Entropy solutions for nonlinear parabolic inequalities involving measure data in Musielak-Orlicz-Sobolev spaces

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 36 2 (2018): 199–229. ISSN-00378712 in press doi:10.5269/bspm.v36i2.31818 Entropy Solutions For Nonlinear Parabolic Inequalities Involving Measure Data In Musielak-Orlicz-Sobolev Spaces A.Talha, A. Benkirane, M.S.B. 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. Key Words: Musielak-Orlicz-Sobolev spaces, parabolic equations, entropy solutions, truncations. Contents 1 Introduction 199 2 Preliminary 200 2.1 Musielak-Orlicz-Sobolev spaces : . . . . . . . . . . . . . . . . . . . . 201 2.2 Inhomogeneous Musielak-Orlicz-Sobolev spaces : . . . . . . . . . . . 204 3 Essential assumptions 205 4 Some technical Lemmas 206 5 Approximation and trace results 209 6 Compactness Results 212 7 Main results 214 1. Introduction Let Ω a bounded open subset of RN and let Q be the cylinder Ω × (0, T ) with some given T > 0. We consider the strongly nonlinear parabolic problem  ∂u   ∂t + A(u) + g(x, t, u, ∇u) = f − div(F ) in Q, (P) u ≡ 0 on ∂Q = ∂Ω × [0, T ]   u(·, 0) = u0 on Ω, where A : D(A) ⊂ W01,x Lϕ (Q) −→ W −1,x Lψ (Q) (see section 2) defined by A(u) = −div(a(x, t, u, ∇u)) is an operator of Leray-Lions type, where a is a Carathéodory function such that   −1 |a(x, t, s, ξ)| ≤ β h1 (x, t) + ψ −1 γ(x, ν|s|) + ψ ϕ(x, ν|ξ|) x x Submitted May 03, 2016. Published July 06, 2016 199 Typeset by BSP style. M c Soc. Paran. de Mat. 200 A.Talha, A. Benkirane, M.S.B. Elemine Vall   a(x, t, s, ξ) − a(x, t, s, ξ ′ ) (ξ − ξ ′ ) > 0 a(x, t, s, ξ).ξ ≥ αϕ(x, |ξ|) with h1 ∈ L1 (Q), β, ν, α > 0 and γ a Musielak function such that γ ≪ ϕ. Let g be a Carathéodory function such that   |g(x, t, s, ξ)| ≤ b(|s|) h2 (x, t) + ϕ(x, |ξ|) , g(x, t, s, ξ)s ≥ 0, is satisfied, where b a positive function in L1 (R+ ) and h2 ∈ L1 (Q), and f ∈ L1 (Q) and F ∈ (Eψ (Q))N . Under these assumptions, 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. This notion was introduced by Bénilan and al. [4] for the study of nonlinear elliptic problems. In the classical Sobolev spaces, the authors in [9, 17] proved the existence of solutions for the problem (P) in the case where F ≡ 0, in [7] the authors had proved the existence of solutions for the problem (P) in the elliptic case. In the setting of Orlicz spaces, the solvability of (P) was proved by Donaldson [10] and Robert [18], and by Elmahi [12] and Elmahi-Meskine [13]. In Musielak framework, recently M. L. Ahmed Oubeid, A. Benkirane and M. 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. It is our purpose in this paper to prove the existence of entropy solutions for problem (P) in the setting of Musielak Orlicz spaces for general Musielak function ϕ with a nonlinearity g(x, t, u, ∇u) having natural growth with respect to the gradient. Our result generalizes that of [13, 1, 2] to the case of inhomogeneous Musielak Orlicz Sobolev spaces. The plan of the paper is as follows. Section 2 presents the mathematical preliminaries. Section 3 we make precise all the assumptions on a, g, f and u0 . Section 4 is devoted to some technical lemmas with be used in this paper. Section 5 we establish some compactness and approximation results. Final section is consecrate to define the entropy solution of (P) and to prove existence of such a solution. 