Starting from the model proposed in [15], we study a stationary flow of fluid, governed by a linear Darcy's law, through an inhomogeneous porous medium with leaky boundary conditions. We establish properties of solutions and regularity of the free boundary. Finally, we prove a comparison result from which we deduce some uniqueness results.

Benjamin recently put forward a model equation for the evolution of waves on the interface of a two-layer system of fluids in which surface tension effects are not negligible. It is our purpose here to investigate the solitary-wave solutions of Benjamin's model. For a class of equations that includes Benjamin's model featuring conflicting contributions to dispersion from dynamic effects on the interface and surface tension, we establish existence of travelling-wave solutions. Using the recently developed theory of Li and Bona, we are also able to determine rigorously the spatial asymptotics of these solutions.

Properties of solutions of a bistable nonlinear convolution equation in higher space dimensions are studied. The nonlinearity is the derivative of a double well function. The theory of traveling waves for this equation was given in a previous publication ([4]). Here we consider spreading phenomena for state regions, in some cases by means of the motion of domain walls, which are modeled by internal layers. These phenomena are analogous to those known for the bistable nonlinear diffusion equation, and in particular, for the Allen--Cahn equation, which is a model for the motion of some grain boundaries in solid materials. Cases when the two wells have unequal depth are considered, as well as when they have equal depth. In the latter case a motion-by-curvature law is derived formally in two space dimensions.

We consider the class of equations $$ -\Delta u={A\over {|x|^{\alpha}}}u+u^{\theta} \qquad\qquad x\in\Bbb R^n\setminus\{0\} $$ where $A\in \Bbb R$, $\alpha>0$ and $\theta>1$. Depending on the values of the three parameters involved, we obtain both results of existence and nonexistence of positive solutions by combining the moving planes and the moving spheres methods through the Kelvin's inversion map and classical arguments on ODE's.

We prove an abstract existence and uniqueness theorem in the little for an inverse problem related to the combustion of a material with memory. Then, we give an application to a specific case.

In a previous work, we have shown the existence of weak solutions for a shallow-water problem (or compressible two-dimensional Navier-Stokes problem) in a depth-mean velocity formulation. Some results have been proved by Kazhikov in the case of domain equal to $\mathbb{R}^N$ and linearized momentum equation, which allows him to look for a velocity of the form $u=\nabla p$. We present some smoothness and uniqueness results whatever a smooth domain and with the complete momentum equation and boundary conditions on $u \cdot n$ and $\mathcal{u}$.