Équations faiblement hyperboliques du deuxième ordre et classes de fonctions ultradifférentiables

We consider a second order weakly hyperbolic equation and we study in which classes the corresponding Cauchy problem is well posed. We consider operators with coefficients depending only on the t variable and belonging to a class X between C^∞ and the real analytic class. We find then a class, strictly related to X, where the Cauchy problem is well posed. Finally, we prove by some counterexamples that these results are almost optimal.     

Quasilinear hyperbolic operators with Log-Lipschitz coefficients

We know that the Cauchy problem for a linear strictly hyperbolic operator with Log-Lipschitz in time coefficients is well posed in C^∞. Here we show that the same result is valid also in the case of a quasilinear operator, but only locally in time.     

Breakdown of classical solutions to Cauchy problem for inhomogeneous quasilinear hyperbolic systems

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for inhomogeneous quasilinear strictly hyperbolic systems with weaker decaying initial data. Under the assumption that the source term satisfies the corresponding matching condition, we obtains a blow-up result for C ¹ solution to the Cauchy problem.     

Loss of regularity for the solutions to hyperbolic equations with non‐regular coefficients—an application to Kirchhoff equation

We consider the Cauchy problem for second‐order strictly hyperbolic equations with time‐depending non‐regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C^∞ well‐posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Log-Lipschitz regularity for SG hyperbolic systems

We consider the Cauchy problem for linear and quasilinear symmetrizable hyperbolic systems with coefficients depending on time and space, not smooth in t and growing at infinity with respect to x. We discuss well-posedness in weighted Sobolev spaces, showing that the non-Lipschitz regularity in t has an influence not only on the loss of derivatives of the solution but also on its behaviour for |x|→∞. We provide examples to prove that the latter phenomenon cannot be avoided.     

Levi condition for hyperbolic equations with oscillating coefficients

In the present paper we explain new Levi conditions of C^∞ type for second-order hyperbolic Cauchy problems. Our goal is to explain the special influence of oscillations in the coefficients. It turns out that such oscillations have an essential influence coupled with the asymptotic behavior of characteristics around multiple points.     

A degenerate hyperbolic equation under Levi conditions

Abstract We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order. Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C^∞ well posedness of the Cauchy problem. Keywords: Cauchy problem, Hyperbolic equations, Levi conditions     

Malliavin calculus with time dependent coefficients and application to nonlinear filtering

Summary In this paper, we prove, using Malliavin calculus, that under a local Hörmander condition the solution of a stochastic differential equation with time depending coefficients admits aC density with respect to Lebesgue measure. An application of this result to nonlinear filtering is developed in this paper to prove the existence of aC density for the filter associated with a correlated system whose observation is one dimensional with unbounded coefficients.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C¹ solutions with the bounded L¹∩L^∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C¹ travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R^1+(1+n).     

A non-local regularization of first order Hamilton-Jacobi equations

In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C² in space and C¹ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W^1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C^∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L^∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.     

Mechanism of singularity formation for quasilinear hyperbolic systems with linearly degenerate characteristic fields

For the Cauchy problem of 1-D first order quasilinear hyperbolic linearly degenerate systems, a new mechanism of singularity formation is given to show that all the W1,p(1<p?+∞) norms of the C¹ solution should blow up simultaneously. It gives a way to verify the property of ODE singularity by directly using the energy method in the framework of C¹ solution.     

Energy estimates at infinity for hyperbolic equations with oscillating coefficients

We study the behaviour, for t→∞, of the energy of the solutions to the Cauchy problem for some strictly hyperbolic second order equations with coefficients very rapidly oscillating.     

Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients.

Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L ² space for solution to the inverse problem under consideration.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

Development of singularities for nonlinear hyperbolic systems with periodic data

We prove two blow-up results for classical (C ¹) solutions of the Cauchy problem for nonlinear hyperbolic systems in one space variable. The initial conditions are periodic and the systems are supposed to be in diagonal form. The first result concernsN×N genuinely nonlinear systems, and the second one is devoted to 2×2 systems with weak nonlinearities. In both cases we give an estimate of the lifespan of the classical solutions.     

Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries

In this paper, we investigate the asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries. Under some suitable assumptions, we prove that the solution approaches a combination of Lipschitz continuous and piecewise C¹ traveling wave solution. As an application, we apply the result to the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n).     

Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients

The goal of this paper is to study well-posedness to strictly hyperbolic Cauchyproblems with non-Lipschitz coefficients with low regularity with respect to timeand smooth dependence with respect to space variables. The non-Lipschitz conditionis described by the behaviour of the time-derivative of coefficients. This leads to a classification of oscillations, which has a strong influence on the loss of derivatives. To study well-posednesswe propose a refined regularizing technique. Two steps of diagonalizationprocedure basing on suitable zones of the phase spaceand corresponding nonstandard symbol classes allow to applya transformation corresponding to the effect of loss of derivatives.Finally, the application of sharp Gårding's inequality allows to derive a suitable energy estimate. From this estimatewe conclude a result about C well-posedness of the Cauchy problem.     

A Remark About Hyperbolic Equations with Log-Lipschitz Coefficients

We prove the well-posedness of the Cauchy problem for strictly hyperbolic equations and systems with Log-Lipschitz coefficients in the time variable.     

Inverse Hyperbolic Problems with Time-Dependent Coefficients

We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in R ⁿ with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation. As a by-product we prove a similar result for the nonself-adjoint hyperbolic operator with time-independent coefficients.