Existence of entropy solutions in Musielak Orlicz spaces via a sequence of penalized equations

Abstract

This paper, is devoted to an existence result of entropy unilateral solutions for the nonlinear parabolic problems with obstacle in Musielak- Orlicz--spaces:$$ \partial_{t}u + A(u) + H(x,t,u,\nabla u) =f + div(\Phi(x,t,u))$$and $$ u\geq \zeta \,\,\mbox{a.e. in }\,\,Q_T.$$Where $A$ is a pseudomonotone operator of Leray-Lions type defined in the inhomogeneous Musielak-Orlicz space $W_{0}^{1,x}L_{\varphi}(Q_{T})$,$H(x,t,s,\xi)$ and $\phi(x,t,s)$ are only assumed to be Crath\'eodory's functions satisfying only the growth conditions prescribed by Musielak-Orlicz functions $\varphi$ and $\psi$ which inhomogeneous and does not satisfies $\Delta_2$-condition. The data $f$ and $u_{0}$ are still taken in $L^{1}(Q_T)$ and $L^{1}(\Omega)$.

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

“…Consequently there exist α andβ in (L ϕ (Q)) d such that a(•, ∇u ℓ ′ ) ⇀ α and K(u ℓ ′ ) ⇀ β weakly in (L ϕ (Q)) d for a subsequence. And by Lemma 5, α ∈ (L ϕ (Q)) d and β ∈ (L ϕ (Q)) d .On the other hand using(14), |K(u ℓ ) − K(u)| ν 1 |u ℓ − u| and as u ℓ → u a.e. in Q and K is continuous, we obtain β = K(u).…”

“…Proposition 1 (Ref. [14]). Let γ ≪ ϕ near infinity and for all t > 0, sup x∈Ω γ(x, t) < ∞, then for all ε > 0, there exists C ε > 0 such that γ(x, t) ϕ(x, εt) + C ε , ∀t > 0.…”

Existence and uniqueness of renormalized solution for nonlinear parabolic equations in Musielak Orlicz spaces

“…Lemma 2.5. [13] Let a < b ∈ IR and Ω be a bounded open subset of IR N with the segment property, then…”

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

“…near infinity) and we write P ≺≺ M, if for every positive constant c, we have lim Proposition 2.1. (See [10]) Let P ≺≺ M near infinity and for all t > 0, sup x∈Ω P(x, t) < ∞, then for all > 0, there exists C > 0 such that…”