On a doubly degenerate quasilinear anisotropic parabolic equation

Sango, Mizuki

doi:10.1524/anly.2003.23.3.249

Cited by 12 publications

(6 citation statements)

References 10 publications

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“…It should be noted that the presence of the Hilbert space-valued cylindrical Wiener process W requires skillful handling because of additional difficulties absent in the finite dimensional case, in particular, when proving the tightness of the probability measures generated by the sequence of solution of the Galerkin scheme. Closely related to our work are other previous papers [10][11][12][13][14][15][16][17][18][19][20] that considered various SPDEs driven by finite-dimensional Wiener processes.…”

Section: Introductionmentioning

confidence: 61%

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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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Section: Introductionmentioning

confidence: 61%

“…where r = ess inf p(·) satisfies condition (18). It follows from the previous estimates (24) and (45) that…”

Section: A Priori Estimates For the Approximate Solutionsmentioning

confidence: 84%

Stochastic system for generalized polytropic filtration

Idriss

2019

Math Methods in App Sciences

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“…In the general studies of nonlinear diffusion the case where the exponents p i are different falls into the category of "structure conditions with nonstandard growth". The anisotropic equation was also studied in a number of references like [35,48]. Actually, a more general doubly nonlinear model was introduced in those references, see also [1].…”

Section: Introductionmentioning

confidence: 99%

Anisotropic p-Laplacian Evolution of Fast Diffusion type

Feo¹,

Luis²,

Volzone³

2021

Preprint

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We study an anisotropic, possibly non-homogeneous version of the evolution p-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive L 1 to L ∞ estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.

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“…Existence, uniqueness and boundedness of solutions to anisotropic parabolic equations of the type (1.1) have been studied by many authors under various conditions on the data and with different methods -see, for example, [1][2][3]6,[13][14][15][16]21,22,26,27] and the further references therein. In the present paper we are interested in the energy solutions of problem (1.1) -see Definition 2.1 below.…”

Section: Introductionmentioning

confidence: 99%

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

Antont︠s︡ev

Shmarev

2010

Journal of Mathematical Analysis and Applications

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We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equationsshow that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of the data. If the solution does not identically vanish in a finite time, we estimate the rate of vanishing of the solution as t → ∞. We establish conditions on the nonlinearity exponents which guarantee vanishing of the solution at a finite instant even if the equation eventually transforms into the linear one.

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On a doubly degenerate quasilinear anisotropic parabolic equation

Abstract: We investigate a quasilinear anisotropic parabolic equation with double degeneration for which we establish the existence of a Solution by combining a semidiscretisation procedure with a suitable time-discretised compactness result.

Cited by 12 publications

References 10 publications

Stochastic system for generalized polytropic filtration

Stochastic system for generalized polytropic filtration

Anisotropic p-Laplacian Evolution of Fast Diffusion type

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

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