arXiv subjects

Marina Ghisi

Publications and source records attributed to Marina Ghisi.

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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↗

arXiv2019

Critical counterexamples for linear wave equations with time-dependent propagation speed

We investigate an abstract wave equation with a time-dependent propagation speed, and we consider both the non-dissipative case, and the case with a strong damping that depends on a power of the elastic operator. Previous results show that, depending on the values of the parameters and on the time regularity of the propagation speed, this equation exhibits either well-posedness in Sobolev spaces, or well-posedness in Gevrey spaces, or ill-posedness with severe derivative loss. In this paper we examine some critical cases that were left open by the previous literature, and we show that they fall into the pathological regime. The construction of the counterexamples requires a redesign from scratch of the basic ingredients, and a suitable application of Baire category theorem in place of the usual iteration scheme.

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arXiv2019

Universal bounds for a class of second order evolution equations and applications

We consider a class of abstract second order evolution equations with a restoring force that is strictly superlinear at infinity with respect to the position, and a dissipation mechanism that is strictly superlinear at infinity with respect to the velocity. Under the assumption that the growth of the restoring force dominates the growth of the dissipation, we prove a universal bound property, namely that the energy of solutions is bounded for positive times, independently of the initial condition. Under a slightly stronger assumption, we show also a universal decay property, namely that the energy decays (as time goes to infinity) at least as a multiple of a negative power of $t$, again independent of the boundary conditions. We apply the abstract results to solutions of some nonlinear wave, plate and Kirchhoff equations in a bounded domain.

math.AP↗

arXiv2019

Small perturbations for a Duffing-like evolution equation involving non-commuting operators

We consider an abstract evolution equation with linear damping, a nonlinear term of Duffing type, and a small forcing term. The abstract problem is inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force. The main feature is that very natural choices of the boundary conditions lead to equations whose linear part involves two operators that do not commute. We extend to this setting the results that are known in the commutative case, namely that for asymptotically small forcing terms all solutions are eventually close to the three equilibrium points of the unforced equation, two stable and one unstable.

math.AP↗

arXiv2017

An infinite dimensional Duffing-like evolution equation with linear dissipation and an asymptotically small source term

We consider an abstract nonlinear second order evolution equation, inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force. When there is no external force, the system has three stationary positions, two stable and one unstable, and all solutions are asymptotic for $t$ large to one of these stationary solutions.We show that this pattern extends to the case where the external force is bounded and small enough, in the sense that solutions can exhibit only three different asymptotic behaviors.

math.AP↗

arXiv2017

Time-dependent propagation speed vs strong damping for degenerate linear hyperbolic equations

We consider a degenerate abstract wave equation with a time-dependent propagation speed. We investigate the influence of a strong dissipation, namely a friction term that depends on a power of the elastic operator. We discover a threshold effect. If the propagation speed is regular enough, then the damping prevails, and therefore the initial value problem is well-posed in Sobolev spaces. Solutions also exhibit a regularizing effect analogous to parabolic problems. As expected, the stronger is the damping, the lower is the required regularity. On the contrary, if the propagation speed is not regular enough, there are examples where the damping is ineffective, and the dissipative equation behaves as the non-dissipative one.

math.AP↗

arXiv2017

Symmetry-breaking in a generalized Wirtinger inequality

The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the $p$-norm of the derivative among all functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero average. Symmetry properties of minimizers have attracted great attention in mathematical literature in the last decades, leading to a precise characterization of symmetry and asymmetry regions. In this paper we provide a proof of the symmetry result without computer assisted steps, and a proof of the asymmetry result which works as well for local minimizers. As a consequence, we have now a full elementary description of symmetry and asymmetry cases, both for global and for local minima. Proofs rely on appropriate nonlinear variable changes.

math.CA↗