Marina Ghisi

An Example of Global Classical Solution for the Perona-Malik Equation

We consider the Perona-Malik equation in an open set Ω ⊆ ℝ n , with initial and Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C 1 solution if the initial condition u 0 is transcritical, namely when |∇u 0(x)| −1 is a sign changing function in Ω. In this paper we show that this result cannot be extended to higher dimension. We show indeed that for n ≥ 2 the problem admits radial solutions of class C 2, 1 with a transcritical initial condition.

Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations,

(t) + Au(t) + f(u(t)) = 0, (1.1)where H is a real Hilbert space, A is a nonnegative self-adjoint linear operator on Hwith dense domain, and f is a nonlinearity tangent to 0 at the origin.When f ≡ 0, then for rather general classes of strongly positive operators A it isknown that all solutions decay to 0 (as t → +∞) exponentially in the energy norm.Therefore, by perturbation theory it is reasonable to expect that also all solutions of(1.1) which decay to 0 have an exponential decay rate. The situation is different whenA has a non-trivial kernel. In this case solutions tend to 0 if f fulfils suitable signconditions, but we do not expect all solutions to have an exponential decay rate. Letus consider for example the hyperbolic equationu

Stability of simple modes of the Kirchhoff equation

It is well known that the Kirchhoff equation admits infinitely many simple modes, i.e. time-periodic solutions with only one Fourier component in the space variable(s). We prove that these simple modes are stable provided that their energy is small enough. Here stable means orbitally stable as solutions of the two-mode system obtained considering initial data with two Fourier components.

Higher order Glaeser inequalities and optimal regularity of roots of real functions

We prove a higher order generalization of Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself, and the Holder constant of its k-th derivative. We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k+alpha)-th root of a function of class C^{k} whose derivative of order k is alpha-Holder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent. Some examples show that our results are optimal.

Kirchhoff equations from quasi-analytic to spectral-gap data

m(σ) > 0. (1.4)We also assume that m is locally Lipschitz continuous. We never assume that theoperator is coercive or that its inverse is compact.We refer to the survey [9] and to the references quoted therein for more details onthis equation and its history. Here we just recall that, under our assumptions on m(σ),problem (1.1), (1.2) has a local solution for all initial data (u

On the evolution of subcritical regions for the Perona-Malik equation

The Perona-Malik equation is a celebrated example of forward-backward parabolic equation. The forward behavior takes place in the so-called subcritical region, in which the gradient of the solution is smaller than a fixed threshold. In this paper we show that this subcritical region evolves in a different way in the following three cases: dimension one, radial solutions in dimension greater than one, general solutions in dimension greater than one. In the first case subcritical regions increase, but there is no estimate on the expansion rate. In the second case they expand with a positive rate and always spread over the whole domain after a finite time, depending only on the (outer) radius of the domain. As a by-product, we obtain a non-existence result for global-in-time classical radial solutions with large enough gradient. In the third case we show an example where subcritical regions do not expand. Our proofs exploit comparison principles for suitable degenerate and non-smooth free boundary problems. Mathematics Subject Classification 2000 (MSC2000): 35K55, 35K65, 35R35.

The Remarkable Effectiveness of Time-Dependent Damping Terms for Second Order Evolution Equations

We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover that constant coefficients do not achieve the goal and neither do time-dependent coefficients, if they are uniformly too big. On the contrary, pulsating coefficients which alternate big and small values in a suitable way prove to be more effective. Our theory applies to ordinary differential equations, systems of ordinary differential equations, and partial differential equations of hyperbolic type.

A concrete realization of the slow-fast alternative for a semilinear heat equation with homogeneous Neumann boundary conditions

Abstract We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infinity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.

Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations

Abstract We consider a second-order equation with a linear “elastic” part and a nonlinear damping term depending on a power of the norm of the velocity. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take into account the decay rate and bound their energy away from zero. We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fourier components are asymptotic to solutions of the linearized equation without damping and exhibit some sort of equipartition of the total energy among the components. Solutions with infinitely many Fourier components tend to zero weakly but not strongly. We show also that the limit of the energy of the solutions depends only on the number of their Fourier components. The proof of our results is inspired by the analysis of a simplified model, which we devise through an averaging procedure, and whose solutions exhibit the same asymptotic properties as the solutions to the original equation.

On the Parabolic Regime of a Hyperbolic Equation with Weak Dissipation: The Coercive Case

We consider a family of Kirchhoff equations with a small parameter e in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 as t→+∞.