Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces

Gwiazda, Piotr; Wittbold, Petra; Wróblewska-Kamińska, Aneta; Zimmermann, Aleksandra

doi:10.1016/j.na.2015.08.017

Cited by 39 publications

(52 citation statements)

References 46 publications

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“…Let us denote it by u k,∞ . On the other hand, taking into account (39) we infer that u k ′ ,∞ ≤ u k,∞ a.e. in Ω T .…”

Section: Almost Everywhere Limitmentioning

confidence: 89%

“…We use the framework developed in [32,18,37,38,39], where elliptic and parabolic problems in the Musielak-Orlicz spaces were studied, and apply the results of [17]. Since in general M * ∈ ∆ 2 , the understanding of the dual pairing is not intuitive.…”

Section: The Methodsmentioning

confidence: 99%

“…Indeed, A(·, ·, ∇(T k (u))) and ∇(T k (u)) do not belong to the dual spaces. Relaxing growth condition on the modular function restricts the admissible classical tools, such as the Sobolev embeddings, the Rellich-Kondrachov compact embeddings, or Aubin-Lions Lemma (applied in [39] to prove almost everywhere convergence). The proof of the existence of a renormalized solution involves the classical truncation ideas, the Young measures methods and monotonicity arguments.…”

Section: The Methodsmentioning

confidence: 99%

“…Let us stress that this is the part that is essentially more complex here, than in the case of the space not changing with time, e.g. [18,39]. Indeed, the classical tool of Landes regularization cannot be applied anymore and we need to use a much more subtle one coming from Theorem 3.1.…”

Section: Almost Everywhere Limitmentioning

confidence: 99%

“…For very recent results on entropy and renormalised solutions, we refer also to [14,27,48,64]. Parabolic problems in non-reflexive Orlicz-Sobolev spaces are studied in this context in [40,48,58,64], while in the nonhomogeneous and non-reflexive Musielak-Orlicz spaces in [18,39]. See [15] for deeper considerations on the problems with data below duality in various instances of Musielak-Orlicz spaces.…”

Section: Introductionmentioning

confidence: 99%

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Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon

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We provide existence and uniqueness of renomalized solutions to a general nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in R N . Namely we studyThe growth of the monotone vector field A is assumed to be controlled by a generalized nonhomogeneous and anisotropic N -Existence and uniqueness of renormalized solutions are proven in absence of Lavrentiev's phenomenon. The condition we impose to ensure approximation properties of the space is a certain type of balance of interplay between the behaviour of M for large |ξ| and small changes of time and space variables. Its instances are log-Hölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (changing in time).The noticeable challenge of this paper is considering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time. New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions. The problems similar to (7) with A depending on ∇u only and with polynomial growth are very well understood. There are countless deep results concerning the corresponding problems involving the p-Laplace operator, A(t, x, ξ) = |ξ| p−2 ξ, stated in the Lebesgue space setting (the modular function is then M (t, x, ξ) = |ξ| p ). There is a wide range of directions in which the polynomial growth case has been developed, including the variable exponent, Orlicz, and double-phase spaces unified. Survey [15] describes how they can be unified in the framework of Musielak-Orlicz spaces and used as a setting for differential equations.The study of nonlinear boundary value problems in non-reflexive Orlicz-Sobolev-type setting originated in the work of Donaldson [22] and Gossez [28,29,30]. We refer to the paper of Mustonen and Tienari [55] for a summary of the results. The case of vector Orlicz spaces with an anisotropic modular function, but independent of spacial or time variables, was investigated in [35].The Musielak-Orlicz setting in full generality has been studied systematically starting from [54, 59, 60] and developed inter alia around the theory arising from fluid mechanics [33,34,36,62]. For other recent developments of the framework of the spaces let us refer e.g. to [1,41,42,43,49,50]. Typically the research concentrates, however, mostly on the ∆ 2 /∇ 2 -case, or -even if without structural conditions of ∆ 2 -type (and thus done in nonreflexive spaces) -when the modular function was trapped between some power-type functions usually briefly described as p, q-growth. This direction comes from the fundamental papers [51,52] by Marcellini and despite it is well understood area it is still an active field especially from the point of view of modern calculus of variations and potential theory, see e.g. [26,41,24,25,4,19,2,44].However, there is a vast range of N -functions that do not satisfy the ∆ 2 condition, e.g.• M (t, x, ξ) = a(t, x) (exp(|ξ|) − 1 + |ξ|);• M (t, x, ξ) = a(t, x)|ξ 1 | p1(...

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“…Let us denote it by u k,∞ . On the other hand, taking into account (39) we infer that u k ′ ,∞ ≤ u k,∞ a.e. in Ω T .…”

Section: Almost Everywhere Limitmentioning

confidence: 89%