Existence of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces

Renormalized Solution for a nonlinear parabolic problems with noncoercivity in divergence form in Orlicz Spaces

“…We omit the proof of (5.23), which is the same as the corresponding result in [5] and On the other hand, by (3.7), for n P m þ 1 we obtain Z fm6un6mþ1g Mðj/ n ðx; t; T mþ1 ðu n ÞÞjÞ dx dt 6 Z fm6un6mþ1g Mð P À1 PðjT mþ1 ðu n ÞjÞÞ dx dt:…”

“…Proof of (5.52): Using the same technics in the proof of Proposition 5.6 p. 49 in [5]. We obtain, It what follows we pass to the limit as n tends to þ1 in each term of (5.63).…”

“…The main difficulty in this kind of problem that is studied in the Theorem 5.1 is due to the fact that U is not assumed to be continuous on R. Thus, it is not possible to use Stokes Theorem to handle this term as in the literature [4,5,3]. .…”

“…P ( M means that P grows essentially less rapidly than M, i.e., for each e > 0, For more details (see [5,9,8]). For K > 0, we define the truncation at height K; T K : R !…”

Rothe time-discretization method for a nonlinear parabolic problem in Orlicz-Sobolev spaces

“…We have as in the paper [3], We can easily show that they form a complementary system when Ω satisfies the segment property. These spaces are considered as subspaces of the product space ΠL M (Q T ) which have as many copies as there is -order derivatives, | | ≤ 1 .…”