Asymptotic Analysis

We study the asymptotic behavior of a diffusion process with small diffusion in a domain D. This process is reflected at ∂D with respect to a co-normal direction pointing inside D. Our asymptotic result is used to study the long time behavior of the solution of the corresponding parabolic PDE with Neumann boundary condition.

In this paper, we explicitly calculate the quasi-potentials for the damped semilinear stochastic wave equation when the system is of gradient type. We show that in this case the infimum of the quasi-potential with respect to all possible velocities does not depend on the density of the mass and does coincide with the quasi-potential of the corresponding stochastic heat equation that one obtains from the zero mass limit. This shows in particular that the Smoluchowski–Kramers approximation can be used to approximate long time behavior in the zero noise limit, such as exit time and exit place from a basin of attraction.

In Quart. Appl. Math. 71 (2013), 183–199, the authors find sharp exponential rates for the energy decay of nontrivial solutions to the abstract telegraph equation

u_tt+2au_t+S²u=0,

where S is a strictly positive self-adjoint operator in a (complex) Hilbert space and a is a positive constant. The aim of this paper is a further extension of these results by considering equations of the form

u_tt+2F(S)u_t+S²u=0,

where the damping term involves the action of the positive self-adjoint operator F(S). The main assumption on the continuous function F :(0,+∞)→(0,+∞) is that g(x)=F(x)−x changes sign only once, being positive close to zero. We obtain sharp estimates of the form

E(t)≤Ce^−2αt,

where α>0 depends on the relative position of the bottom of the spectrum of S and the point where g vanishes, as well as on the specific behavior of F on the spectrum of S. The general result is then applied to some particular classes of functions F. We also provide a number of applications.

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.