This work deals with the study of some stratigraphic models for the formation of geological basins under a maximal erosion rate constrain. It leads to introduce differential inclusions of degenerated hyperbolic-parabolic type $0\in {\partial }_{t}u-div\left\{H({\partial }_{t}u+E)\nabla u\right\}$, where H is the maximal monotonous graph of the Heaviside function and E is a given non-negative function. Firstly, we present the new and realistic models and an original mathematical formulation, taking into account the weather-limited rate constraint in the conservation...

De Pablo et al. [Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 513-530] considered a nonlinear boundary value problem for a porous medium equation with a convection term, and they classified exponents of nonlinearities which lead either to the global-in-time existence of solutions or to a blow-up of solutions. In their analysis they left open the case of a certain critical range of exponents. The purpose of this note is to fill this gap.

The goal of this note is to present the results of the references [5] and [4]. We study the null controllability of the parabolic equations associated with the Grushin-type operator ${\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_y^2$ ($\gamma \>0$) in the rectangle $(x,y)\in (-1,1)\times (0,1)$ or with the Kolmogorov-type operator $v^{\gamma }{\partial }_{x}f+{\partial }_v^{2}f$ ($\gamma \in \{1,2\}$) in the rectangle $(x,v)\in 𝕋\times (-1,1)$, under an additive control supported in an open subset $\omega$ of the space domain.We prove that the Grushin-type equation is null controllable in any positive time for $\gamma \<1$ and that there is no time for which it is null controllable for $\gamma \>1$....

We study the null controllability of the parabolic equation associated with the Grushin-type operator $A={\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_{\gamma }^2,(\gamma >0)$, in the rectangle $\Omega =(-1,1)\times (0,1)$, under an additive control supported in an open subset $\omega$ of $\Omega$. We prove that the equation is null controllable in any positive time for $\gamma <1$ and that there is no time for which it is null controllable for $\gamma >1$. In the transition regime $\gamma =1$ and when $\omega$ is a strip $\omega =(a,b)\times (0,1)(0<a,b\le 1)$), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular...

Stability and convergence of the linear semi-implicit discrete duality finite volume (DDFV) numerical scheme in 2D for the solution of the regularized curvature driven level set equation is proved. Numerical experiments concerning comparison with exact solution and image filtering problem using proposed scheme are included.

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern’s iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant),...