Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations

Abstract

Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations F. Ragnedda · S. Vernier Piro · V. Vespri Abstract We consider a class of non-autonomous, degenerate parabolic equations and we study the asymptotic behaviour of the solutions. Even if the equation depends explicitly upon the time, we prove that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. To our knowledge, it is the first time that the asymptotic behaviour of solutions to non-autonomous equations is studied.

Coexistence and optimal control problems for a degenerate predator–prey model

“…In particular, in [49] an intrinsic Harnack estimate is proved for degenerate parabolic equations with degeneracy of two types: p-Laplacean and porous media (and it has been extended to doubly degenerate equations in [50]). The results in those papers, on the one hand, imply that, if u(x 0 , t 0 ) > 0, then u(x 0 , t) > 0 for all t > t 0 and, on the other, have been used in [51,52] to show that the expansion of the positivity set does occur also with lower order terms in the case of p-Laplace type of degenerate equations. The validity of the analogous result for the porous medium equations still seems to be an open problem.…”

Asymptotic time behaviour for non-autonomous degenerate parabolic problems with forcing term

“…If, from one side, this makes the proof simple and very elegant, on the other hand it looks as if this method cannot be applied in the case of time-dependent coefficients. In a recent paper [3] the authors followed an alternative approach introduced by Berryman-Holland [4] and used in the context of the asymptotic behaviour of solutions to degenerate parabolic equations in [5,6]. This approach is more parabolic of the previous one, namely, relying on the properties of the evolution equations, it is possible to study the asymptotic behaviour of the solutions and derive the elliptic properties of the asymptotic limit as a by-product.…”

“…This approach is more parabolic of the previous one, namely, relying on the properties of the evolution equations, it is possible to study the asymptotic behaviour of the solutions and derive the elliptic properties of the asymptotic limit as a by-product. Using this alternative approach and under the assumption B = 0, in [3] the authors studied the asymptotic behaviour of the solutions of a degenerate parabolic equations with time-dependent coefficients. In this note we are able to remove the condition B = 0 even if we are compelled to assume the condition u 0 ∈ L 2 (Ω).…”

The Asymptotic Profile of Solutions of a Class of Singular Parabolic Equations

“…If, from one side, this makes the proof simple and very elegant, on the other hand it looks like this method is not flexible and cannot be applied for more general operators. In recent papers ( [8]), ( [9]) the Authors followed an alternative approach introduced by Berryman-Holland ( [1]) and used in the context of the asymptotic behaviour of solutions to degenerate parabolic equations in [3], [7] and [10]. This approach is more parabolic than the previous one, namely, relying on the properties of the evolution equations, it is possible to study the asymptotic behaviour of the solutions and derive the elliptic properties of the asymptotic limit as a by-product.…”

“…This generalization is based on recent techniques developed in [5], that allow us to avoid the use of comparison functions as in [3] and [10]. With respect the results proved in [8] and [9], here we use the Rayleigh quotient. Remark 1.1.…”