Stabilization of solutions of a degenerate nonlinear diffusion problem

Convergence to equilibrium in degenerate parabolic equations with delay

“…There is a large literature relating to problem (D) when delays are absent (r = 0), see [3,23,27] for an overview and extensive bibliographies. We will make use of several key results from this literature on degenerate parabolic equations, using it mainly to provide suitable comparison solutions for the solutions of (D).…”

Stability and uniqueness for cooperative degenerate Lotka–Volterra model

“…In the prey-predator (b < 0 and c > 0) and competition (b < 0 and c < 0) cases, Pozio and Tesei [6] proved the existence of a unique nonnegative global solution (w, z) of (2) with (w, z) ∈ (C(IR + ; L 1 (Ω)) ∩ L ∞ (D T )) 2 for any T > 0. In both cases, they proved that if we have a pair of sub-supersolutions (w, z), (w, z) of the stationary problem associated to (2), then the interval I = [(w, z), (w, z)] is stable in L p (Ω) norm in the following sense: there exists a set K containing a neighbourhood of I such that for any (w 0 , z 0 ) ∈ K, the distance from I to (w, z) goes to zero in the L p (Ω) norm as t diverges.…”

“…This kind of equations was introduced by Gurtin and MacCamy [1] to model the evolution of a biological population whose density is w. It is well known, see for example [2], that problem (1) In [3] and [4] the large time behaviour of the nonnegative solutions of (1) was studied. It was shown that if λ > 0 the unique positive steady-state solution of (1), w λ , attracts in L p (Ω) norm (p = +∞ if N = 1 and p ∈ [1, +∞) if N ≥ 2) any solution of (1) for any w 0 in a suitable subset of L ∞ (Ω).…”

“…In [7] it was proved that under the assumptions of the existence of a pair of sub-supersolutions (w, z), (w, z) of the stationary problem associated to (2), this problem possesses a unique nonnegative global solution. In the symbiotic case (b > 0 and c > 0) the authors showed that for (w 0 , z 0 ) = (w, z) (resp.…”

“…decreasing) to the minimal (w * , z * ) (resp. maximal (w * , z * )) solution of the stationary problem of (2) in the L p (Ω) norm. Moreover, (w * , z * ) (resp.…”