Strongly nonlinear parabolic inequality in orlicz spaces via a sequence of penalized equations

Nonlinear parabolic problem with lower order terms in Musielak-Orlicz spaces

“…We can easily show as in [8],that when Ω has the segment property, then each element u of the closure of D(Q T ) with respect of the weak* topology…”

“…Démonstration: We adopt the same technics in the proof in [8]. Proof of (72): If we take n > m + 1, we get…”

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

“…In the Orlicz-Sobolev spaces, Rhoudaf et al in [19] proved the existence of entropy solutions of the problem (1.1) where H(x, t, u, ∇u) ≡ 0 and the growth of the first lower order Φ prescribed by an anisotropic N-function ϕ defining space does not satisfy the △ 2 -condition. To our knowledge, differential equations in general Musielak-Sobolev spaces have been studied rarely see [3,10,17,20], then our aim in this paper is to overcome some difficulties encountered in these spaces and to generalize the result of [2,5,19,22], and we prove an existence result of entropy solutions for the obstacle parabolic problem (1.1), with less restrictive growth, and no coercivity condition in the first lower order term Φ, and without sign condition in the second lower order H, in the general framework of inhomogeneous Musielak-Orlicz-Sobolev spaces W 1,x 0 L ϕ (Q T ), and the anisotropic N-function ϕ, defining space does not satisfy the △ 2 -condition.…”

Existence of a renormalized solution of nonlinear parabolic equations with general measure data

“…Both renormalized and entropy solutions provide a convenient framework to deal with elliptic or parabolic equations with L 1 data. A large number of papers was then devoted to the study the existence of renormalized (or entropy) solution of parabolic problems with rough data under various 2 A. Marah and H. Redwane assumptions and different contexts: in addition to the references already mentioned see, among others, [1,2,9,10,11,12,13,18,19,32,40].…”