Time-dependent Propagation Speed Research Articles

Scale-invariant semilinear damped wave models with mass term and integrable in time speed of propagation

In this paper, we consider the Cauchy problem for scale-invariant semilinear wave models with time-dependent mass, dissipation and integrable time-dependent propagation speed. The goal is to study the interplay between the coefficients appearing in the mass and dissipation term and the exponent in the speed of propagation to prove global (in time) existence of small data Sobolev solutions and blow-up results.

On the energy estimates of semi-discrete wave equations with time dependent propagation speed

Discretization is a fundamental step in numerical analysis for the problems described by differential equations, and the difference between the continuous model and discrete model is one of the most important problems. In this paper, we consider the difference in the effect of the time-dependent propagation speed on the energy estimate of the solutions for the wave equation and the semi-discrete wave equation which is a discretization with respect to space variables.

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.

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.

Very weak solutions to hypoelliptic wave equations

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of “very weak solutions” to the Cauchy problem, that was already successfully used in similar contexts in [12] and [20]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p-evolution equations for higher order operators on Rn or on groups, the results already being new in all these cases.

Analog of cosmological particle creation in electromagnetic waveguides

We consider an electromagnetic waveguide with a time-dependent propagation speed $v(t)$ as an analog for cosmological particle creation. In contrast to most previous studies which focus on the number of particles produced, we calculate the corresponding two-point correlation function. For a small step-like variation $\delta v(t)$, this correlator displays characteristic signatures of particle pair creation. As another potential advantage, this observable is of first order in the perturbation $\delta v(t)$, whereas the particle number is second order in $\delta v(t)$ and thus stronger suppressed for small $\delta v(t)$.

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.

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

Given a Hilbert space {mathcal{H}}, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on {mathcal{H}}. We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on {mathbb{R}^n}, uniformly elliptic operators of different orders on domains, Hörmander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

Theory of damped wave models with integrable and decaying in time speed of propagation

We study the Cauchy problem for damped wave equations with a time-dependent propagation speed and dissipation. The model of interest is [Formula: see text] We assume [Formula: see text]. Then we propose a classification of dissipation terms in non-effective and effective. In each case we derive estimates for kinetic and elastic type energies by developing a suitable WKB analysis. Moreover, we show optimality of results by the aid of scale-invariant models. Finally, we explain by an example that in some estimates a loss of regularity appears.

Linear wave 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.

On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions

We consider the energy estimates for the wave equation with time dependent propagation speed. It is known that the asymptotic behavior of the energy is determined by the interactions of the properties of the propagation speed: smoothness, oscillation and the difference from an auxiliary function. The main purpose of the article is to show that if the propagation speed behaves asymptotically as a monotone decreasing function, then we can extend the preceding results to allow faster oscillating coefficients. Moreover, we prove that the regularity of the initial data in the Gevrey class can essentially contribute for the energy estimate.

On the asymptotic boundedness of the energy of solutions of the wave equation u tt − a(t)Δu = 0

Applying a class of quadratic forms introduced by Manfrin (Port. Math. 65(4):447–484, 2008), we study the asymptotic behavior, as t → +∞, of the energy of the solutions to the Cauchy problem for wave equations with time dependent propagation speed.

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples.

Infants are not distractable during physiologically defined periods of sustained attention

Discretization is a fundamental step in numerical analysis for the problems described by differential equations, and the difference between the continuous model and discrete model is one of the most important problems. In this paper, we consider the difference in the effect of the time-dependent propagation speed on the energy estimate of the solutions for the wave equation and the semi-discrete wave equation which is a discretization with respect to space variables.