For a nonlinear hyperbolic equation defined in a thin domain we prove the existence of a periodic solution with respect to time both in the non-autonomous and autonomous cases. The methods employed are a combination of those developed by J. K. Hale and G. Raugel and the theory of the topological degree.

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution in the case of onset of singularities, which can be interpreted as a local collapse of the medium, in the general...

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 $\alpha \ge 0$.

In this paper we prove the existence of a weak solution for a given semilinear singular real hyperbolic system.

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.