2. Preliminary In this section we list briefly some definitions and facts about Musielak-OrliczSobolev spaces. Standard reference is [16]. We also include the definition of inhomogeneous Musielak-Orlicz-Sobolev spaces and some preliminaries Lemmas to be used later. Entropy solutions for nonlinear parabolic inequalities . . . 201 2.1. Musielak-Orlicz-Sobolev spaces : Let Ω be an open set in RN and let ϕ be a real-valued function defined in Ω × R+ , and satisfying the following conditions :  a) ϕ(x, ·) is an N-function convex, increasing, continuous, ϕ(x, 0) = 0, ϕ(x, t) > 0,  ϕ(x,t) ∀t > 0, supx∈Ω ϕ(x,t) −→ 0 as t −→ 0, inf −→ ∞ as t −→ ∞ . x∈Ω t t b) ϕ(·, t) is a measurable function. A function ϕ, which satisfies the conditions a) and b) is called Musielak-Orlicz function. For a Musielak-orlicz function ϕ we put ϕx (t) = ϕ(x, t) and we associate its nonnegative reciprocal function ϕ−1 x , with respect to t that is −1 ϕ−1 x (ϕ(x, t)) = ϕ(x, ϕx (t)) = t. The Musielak-orlicz function ϕ is said to satisfy the ∆2 -condition if for some k > 0 and a non negative function h integrable in Ω, we have ϕ(x, 2t) ≤ kϕ(x, t) + h(x) for all x ∈ Ω and t ≥ 0. (2.1) When (2.1) holds only for t ≥ t0 > 0; then ϕ said to satisfy ∆2 near infinity. Let ϕ and γ be two Musielak-orlicz functions, we say that ϕ dominate γ, and we write γ ≺ ϕ, near infinity (resp. globally) if there exist two positive constants c and t0 such that for almost all x ∈ Ω γ(x, t) ≤ ϕ(x, ct) for all t ≥ t0 , ( resp. for all t ≥ 0 i.e. t0 = 0). We say that γ grows essentially less rapidly than ϕ at 0 (resp. near infinity), and we write γ ≺≺ ϕ, If for every positive constant c we have     γ(x, ct) γ(x, ct) = 0, (resp. lim sup = 0). lim sup t−→∞ t−→0 x∈Ω ϕ(x, t) x∈Ω ϕ(x, t) Remark 2.1. [6] If γ ≺≺ ϕ near infinity, then ∀ε > 0 there exist k(ε) > 0 such that for almost all x ∈ Ω we have γ(x, t) ≤ k(ε)ϕ(x, εt), for all t ≥ 0. (2.2) We define the functional ρϕ,Ω (u) = Z ϕ(x, |u(x)|)dx. Ω where u : Ω −→ R a Lebesgue measurable function. In the following, the measurability of a function u : Ω −→ R means the Lebesgue measurability. The set n o Kϕ (Ω) = u : Ω −→ R measurable : ρϕ,Ω (u) < +∞ . 202 A.Talha, A. Benkirane, M.S.B. Elemine Vall is called the generalized Orlicz class. The Musielak-Orlicz space (or the generalized Orlicz spaces) Lϕ (Ω) is the vector space generated by Kϕ (Ω), that is, Lϕ (Ω) is the smallest linear space containing the set Kϕ (Ω). Equivalently   u < +∞, for some λ > 0 . Lϕ (Ω) = u : Ω −→ R measurable : ρϕ,Ω λ Let ψ(x, s) = sup {st − ϕ(x, t)}. t≥0 that is, ψ is the Musielak-Orlicz function complementary to ϕ in the sens of Young with respect to the variable s. We define in the space Lϕ (Ω) the following two norms   Z  |u(x)|  ϕ x, kukϕ,Ω = inf λ > 0/ dx ≤ 1 . λ Ω which is called the Luxemburg norm and the so called Orlicz norm by : Z k|u|kϕ,Ω = sup |u(x)v(x)|dx. kvkψ ≤1 Ω where ψ is the Musielak Orlicz function complementary to ϕ. These two norms are equivalent [16]. The closure in Lϕ (Ω) of the bounded measurable functions with compact support in Ω is denoted by Eϕ (Ω). A Musielak function ϕ is called locally integrable on Ω if ρϕ (tχE ) < ∞ for all (...truncated)