In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $$u_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta u(t,x)+\delta |u_t{(t,x)|}^{p-1}{u}_t(t,x)={\mu \left|u(t,x)\right|}^{q-1}u(t,x),x\in \Omega ,t\ge 0,v_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta v(t,x)+\delta |v_t{(t,x)|}^{p-1}{v}_t(t,x)={\mu \left|v(t,x)\right|}^{q-1}v(t,x),x\in \Omega ,t\ge 0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in \Omega ,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),x\in \Omega ,u{{|}_{{}_{\partial \Omega }}=v|}_{{}_{\partial \Omega }}=0$$ where $q>1$, $p\ge 1$, $\delta >0$, $\alpha >0$, $\beta \ge 0$, $\mu \in ℝ$ and $\Delta$ is the Laplacian in ${ℝ}^N$.

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing...

2000 Mathematics Subject Classification: 35L15, Secondary 35L30.In this paper we prove that for non effectively hyperbolic operators with smooth double characteristics with the Hamilton map exhibiting a Jordan block of size 4 on the double characteristic manifold the Cauchy problem is well posed in the Gevrey 6 class if the strict Ivrii-Petkov-Hörmander condition is satisfied.

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 study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i\theta }$ for all $\theta \in ℝ$. We investigate about the persistence of the decorrelation between the...

We consider the coupling between three-dimensional (3D) and one-dimensional (1D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolic system of partial differential equations. The 3D model consists of the Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulation for the Navier-Stokes equations is adopted to have suitable boundary conditions for the...