M\left({\int }_{\Omega }{\left|\nabla y\right|}^2\mathrm{d}x\right)\frac{\partial y}{\partial \nu }+\frac{\partial }{\partial t}\left(\frac{\partial y}{\partial \nu }\right)=g\text{in}{\Sigma }_0={\Gamma }_0\times (0,\infty ),y\left(0\right)=y_0,\frac{\partial y}{\partial t}\left(0\right)=y_1\text{in}\Omega ,\left(1\right)$$ where $M$ is a $C^1$-function such that $M\left(\lambda \right)\ge {\lambda }_0>0$ for every $\lambda \ge 0$ and $f\left(y\right)={\left|y\right|}^{\alpha }y$ for $\alpha \ge 0$.

Following the ideas of D. Serre and J. Shearer (1993), we prove in this paper the existence of a weak solution of the Cauchy problem for a given second order quasilinear hyperbolic equation.

We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.

B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation $z_{xy}=f(x,y,z,z_{xy})$ on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation $z_{xy}=f(x,y,z,z_x,z_{xy})$ using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].

We consider a class of nonconvex and nonclosed hyperbolic differential inclusions and we prove the arcwise connectedness of the solution set.

In this paper, we first investigate the existence and uniqueness of solution for the Darboux problem with modified argument on both bounded and unbounded domains. Then, we derive different types of the Ulam stability for the proposed problem on these domains. Finally, we present some illustrative examples to support our results.

Optimization problem for a structural acoustic model with controls governed by unbounded operators on the state space is considered. This type of controls arises naturally in the context of "smart material technology". The main result of the paper provides an optimal synthesis and solvability of associated nonstandard Riccati equations. It is shown that in spite of the unboundedness of control operators, the resulting gain operators (feedbacks) are bounded on the state space. This allows to provide...