Abstract
In this paper, we study an existence result of entropy solutions for some nonlinear parabolic problems in the Musielak-Orlicz-Sobolev spaces.
Highlights
Elemine Vall abstract: In this paper, we study an existence result of entropy solutions for some nonlinear parabolic problems in the Musielak-Orlicz-Sobolev spaces
The above problem does not admit, in general, a weak solution since the field a(x, t, u, ∇u) does not belong to (L1loc(Q))N in general. To overcome this difficulty we use in this paper the framework of entropy solutions
Sidi El Vally in [2] had studied the problem (P) in the Inhomogeneous case and the data belongs to L1(Q), in the elliptic case the authors in [1] proved the existence of weak solutions for the problem (P) where the data assume to be measure and g ≡ 0
Summary
We list briefly some definitions and facts about Musielak-OrliczSobolev spaces. |α|≤m for u ∈ W mLφ(Ω), these functionals are a convex modular and a norm on W mLφ(Ω), respectively, and the pair W mLφ(Ω), m φ,Ω is a Banach space if φ satisfies the following condition [16] : there exist a constant c > 0 such that inf φ(x, 1) ≥ c. Let W mEφ(Ω) the space of functions u such that u and its distribution derivatives up to order m lie to Eφ(Ω), and W0mEφ(Ω) is the (norm) closure of D(Ω) in W mLφ(Ω). We say that a sequence of functions un ∈ W mLφ(Ω) is modular convergent to u ∈ W mLφ(Ω) if there exists a constant k > 0 such that lim n→∞ For φ and her complementary function ψ, the following inequality is called the Young inequality [16]: ts ≤ φ(x, t) + ψ(x, s), ∀t, s ≥ 0, x ∈ Ω.
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