Asymptotic Analysis

The homogenization of neutron transport source problems in a finite domain with periodic structure is considered. It is known that the solution of such problems can be factored asymptotically as the product of two terms. The first one gives the local behavior of the neutron transport and is a solution of a periodic transport equation. The second term contains the large scale fluctuations of the neutron density and is a solution of a homogeneous diffusion equation in finite domain. In this paper we present a detailed analysis of the first‐order correction to the expansion that accounts for the neutron leakage at the boundary of the domain. We study a multi‐dimensional boundary layer equation by considering half‐space problems with heterogeneous boundary conditions and exponentially decaying source terms. Modified boundary conditions are then obtained for the homogeneous diffusion equation.

The aim of this paper is to establish existence and uniqueness of the solution to a diffusive phase transition problem for an integrodifferential energy balance equation of hyperbolic type. We also examine some asymptotic relations with the related phase‐field problem and the limiting case of the hyperbolic Stefan problem with memory.

A strongly damped semilinear wave equation with linear memory is considered in a history space setting. Namely, the evolution of the past history of the displacement vector is contained in the dynamical system. Existence, uniqueness and continuous dependence results are discussed. Under proper assumptions on the memory kernel, the existence of uniform absorbing sets is achieved. Moreover, when the source term is translation compact in a suitable functional space, the system is shown to possess a uniform attractor.

We consider existence of solutions of ergodic type Bellman equations of risk‐sensitive control with large parameters. Then we deduce as the singular limits first‐order nonlinear partial differential equations, which are considered to relate to $H_{\infty}$ control. This is a generalization of our previous work that appeared in Asymptotic Analysis 16 (1998), 347–362.

The initial–boundary problem is considered for a nonsmooth inhomogeneous string with the left end fixed and the right one equipped with a massive ring moving with damping in the direction orthogonal to the length of the string. The asymptotic behaviour of the eigenvalues of the corresponding boundary problem is investigated. The eigenvectors of the problem are those of a dissipative operator acting in the Hilbert space $H=\hat{W}_2^1[0,l]\oplus L_2[0,l]\oplus\mathbb{C},$ where $\hat{W}_2^1[0,l]$ is the subspace of functions in Sobolev space $W_2^1[0,l],$ which vanish at the origin. It is proved that the set of the normalized rootvectors of this operator is a Riesz basis in $H.$ Under certain conditions the initial–boundary problem admits a unique solution.

We consider the one‐dimensional wave equation with periodic density $\rho$ of period $\varepsilon\to0$ in a bounded interval. By a counterexample due to Avellaneda, Bardos and Rauch we know that the exact controllability property does not hold uniformly as $\varepsilon\to0$ when the control acts on one of the extremes of the interval. The reason is that the eigenfunctions with wavelength of the order of $\varepsilon$ may have a singular behavior so that their total energy cannot be uniformly estimated by the energy observed on one of the extremes of the interval. We give partial controllability results for the projection of the solutions over the subspaces generated by the eigenfunctions with wavelength larger and shorter than $\varepsilon$ . Both results are sharp.

We use recent results on the asymptotic behavior of the spectrum with respect to the oscillation parameter $\varepsilon$ , the theory of nonharmonic Fourier series and the Hilbert uniqueness method (HUM).

We study the large time asymptotic behavior of solutions of the Cauchy problem for the viscous scalar conservation laws. If there exists a travelling wave solution, then it is well known that solutions of the Cauchy problem converge to it. In the case where a wave does not exist we introduce a notion of system of waves, which is a set of waves propagating with different velocities. We show that solutions of the Cauchy problem converge to the system of waves.