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.
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...
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.
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.
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...
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...
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...
We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is . The problem is posed in with nonnegative initial data that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and regularity of such weak solutions. Finally, we extend the existence...
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