Reverse smoothing effects, fine asymptotics, and Harnack inequalities for fast diffusion equations.
Sets of admissible initial data for porous-medium equations with absorption.
Solutions of nonlinear parabolic equations without growth restrictions on the data.
A mixed semilinear parabolic problem from combustion theory.
The nonlinearly damped oscillator
We study the large-time behaviour of the nonlinear oscillator where and is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case with real, . We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.
Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type
We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form with suitable and . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection...
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 Nonlinearly Damped Oscillator
We study the large-time behaviour of the nonlinear oscillator where and is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case with real, . We characterize the existence and behaviour of fast orbits, , orbits that stop in finite time.
Singularities of Elliptic Equations with an Exponential Nonlinearity.
Removable Singularities of Some Strongly Nonlinear Elliptic Equations.
Non-uniqueness of solutions of nonlinear heat equations of fast diffusion type
Asymptotic analysis of the -laplacian flow in an exterior domain
Asymptotic behaviour of the porous media equation in an exterior domain
The dipole solution for the porous medium equation in several space dimensions
Fundamental solutions and asymptotic behaviour for the p-Laplacian equation.
We establish the uniqueness of fundamental solutions to the p-Laplacian equation ut = div (|Du|p-2 Du), p > 2, defined for x ∈ RN, 0 < t < T. We derive from this result the asymptotic behavoir of nonnegative solutions with finite mass, i.e., such that u(*,t) ∈ L1(RN). Our methods also apply to the porous medium equation ut...
The pressure equation in the fast diffusion range.
Blow-up solutions of some nonlinear elliptic problems.
A symmetrization result for elliptic equations with lower-order terms
A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media
In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution of an elliptic equation, that we write , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to...
The relation between the porous medium and the eikonal equations in several space dimensions.
We study the relation between the porous medium equation ut = Δ(um), m > 1, and the eikonal equation vt = |Dv|2. Under quite general assumtions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as m↓1 to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions...
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