Strong solutions of the double phase parabolic equations with variable growth

Arora, Rakesh; Shmarev, Sergey

doi:10.48550/arxiv.2010.08306

Cited by 3 publications

(7 citation statements)

References 20 publications

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“…For this equation, the questions of global higher integrability of the gradient and the second-order spatial regularity were studied in [8]. The assertions of Theorem 2.1 and 2.3 improve the corresponding results in [8] and, by the same token, complete the results of [9] for another special case (1.4).…”

Section: Assumptions and Resultssupporting

confidence: 53%

“…we may transform the first term on the right-hand side of (6.7) using the Green formula two times (see [9,Lemma 5.2] or [8,Lemma 3.2] for the details):…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…Lemma 3.3,[9]). Let Ω ⊂ R N , N ≥ 2 be a bounded domain with the boundary ∂Ω ∈ C 2 , and a, b ∈ W 1,∞ (Ω) be given nonnegative functions.…”

mentioning

confidence: 94%

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Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

Arora¹,

Shmarev²

2021

Preprint

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We study the homogeneous Dirichlet problem for the equationwhere Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω ∈ C 2 . The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions of the argument z = (x, t) ∈ Q T . It is assumed that 2N N+2 < p(z), q(z) and that the modulating coefficients and growth exponents satisfy the balance conditionsWe find conditions on the source f and the initial data u(•, 0) that guarantee the existence of a unique strong solution u with ut ∈ L 2 (Q T ) and a|∇u| p + b|∇u| q ∈ L ∞ (0, T ; L 1 (Ω)). The solution possesses the property of global higher integrability of the gradient, |∇u| min{p(z),q(z)}+r ∈ L 1 (Q T ) with any r ∈ 0, 4 N + 2 , which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The secondorder differentiability of the strong solution is proven:Dx i a|∇u| p−2 + b|∇u| q−2 1 2 Dx j u ∈ L 2 (Q T ), i, j = 1, 2, . . . , N.1,p(•) 0

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Section: Assumptions and Resultssupporting

confidence: 53%

“…we may transform the first term on the right-hand side of (6.7) using the Green formula two times (see [9,Lemma 5.2] or [8,Lemma 3.2] for the details):…”

Section: A Priori Estimatesmentioning

confidence: 99%

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Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

Arora¹,

Shmarev²

2021

Preprint

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“…|B| ], for a. a. x, y ∈ B ∩ Ω and for every ball B with |B| ≤ 1. (v) We say that ϕ ∈ Φ w (Ω) satisfies (A1'), if there exists β ∈ (0, 1) such that ϕ(x, βt) ≤ ϕ(y, t) for every ϕ(y, t) ∈ [1, 1 |B| ], for a. a. x, y ∈ B ∩ Ω and for every ball B with |B| ≤ 1. (vi) We say that ϕ ∈ Φ w (Ω) satisfies (A2), if for every s > 0 there exist β ∈ (0, 1] and…”

Section: A New Musielak-orlicz Sobolev Space and Some Preliminariesmentioning

confidence: 99%

“…Very recently, Arora-Shmarev [1] (see also Arora [2]) treated a parabolic problem of double phase type with variable growth of the form…”

Section: Introductionmentioning

confidence: 99%

A new class of double phase variable exponent problems: Existence and uniqueness

Crespo-Blanco¹,

Gasiński²,

Harjulehto³

et al. 2021

Preprint

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In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (completeness, reflexivity, an equivalent norm) and the properties of the new double phase operator (continuity, strict monotonicity, (S + )-property). Finally we show the existence and uniqueness of corresponding elliptic equations with right-hand sides that have gradient dependence (so-called convection terms) under very general assumptions on the data. As a result of independent interest, we also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded.

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The weak solutions of a nonlinear parabolic equation from two-phase problem

Huang

2021

J Inequal Appl

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A nonlinear parabolic equation from a two-phase problem is considered in this paper. The existence of weak solutions is proved by the standard parabolically regularized method. Different from the related papers, one of diffusion coefficients in the equation, $b(x)$ b ( x ) , is degenerate on the boundary. Then the Dirichlet boundary value condition may be overdetermined. In order to study the stability of weak solution, how to find a suitable partial boundary value condition is the foremost work. Once such a partial boundary value condition is found, the stability of weak solutions will naturally follow.

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Strong solutions of the double phase parabolic equations with variable growth

Cited by 3 publications

References 20 publications

Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

A new class of double phase variable exponent problems: Existence and uniqueness

The weak solutions of a nonlinear parabolic equation from two-phase problem

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