Asymptotic Analysis

We provide complete structural theorems for the so-called quasiasymptotic behavior of non-quasianalytic ultradistributions. As an application of these results, we obtain descriptions of quasiasymptotic properties of regularizations at the origin of ultradistributions and discuss connections with Gelfand–Shilov type spaces.

In this report we obtain higher order asymptotic expansions of solutions to wave equations with frictional and viscoelastic damping terms. Although the diffusion phenomena are dominant, differences between the solutions we deal with and those of heat equations can be seen by comparing the second order expansions of them. In order to analyze such effects we consider the weighted L 1 initial data. We also give some lower bounds which show the optimality of obtained expansions.

We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

We consider a singular limit for the compressible Euler system in the low Mach number regime driven by a large external force. We show that any dissipative measure-valued solution approaches a solution of the lake equation in the asymptotic regime of low Mach and Froude numbers. The result holds for the ill-prepared initial data creating rapidly oscillating acoustic waves. We use dispersive estimates of Strichartz type to eliminate the effect of the acoustic waves in the asymptotic limit.

In this paper, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We first investigate the stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau-Boussouira’s energy comparison principle introduced in (J. Differ. Equations 248 (2010), 1473–1517) (see also NoDEA Nonlinear Differential Equations Appl. 18(5) (2011), 571–597). We extend to our model the results achieved in (NoDEA Nonlinear Differential Equations Appl. 18(5) (2011), 571–597) for the case of nonlinearly damped Timoshenko system with thermoelasticity. The proof of our results relies on the approach in (NoDEA Nonlinear Differential Equations Appl. 14(5–6) (2007), 643–669 and Appl. Math. Optim. 51(1) (2005), 61–105).

This paper is concerned with a non-conserved phase field system of Caginalp type in which the main operators are fractional versions of two fixed linear operators A and B. The operators A and B are supposed to be densely defined, unbounded, self-adjoint, monotone in the Hilbert space L 2 ( Ω ) , for some bounded and smooth domain Ω, and have compact resolvents. Our definition of the fractional powers of operators uses the approach via spectral theory. A nonlinearity of double-well type occurs in the phase equation and either a regular or logarithmic potential, as well as a non-differentiable potential involving an indicator function, is admitted in our approach. We show general well-posedness and regularity results, extending the corresponding results that are known for the non-fractional elliptic operators with zero Neumann conditions or other boundary conditions like Dirichlet or Robin ones. Then, we investigate the longtime behavior of the system, by fully characterizing every element of the ω-limit as a stationary solution. In the final part of the paper we study the asymptotic behavior of the system as the parameter σ appearing in the operator B 2 σ that plays in the phase equation decreasingly tends to zero. We can prove convergence to a phase relaxation problem at the limit, in which an additional term containing the projection of the phase variable on the kernel of B appears.