Existence results for a nonlinear parabolic problems with lower order terms

Abstract

In this paper, we study the nonlinear parabolic problem: ∂b(x, u) ∂t − div(a(x, t, u, ∇u)) + div(φ(x, t, u)) = f where b(x, u) is unbounded function of u, the term −div a(x, t, u, ∇u) is a Leray-Lions operator and the function φ is a nonlinear lower order and satisfy only the growth condition. The second term f belongs to L 1 (Ω × (0, T)). The main contribution of our work is to prove the existence of a renormalized solution.

Discrete solution for the nonlinear parabolic equations with diffusion terms in Museilak-spaces

“…In classical Sobolev spaces, starting with the paper [3], the authors proved an existence result of a weak solution for the non coercive problem (1) in the stationary case b(u) = 0 using the symmetrization method. More later Di Nardo et al [4] has shown the existence of renormalized solution for the parabolic version, more precisely in the linear case b(u) = u, and the uniqueness for such solutions in the paper [5], A. Aberqi et al [6,7] has proved the existence of a renormalized solution for (1) with more general parabolic terms b(x, s).…”

“…The diffusion terms K : R → R d is a continuous function such that |K(s)| ν 0 ϕ −1 ϕ x, s λ for all s in R, for some ν 0 > 0, (6) and f ∈ L 1 0, T ; L 2 (Ω) .…”

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

“…Starting with the paper [8] where g = 0, the existence results have been proved in the framework of Classical Sobolev spaces in ( [5], [7], [15]) where g(x, t, u) = g(u) continuous function on u in the Orlicz spaces. For the lower order g = 0 depending on x, t and u and without coercivity condition, the problem (1.1) was treated firstly in [14] and recently in ( [1]), [2], [9]) using the framework of renormalized solutions. In Musielak spaces Gwiazda et al in [11], have been proved the renormalized solution where the conjugate of Musielak-Orlicz function satisfies the ∆ 2 -condition and in [12] where b(x, u) = u and g = 0.…”

Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form

“…In the classical Sobolev spaces, Aberqi et al in [1] have proved the existence of renormalized solutions (1.1) in the case where b(u) ≡ b(x, u) and Θ satisfies a growth condition (for the definition of this notion of solution see [1], [20]), Redwane in [19] has proved the existence of renormalized solutions of (1.1), where Θ(x, t, u) = Θ(u).…”