Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case

Abstract

Abstract. We consider the Dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients. Under such minimal regularity assumptions, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. As a special case of our main theorem, assuming the coefficients are time-periodic, we obtain a new result on the existence of a principal eigenvalue of an associated (time-periodic) parabolic eigenvalue problem. We also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time-periodic uniformly parabolic operators. Finally, we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity.

Effects of diffusion and advection on the principal eigenvalue of a periodic-parabolic problem with applications

“…Besides, when no advection appears, Du and Peng in [27,28] considered the case that V (x, t) = μv(x, t), where v(x, t) ≥ 0, ≡ 0 may vanish somewhere, and studied the asymptotic behavior of λ 1 as μ → ∞, which turns out to be essentially different from that of the autonomous elliptic counterpart. For some other recent work on the periodic-parabolic eigenvalue problems and related applications, one may further refer to [14,18,19,37,40,48,50]; on the other hand, the study on the principal eigenvalue for cooperative systems can be found in [2][3][4]12,13,15] and the references therein.…”

Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$

“…This is hardly an optimal result, although restricting the class of admissible solutions is probably necessary for the uniqueness. Similar uniqueness problems for bounded domains have been addressed in several papers, see for example [11,13,14,16,23,24,29]. A different result for nonautonomous equations on time-dependent domains can be found in [28].…”

Qualitative properties of coexistence and semi-trivial limit profiles of nonautonomous nonlinear parabolic Dirichlet systems

“…An approach based on maximum principles for small domains as in [6] would have the advantage of requiring weaker regularity assumptions on the nonlinearity, but this technique typically only guarantees asymptotic positivity of (v e ′ n , u e ′ 2 ) and therefore, to perform a perturbation argument, one needs that the coefficients of the linearization are uniformly bounded independently of (v e ′ n , u e ′ 2 ) for all times, which, as explained above, does not happen in general. For scalar equations, eventual positivity using maximum principles for small domains was obtained in [8,Proposition 6.11], but an extension of this result to the case of linear systems remains, up to our knowledge, an open question.…”