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

Abstract

We consider a non-autonomous, degenerate parabolic problem with Dirichlet boundary\ud condition. We study the asymptotic behaviour of solutions, extending an earlier result of\ud the authors, where the forcing term was taken zero

Global estimates for generalized Forchheimer flows of slightly compressible fluids

“…In this article, we focus on the spatially global estimates, i.e., on the entire domain. Since the equation for pressure is of degenerate parabolic type, finding L ∞ -bounds for its gradient, in general, is a difficult task and requires much work, see, for e.g., [5,18,19,24]. We will show that the gradient estimates for the particular flows under the current study can be obtained in a relatively simple way thanks to their equations' specific structure.…”

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.…”

Convergence to Separate Variables Solutions for a Degenerate Parabolic Equation with Gradient Source

“…is the unique solution to (1.1)-(1.2) with an initial condition identically infinite in Ω, see Theorem 1.2 below. The availability of solutions having infinite initial value in Ω (also called friendly giants) and their stability are well-known for the porous medium equation ∂ t z − ∆z m = 0, m > 1, the p-Laplacian equation ∂ t z − ∆ p z = 0, p > 2, and some related equations sharing a similar variational structure, see [18,20,25] for instance, but also for the semilinear diffusive Hamilton-Jacobi equation with gradient absorption ∂ t z − ∆z + |∇z| q = 0, q > 1 [10].…”