arXiv subjects

Massimo Gobbino

Publications and source records attributed to Massimo Gobbino.

At least 19 recordsLinked to original sources

arXiv2026

Billiards, Refraction, and Blow-Up for Kirchhoff Equations

We start from a simple problem in geometric optics. A particle moves between two homothetic ellipses, with refraction at the inner interface and reflection at the outer one. In a suitable nearly circular regime, the small geometric anisotropy of the ellipses is amplified by the strong refraction, and the corresponding return map develops a transverse heteroclinic connection between the two axial motions. We turn this geometric mechanism into a construction for a two-mode Kirchhoff system while keeping the standard quadratic elastic variable throughout. The singular optical model is first reduced to an explicit kick--drift map, for which the heteroclinic connection is obtained by a contraction argument. The same argument also gives exponential convergence along the two tails and transversality. We then show that the connection persists for the genuine elliptic billiard and, subsequently, through a smooth regularization of the reflecting and refracting interfaces. Finally, a small positive background is added to the Kirchhoff coefficient, making it uniformly positive without destroying the hyperbolicity of the axial modes or the transverse connection. This produces a smooth and uniformly positive Kirchhoff coefficient admitting a transverse heteroclinic orbit between two simple modes. Combined with the road-map theorem developed in our previous work, the construction yields a finite-time blow-up example for a forced abstract Kirchhoff equation with a regular forcing term.

math.AP↗

arXiv2026

Fifty Years of Wave Equations with Time-Dependent Propagation Speeds: A Variational Perspective and New Frontiers

We study abstract wave equations with time-dependent propagation speed under strict hyperbolicity. Our starting point is the observation that a substantial part of the classical energy theory, developed through different constructions, can be organized around a common variational mechanism. Approximate energies naturally lead to regularity--fidelity problems in which a smooth approximation of the reciprocal propagation speed is chosen by balancing derivative size against distance from the original coefficient. We combine this approximation mechanism with higher-order energy corrections and obtain a hierarchy of variational problems together with a cascade of almost conserved energies of arbitrary order. The resulting variational problems are then studied independently of the evolution equation. A qualitative distinction emerges between the first and higher orders: for arbitrary prescribed growth, all higher orders generate the same classes, while the first-order problem may behave differently. We also establish connections with fractional and generalized fractional Sobolev regularity. This framework provides a unified interpretation of several classical sufficient conditions for energy control and derivative-loss estimates. At the same time, it identifies a natural limitation of the traditional approach, namely the use of absolute integrability of energy errors. Once this limitation is recognized, a different regime becomes accessible: by retaining their oscillatory structure and exploiting cancellations, we construct strictly hyperbolic propagation speeds beyond the regularity thresholds detected by the variational theory, for which uniform energy estimates and hence no derivative loss nevertheless hold. Thus the variational viewpoint reveals an unexpected common structure behind a substantial part of the classical theory and indicates a genuinely different regime beyond it.

math.AP↗

arXiv2025

Symmetry breaking for local minimizers of a free discontinuity problem

We study a functional defined on the class of piecewise constant functions, combining a jump penalization, which discourages discontinuities, with a fidelity term that penalizes deviations from a given linear function, called the forcing term. In one dimension, it is not difficult to see that local minimizers form staircases that approximate the forcing term. Here we show that in two dimensions symmetry breaking occurs, leading to the emergence of exotic minimizers whose level sets are not simple stripes with boundaries orthogonal to the gradient of the forcing term. The proof relies on a suitable adaptation of the calibration method for free discontinuity problems; as a side benefit, our version requires less regularity than the classical one.

math.AP↗

arXiv2025

The model example of wave equation with oscillating scale-invariant damping

We analyze a simple example of wave equation with a time-dependent damping term, whose coefficient decays at infinity at the scale-invariant rate and includes an oscillatory component that is integrable but not absolutely integrable. We show that the oscillations in the damping coefficient induce a resonance effect with a fundamental solution of the elastic term, altering the energy decay rate of solutions. In particular, some solutions exhibit slower decay compared to the case without the oscillatory component. Our proof relies on Fourier analysis and a representation of solutions in polar coordinates, reducing the problem to a detailed study of the asymptotic behavior of solutions to a family of ordinary differential equations and suitable oscillatory integrals.

