Renormalized solutions of nonlinear degenerated parabolic problems: Existence and uniqueness

Existence results for a nonlinear parabolic problems with lower order terms

“…It is our purpose, in this paper to generalize the result of ( [3], [4], [5], [16]) and we prove the existence of a renormalized solution of (1.1).…”

“…For the degenerated parabolic equations the existence of weak solutions have been proved by L. Aharouch and al [2] in the case where a is strictly monotone, φ = 0 and f ∈ L p (0, T, W −1,p (Ω, ω * )). See also the existence of renormalized solution proved by Y. Akdim and al [4] in the case where a(x, t, s, ξ) is independent of s and φ = 0.…”

Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities

“…Definition 2.15. See [5] A mapping S is called pseudo-monotone with u n ⇀ u and Lu n ⇀ Lu and lim n→∞ sup S(u n ), u n − u ≤ 0, that we have lim n→∞ sup S(u n ), u n − u = 0 and S(u n ) ⇀ S(u) as n → ∞.…”

“…The notion of renormalized solutions was introduced by R. J. Diperna and P. L. Lions [12] for the study of the Boltzmann equation, it was then used by L. Boccardo and al [11] when the right hand side is in W −1,p ′ (Ω) and by J. M Rakoston [16] when the right hand side is in L 1 (Ω). For the degenerated parabolic equations the existence of weak solutions have been proved by L. Aharouch and al [2] in the case where a(x, t, u, ∇u)is strictly monotone H = 0, F = 0 and f ∈ L p ′ (0, T, W −1,p ′ (Ω, W * )), see also the existence and uniqueness of a renormalized solution proved by Y. Akdim and al [5] in the case where a(x, t, s, ξ) is independent of s, H = 0 and F = 0. In the case H(x, t, u, ∇u) = divφ(u) and F = 0, the existence of renormalized solution has been established by H. Redwane in the classical Sobolev space and in Orlicz space [20,22] and by Y. Akdim and al [4] in the degenerate Sobolev space without the sign condition and the coercivity condition on the term H(x, t, u, ∇u) = div(φ(x, t, u)) and F = 0, the existence of renormalized solutions has been established by A.Aberqi and al [1] in the classical Sobolev space.…”

Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems with <em>L</em>¹ Data

“…For the degenerated parabolic equations the existence of weak solutions have been proved by L. Aharouch et al [10] in the case where a is strictly monotone, φ = 0 and f ∈ L p ′ (0,T,W −1, p ′ (Ω,w * )). See also the existence of renormalized solution by Y.Akdim et al [11] in the case where a(x,t,s,ξ) is independent of s and φ = 0.…”