Comparison results for nonlinear anisotropic parabolic problems

“…Proof. We can argue as in Theorem 3.6 of [2] but considering the problem defined in whole space R N and with a smooth datum. In order to obtain the result when the datum is in L 1 (R N ) we argue by approximation (see section 4) and we pass to the limit in the concentration estimate, recalling that the rearrangement application u → u * is a contraction in L r (R N ) for any r ≥ 1 (see [44]).…”

Section: Comparison Results For Stationary Problems In the Wholementioning

confidence: 99%

“…Now, it is well-known that a the pointwise comparison (5.2) need not hold for nonlinear parabolic equations, not even for the heat equation, and has to be replaced by a comparison of integrals known in the literature as Concentration Comparison, and reads (see [2,4,53,54,55])…”

Section: Main Ideas Of the Parabolic Symmetrizationmentioning

confidence: 99%

Anisotropic p-Laplacian Evolution of Fast Diffusion type

Feo¹,

Luis²,

Volzone³

2021

Preprint

Self Cite

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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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“…Indeed, a maximal monotone graph is a natural generalization of the concept of monotone non-decreasing real function; moreover, the inverse of a maximal monotone graph (that appears in the proof of Theorem 3.1) is again a maximal monotone graph (see [33] for more details). Results in this order of idea are contained in [4] when Φ is given by (2.1).…”

Section: Generalizationmentioning

confidence: 99%

“…[5,6,9,13,18,19,22,23,24,25,27]). This interest has led to an extensive investigation also for problems governed by fully anisotropic growth conditions (see [1,2,3,4,16]) and problems related to different type of anisotropy (see e.g. [8,11,17]).…”

Section: Introductionmentioning

confidence: 99%