math.AP↗

arXiv2024

Generalized energy conservation for linear wave equations with time-dependent propagation speed

We consider a wave equation with a time-dependent propagation speed, whose potential oscillations are controlled through bounds on its first and second derivatives and by limiting the integral of the difference with a fixed constant. We investigate when the wave equation exhibits generalized energy conservation (GEC), meaning that the energy of all solutions remains bounded for all times by a multiple of the initial energy. When GEC is not satisfied, we provide upper bounds for the growth of the energy. These upper bounds are derived by analyzing the growth of the Fourier components of the solution. Depending on the frequency and the time interval, different energy inequalities are employed to fully exploit our assumptions on the propagation speed. Finally, we present counterexamples that demonstrate the optimality of our upper bound estimates.

math.AP↗

arXiv2024

Multi-scale analysis of minimizers for a second order regularization of the Perona-Malik functional

We investigate the asymptotic behavior of minimizers for the singularly perturbed Perona-Malik functional in one dimension. In a previous study, we have shown that blow-ups of these minimizers at a suitable scale converge to staircase-like piecewise constant functions. Building upon these findings, we delve into finer scales, revealing that both the vertical and horizontal regions of the staircase steps display cubic polynomial behavior after appropriate rescaling. Our analysis hinges on identifying the dominant terms of the functional within each regime, elucidating the mechanisms driving the observed asymptotic behavior.

math.AP↗

arXiv2024

Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms

We consider a family of non-local and non-convex functionals, and we prove that their Gamma-liminf is bounded from below by a positive multiple of the Sobolev norm or the total variation. As a by-product, we answer some open questions concerning the limiting behavior of these functionals. The proof relies on the analysis of a discretized version of these functionals.

math.FA↗

arXiv2024

Resonance effects for linear wave equations with scale invariant oscillating damping

We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation term is smooth, and we investigate the effect of its oscillations on the decay rate of solutions. We prove a decay estimate that holds true regardless of the oscillations. Then we show that oscillations that are too fast have no effect on the decay rate, while oscillations that are in resonance with one of the frequencies of the elastic part can alter the decay rate. In the proof we first reduce ourselves to estimating the decay of solutions to a family of ordinary differential equations, then by using polar coordinates we obtain explicit formulae for the energy decay of these solutions, so that in the end the problem is reduced to the analysis of the asymptotic behavior of suitable oscillating integrals.

math.AP↗

arXiv2023

Monotonicity properties of limits of solutions to the semi-discrete scheme for the Perona-Malik equation

We consider generalized solutions of the Perona-Malik equation in dimension one, defined as all possible limits of solutions to the semi-discrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times, and in particular many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub/supersolutions. The monotonicity results are proved for a more general class of evolution curves, that we call $uv$-evolutions.

math.AP↗

arXiv2023

Almost global existence for Kirchhoff equations around global solutions

It is well-known that the life span of solutions to Kirchhoff equations tends to infinity when initial data tend to zero. These results are usually referred to as almost global existence, at least in a neighborhood of the null solution. Here we extend this result by showing that the life span of solutions is lower semicontinuous, and in particular it tends to infinity whenever initial data tend to some limiting datum that originates a global solution. We also provide an estimate from below for the life span of solutions when initial data are close to some of the classes of data for which global existence is known, namely data with finitely many Fourier modes, analytic data and quasi-analytic data.

math.AP↗

arXiv2022

A road map to the blow-up for a Kirchhoff equation with external force

It is well-known that the classical hyperbolic Kirchhoff equation admits infinitely many simple modes, namely time-periodic solutions with only one Fourier component in the space variables. In this paper we assume that, for a suitable choice of the nonlinearity, there exists a heteroclinic connection between two simple modes with different frequencies. Under this assumption, we cook up a forced Kirchhoff equation that admits a solution that blows-up in finite time, despite the regularity and boundedness of the forcing term. The forcing term can be chosen with the maximal regularity that prevents the application of the classical global existence results in analytic and quasi-analytic classes.

