The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation
The preventive effect of the convection and of the diffusion in the blow-up phenomenon for parabolic equations
The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation
We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.
The quasilinear parabolic kirchhoff equation
In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.
The role of symmetry and separation in surface evolution and curve shortening.
The speed of propagation for KPP type problems. I: Periodic framework
This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article [7]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain,...
The steepest descent method for forward-backward SDEs.
The Stefan problem in heterogeneous media
The thermistor obstacle problem with periodic data
The Total Variation of Solutions of Parabolic Differential Equations and a Maximum Principle in Unbounded Domains.
The waiting time property for parabolic problems trough the nondiffusion of support for the stationary problems.
In this note we study the waiting time phenomenon for local solutions of the nonlinear diffusion equation through its connection with the nondiffusion of the support property for local solutions of the family of discretized elliptic problems. We show that, under a suitable growth condition on the initial datum near the boundary of its support, a finite part of the family of solutions of the discretized problem maintain unchanged its support.
Time periodic solutions to a nonhomogeneous Dirichlet periodic problem.
Time-delay regularization of anisotropic diffusion and image processing
We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell 12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell 14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.
Time-delay regularization of anisotropic diffusion and image processing
We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.
Time-dependent invariant regions for parabolic systems related to one- dimensional nonlinear elasticity
A parabolic system arisng as a viscosity regularization of the quasilinear one-dimensional telegraph equation is considered. The existence of - a priori estimates, independent of viscosity, is shown. The results are achieved by means of generalized invariant regions.
Time-optimal control of nonlinear parabolic systems with constrained derivative of control, existence theorem
Time-periodic solutions of quasilinear parabolic differential equations II. Oblique derivative boundary conditions
We study boundary value problems for quasilinear parabolic equations when the initial condition is replaced by periodicity in the time variable. Our approach is to relate the theory of such problems to the classical theory for initial-boundary value problems. In the process, we generalize many previously known results.
Traveling wave solutions of the heat flow of director fields
Traveling waves in a one-dimensional heterogeneous medium
Travelling wave for absorption-convection-diffusion equations.