Existence and uniqueness of bounded weak solutions for some nonlinear parabolic problems

Zou, Weilin; Li, Juan

doi:10.1186/s13661-015-0332-6

Cited by 3 publications

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References 27 publications

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“…By some modifications, the existence results may be obtained for the more general case that g also depends on ru, for example, g satisfies the natural growth condition. We also remark that similar problems involving parabolic equations was studied in [16].…”

Section: Remark 32mentioning

confidence: 79%

On a class of nonlinear obstacle problems with nonstandard growth

Zou

Wang

2014

Math Methods in App Sciences

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Section: Remark 32mentioning

confidence: 79%

On a class of nonlinear obstacle problems with nonstandard growth

Zou

Wang

2014

Math Methods in App Sciences

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“…Remark A.4. We note that the comparison principle in Proposition A.2 together with the standard method for proving existence using Galerkin approximation (see [16, pages 466-475] and [19,25]) ensures that: for any u ∈ L ∞ (Q 3 ) ∩ L p (−9, 9; W 1,p (B 3 )) satisfying u(z) ∈ K for a.e. z ∈ Q 3 , the Dirichlet problem…”

Section: Higher Integrability Of Gradientsmentioning

confidence: 99%

Interior Calderón–Zygmund estimates for solutions to general parabolic equations of p-Laplacian type

Nguyen

2017

Calc. Var.

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We study general parabolic equations of the form u t = div A(x, t, u, Du) + div (|F| p−2 F) + f whose principal part depends on the solution itself. The vector field A is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderón-Zygmund estimates for locally bounded weak solutions to the equations when p > 2n/(n + 2). This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto [6] and the maximal function free approach by Acerbi and Mingione [1].

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Stochastic system for generalized polytropic filtration

Idriss

2019

Math Methods in App Sciences

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In this paper, we study a certain class of stochastic quasilinear parabolic equations describing a generalized polytropic elastic filtration in the framework of variable exponents Lebesgue and Sobolev spaces. We establish an existence result in the infinite dimensional framework of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions, and the governing equations are subjected to cylindrical Wiener processes. We use a Galerkin method, derive crucial a priori estimates for the approximate solutions, and combine profound analytic and probabilistic compactness results in order to pass to the limit. Several difficulties arise in obtaining these uniform bounds and passing to the limit since the nonlinear elliptic part of the leading operator admits nonstandard growth. Apart from adapting the above essential tools, we extend classical methods of monotonicity to the present situation. KEYWORDS generalized weak solution, monotone method, non-Newtonian polytropic filtration, stochastic partial differential equations, stochastic systems MSC CLASSIFICATION 60H15; 35D30; 35K59; 76A05; 93E03where Q T = (0, T) × D, the function u = u(t, x) is an unknown defined in Q T , and f (t, u) and G(t, u) are random external forces. Moreover, W is a cylindrical Wiener process evolving on L 2 (D), which enters the equation as an unknown, the function u 0 ∈ L 2 (D), and A is a nonlinear operator in the divergence form

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Existence and uniqueness of bounded weak solutions for some nonlinear parabolic problems

Cited by 3 publications

References 27 publications

On a class of nonlinear obstacle problems with nonstandard growth

On a class of nonlinear obstacle problems with nonstandard growth

Interior Calderón–Zygmund estimates for solutions to general parabolic equations of p-Laplacian type

Stochastic system for generalized polytropic filtration

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