math.AP↗

arXiv2022

Global solutions to the Kirchhoff equation with spectral gap data in the energy space

We prove that the classical hyperbolic Kirchhoff equation admits global-in-time solutions for some classes of initial data in the energy space. We also show that there are enough such solutions so that every initial datum in the energy space is the sum of two initial data for which a global-in-time solution exists. The proof relies on the notion of spectral gap data, namely initial data whose components vanish for large intervals of frequencies. We do not pass through the linearized equation, because it is not well-posed at this low level of regularity.

math.AP↗

arXiv2022

A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional

We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient. We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale. Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models.

math.AP↗

arXiv2022

On the characterization of constant functions through nonlocal functionals

We address a classical open question by H.Brezis and R.Ignat concerning the characterization of constant functions through double integrals that involve difference quotients. Our first result is a counterexample to the question in its full generality. This counterexample requires the construction of a function whose difference quotients avoid a sequence of intervals with endpoints that diverge to infinity. Our second result is a positive answer to the question when restricted either to functions that are bounded and approximately differentiable almost everywhere, or to functions with bounded variation. We also present some related open problems that are motivated by our positive and negative results.

math.FA↗

arXiv2021

Optimal derivative loss for abstract wave equations

We consider an abstract wave equation with a propagation speed that depends only on time. We assume that the propagation speed is differentiable for positive times, continuous up to the origin, but with first derivative that is potentially singular at the origin. We examine the derivative loss of solutions, and in particular we investigate which conditions on the modulus of continuity and on the behavior of the derivative in the origin yield, respectively, no derivative loss, an arbitrarily small derivative loss, a finite derivative loss, or an infinite derivative loss. As expected, we obtain that stronger assumptions on the modulus of continuity can compensate weaker assumptions on the growth of the derivative, and viceversa. Suitable counterexamples show that our results are sharp. We prove indeed that, for every set of conditions, the class of propagation speeds that satisfy the given conditions, and for which the corresponding equation exhibits a derivative loss as large as possible, is nonempty and actually also residual in the sense of Baire category.

math.AP↗

arXiv2020

Residual pathologies

Several counterexamples in analysis show the existence of some special object with some sort of pathological behavior. We present three different examples where the pathological behavior is not an isolated exception, but it is the "typical" behavior of the "generic" object in a suitable class, where here generic means residual in the sense of Baire category. The first example is the revisitation of a classical result concerning approximate differentiation. The second example is the derivative loss for solutions to linear wave equations with time-dependent Holder continuous propagation speed. The third result is the derivative loss for solutions to transport equations with non-Lipschitz velocity field.

math.FA↗

arXiv2020

Finite vs infinite derivative loss for abstract wave equations with singular time-dependent propagation speed

We consider an abstract wave equation with a propagation speed that depends only on time. We investigate well-posedness results with finite derivative loss in the case where the propagation speed is smooth for positive times, but potentially singular at the initial time. We prove that solutions exhibit a finite derivative loss under a family of conditions that involve the blow up rate of the first and second derivative of the propagation speed, in the spirit that the weaker is the requirement on the first derivative, the stronger is the requirement on the second derivative. Our family of conditions interpolates between the two limit cases that were already known in the literature. We also provide the counterexamples that show that, as soon as our conditions fail, solutions can exhibit an infinite derivative loss. The existence of such pathologies was an open problem even in the two extreme cases.

math.AP↗

arXiv2020

Sharp ultimate velocity bounds for the general solution of some linear second order evolution equation with damping and bounded forcing

We consider a class of linear second order differential equations with damping and external force. We investigate the link between a uniform bound on the forcing term and the corresponding ultimate bound on the velocity of solutions, and we study the dependence of that bound on the damping and on the "elastic force". We prove three results. First of all, in a rather general setting we show that different notions of bound are actually equivalent. Then we compute the optimal constants in the scalar case. Finally, we extend the results of the scalar case to abstract dissipative wave-type equations in Hilbert spaces. In that setting we obtain rather sharp estimates that are quite different from the scalar case, in both finite and infinite dimensional frameworks. The abstract theory applies, in particular, to dissipative wave, plate and beam equations.

math.AP↗