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Marina Ghisi

Publications and source records attributed to Marina Ghisi.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2017

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

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 Fouriercomponents are asymptotic to solutions of the linearized equationwithout damping, and exhibit some sort of equipartition of theenergy among the components. Solutions with infinitely manyFourier components tend to zero weakly but not strongly. We showalso that the limit of the energy of solutions depends only on thenumber of their Fourier components.The proof of our results is inspired by the analysis of asimplified model which we devise through an averaging procedure,and whose solutions exhibit the same asymptotic properties as thesolutions to the original equation.

math.AP↗
arXiv2016

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

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.

math.AP↗
arXiv2015

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, as well as time-dependent coefficients that are 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.

math.AP↗
arXiv2014

Finding the exact decay rate of all solutions to some second order evolution equations with dissipation

We consider an abstract second order evolution equation with damping. The "elastic" term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions. We prove the coexistence of slow solutions and fast solutions. Slow solutions live close to the kernel of A, and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator A and the second order time-derivatives in the original equation. Fast solutions live close to the range of A and decay exponentially as solutions of the linear homogeneous equation obtained by neglecting the nonlinear terms in the original equation. The abstract results apply to semilinear dissipative hyperbolic equations.

math.AP↗
arXiv2014

Kirchhoff equations with strong damping

We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the "elastic" operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space we prove local existence, and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces.

math.AP↗
arXiv2014

Linear hyperbolic equations with time-dependent propagation speed and strong damping

We consider a second order linear equation with a time-dependent coefficient c(t) in front of the "elastic" operator. For these equations it is well-known that a higher space-regularity of initial data compensates a lower time-regularity of c(t). In this paper we investigate the influence of a strong dissipation, namely a friction term which depends on a power of the elastic operator. What we discover is a threshold effect. When the exponent of the elastic operator in the friction term is greater than 1/2, the damping prevails and the equation behaves as if the coefficient c(t) were constant. When the exponent is less than 1/2, the time-regularity of c(t) comes into play. If c(t) is regular enough, once again the damping prevails. On the contrary, when c(t) is not regular enough the damping might be ineffective, and there are examples in which the dissipative equation behaves as the non-dissipative one. As expected, the stronger is the damping, the lower is the time-regularity threshold. We also provide counterexamples showing the optimality of our results.

math.AP↗
arXiv2014

Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution. What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term. We also provide counterexamples in order to show the optimality of our results.

math.AP↗
arXiv2014

A description of all possible decay rates for solutions of some semilinear parabolic equations

We consider an abstract first order evolution equation in a Hilbert space in which the linear part is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions. We prove that two different regimes coexist. Close to the kernel of A the dynamic is governed by the nonlinear term, and solutions (when they decay to 0) decay as negative powers of t. Close to the range of A, the nonlinear term is negligible, and solutions behave as solutions of the linearized problem. This means that they decay exponentially to 0, with a rate and an asymptotic profile given by a simple mode, namely a one-frequency solution of the linearized equation. The abstract results apply to semilinear parabolic equations.

math.AP↗
arXiv2013

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

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as t -> +infinity, at least as fast as a suitable negative power of t. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of t. Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.

math.AP↗
arXiv2012

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 epsilon in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 as t -> +infinity. It is well-known that, when the decay of the coefficient is slow enough, solutions behave as solutions of the corresponding parabolic equation, and in particular they decay to 0 as t -> +infinity. In this paper we consider the nondegenerate and coercive case, and we prove optimal decay estimates for the hyperbolic problem, and optimal decay-error estimates for the difference between solutions of the hyperbolic and the parabolic problem. These estimates show a quite surprising fact: in the coercive case the analogy between parabolic equations and dissipative hyperbolic equations is weaker than in the noncoercive case. This is actually a result for the corresponding linear equations with time-dependent coefficients. The nonlinear term comes into play only in the last step of the proof.

math.AP↗
arXiv2012

Optimal decay-error estimates for the hyperbolic-parabolic singular perturbation of a degenerate nonlinear equation

We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter epsilon in front of the second-order time-derivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem. These estimates show in the same time that the difference tends to zero both as epsilon -> 0, and as t -> +infinity. In particular, in that case the difference decays faster than the two terms separately. In this paper we consider the complementary assumption on initial data, and we show that now the optimal decay-error estimates involve a decay rate which is slower than the decay rate of the two terms. In both cases, the improvement or deterioration of decay rates depends on the smallest frequency represented in the Fourier components of initial data.

math.AP↗
arXiv2012

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.

math.CA↗
arXiv2011

Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: decay-error estimates

We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions decay as t -> +infinity with the same rate of solutions of the limit problem (of parabolic type). In this paper we prove decay-error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem. These estimates show in the same time that the difference tends to zero both as epsilon -> 0, and as t -> +infinity. Concerning the decay rates, it turns out that the difference decays faster than the two terms separately (as t -> +infinity). Proofs involve a nonlinear step where we separate Fourier components with respect to the lowest frequency, followed by a linear step where we exploit weighted versions of classical energies.

math.AP↗
arXiv2010

Kirchhoff equations from quasi-analytic to spectral-gap data

In a celebrated paper (Tokyo J. Math. 1984) K. Nishihara proved global existence for Kirchhoff equations in a special class of initial data which lies in between analytic functions and Gevrey spaces. This class was defined in terms of Fourier components with weights satisfying suitable convexity and integrability conditions. In this paper we extend this result by removing the convexity constraint, and by replacing Nishihara's integrability condition with the simpler integrability condition which appears in the usual characterization of quasi-analytic functions. After the convexity assumptions have been removed, the resulting theory reveals unexpected connections with some recent global existence results for spectral-gap data.

math.AP↗
arXiv2009

Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness

In this note we present some recent results for Kirchhoff equations in generalized Gevrey spaces. We show that these spaces are the natural framework where classical results can be unified and extended. In particular we focus on existence and uniqueness results for initial data whose regularity depends on the continuity modulus of the nonlinear term, both in the strictly hyperbolic case, and in the degenerate hyperbolic case.

math.AP↗
arXiv2009

Hyperbolic-parabolic singular perturbation for Kirchhoff equations with weak dissipation

We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the values of the parameters.

math.AP↗