Diffusion phenomenon in Hilbert spaces and applications

Abstract

We prove an abstract version of the striking diffusion phenomenon that offers a strong connection between the asymptotic behavior of abstract parabolic and dissipative hyperbolic equations. An important aspect of our approach is that we use in a natural way spectral analysis without involving complicated resolvent estimates. Our proof of the diffusion phenomenon does not use the individual behavior of solutions; instead we show that only their difference matters. We estimate the Hilbert norm of the difference in terms of the Hilbert norm of solutions to the parabolic problems, which allows us to transfer the decay from the parabolic to the hyperbolic problem. The application of these estimates to operators with Markov property combined with a weighted Nash inequality yields explicit and sharp decay rates for hyperbolic problems with variable (x-dependent) coefficients in exterior domains. Our method provides new insight in this area of extensive research which was not well understood until now.

Higher order asymptotic expansion of solutions to abstract linear hyperbolic equations

“…After that Chill-Haraux [3] discussed the same problem and succeeded in removing the logarithmic correction of the above inequality, which is conjectured in Ikehata-Nishihara [6]. Radu-Todorova-Yordanov [14] studied also diffusion phenomena with respect to stronger norms · D(A k ) (a similar analysis for a linear hyperbolic equation in Hilbert spaces with timedependent damping term b(t)u ′ can be found in Yamazaki [21]). Radu-Todorova-Yordanov [15] discussed a higher order approximation of solutions to Bu ′′ + Au + u ′ = 0 (B is bounded, selfadjoint and positively definite); however, their framework is only valid for semigroup in metric measure spaces L 2 (Ω, µ) with an extra maximal L p -L q regularity.…”

Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

“…We refer the reader to [22,4,31,32,53,14,33,21,5,30,44]. For an exterior domain Ω ⊂ R N , namely, in the case a(x) ≡ 1 in our problem (1.1), the diffusion phenomena was proved by [8,11,2,40].…”

Exponential decay for the damped wave equation in unbounded domains

“…However, when ε = 0, that is when the right-handside is not a negative operator but only a non-positive one, it is known that one cannot hope an exponential decay of the solutions. Indeed, it has been established since a long time (see [25]) that the solutions of ∂ 2 tt u+∂ t u = ∆u in R d asymptotically behave as the ones of ∂ t v = ∆v (see for example [27], [29] and the references therein). It is shown in [8]…”