The well-posedness of renormalized solutions for a non-uniformly parabolic equation

Zhang, Chao; Zhou, Shulin

doi:10.1090/proc/13406

Cited by 7 publications

(11 citation statements)

References 19 publications

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“…Conclusion via the monotonicity trick. Recall(59). Passing there to the limit, due to (62) and (61), we get 0…”

mentioning

confidence: 88%

“…Lately, generalising the setting, renormalized solutions to parabolic problems have been considered in the variable exponent setting [2,41,58] and in the model of thermoviscoelasticity [15]. For very recent results on entropy and renormalised solutions, we refer also to [15,24,42,59]. This issue in parabolic problems in non-reflexive Orlicz-Sobolev spaces are studied in [37,42,53,59], while in the nonhomogeneous and non-reflexive Musielak-Orlicz spaces in [36] (under certain growth conditions on the modular function).…”

Section: State Of Artmentioning

confidence: 99%

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Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions

Chlebicka

Gwiazda

Zatorska–Goldstein

2018

Journal of Differential Equations

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We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consideron a Lipschitz bounded domain in R N . The growth of the weakly monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N -function M . The approach does not require any particular type of growth condition of M or its conjugate M * (neither ∆ 2 , nor ∇ 2 ). The condition we impose on M is continuity of log-Hölder-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.

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“…Conclusion via the monotonicity trick. Recall(59). Passing there to the limit, due to (62) and (61), we get 0…”

mentioning

confidence: 88%

Section: State Of Artmentioning

confidence: 99%

Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions

Chlebicka

Gwiazda

Zatorska–Goldstein

2018

Journal of Differential Equations

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“…For results concerning parabolic problems we refer to [24,47,48,114,142,149,198,199]. Among them the variable exponent setting is employed in [24,142,198] and non-reflexive Orlicz-Sobolev spaces are studied in [149,199]. Problems stated in the anisotropic and non-reflexive Musielak-Orlicz spaces in [114] under certain growth conditions on a modular function and in [47,48] under regularity restrictions only.…”

Section: Parabolic Existencementioning

confidence: 99%

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

Chlebicka¹

2018

Nonlinear Analysis

112

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The Musielak-Orlicz setting unifies variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity.In this survey we present an overview of developments of the theory of existence of PDEs in the setting including reflexive and non-reflexive cases, as well as isotropic and anisotropic ones. Particular attention is paid to problems with data below natural duality in absence of Lavrentiev's phenomenon.

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“…in [7,8,9,23,57], in the variable exponent setting [5,47,63]. 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].…”

Section: Introductionmentioning

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%

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

Chlebicka

Gwiazda

Zatorska–Goldstein

2019

Journal of Differential Equations

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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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The well-posedness of renormalized solutions for a non-uniformly parabolic equation

Cited by 7 publications

References 19 publications

Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions

Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

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

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