Regularizing effect of homogeneous evolution equations with perturbation

Hauer, Daniel

doi:10.1016/j.na.2021.112245

Cited by 5 publications

(1 citation statement)

References 15 publications

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“…The Crandall-Liggett theorem (see[CL71],[Bar72]) says that −(∂ C 0 E +F ) generates a strong continuous semigroup of ω-quasi contractions on C 0 (Ω). Further, since (∂E) | C 0 is homogeneous of order p − 1, and since(∂E) | C 0 ⊆ ∂ L q Efor all 1 ≤ q ≤ ∞, it follows from[Hau21] that for every u 0 ∈ C 0 (Ω) and g ∈ C((0, T ); C 0 (Ω)) ∩ BV (0, T ; C 0 (Ω)), the mild solution u ∈ C([0, T ]; C 0 (Ω)) of the initial boundary value problem (7.1) is a strong solution and satisfies u ∈ W 1,∞ (δ, T ; C 0 (Ω)), moreover u ∈ C lip ([δ, T ]; C 0 (Ω)) for every 0 < δ < T .In particular, u is a weak solution of the non-local Poisson problem (7.3) with h := g(t) − F (u) − u t (t) ∈ C((0, T ); C 0 ).…”

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confidence: 96%

A doubly nonlinear evolution problem involving the fractional p-Laplacian

Collier¹,

Hauer²

2021

Preprint

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In this article we focus on a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional p-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in R d and composed with a continuous, strictly increasing function. We establish well-posedness in L 1 in the sense of mild solutions, a comparison principle, and for restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong. We obtain L q − L ∞ regularity estimates for mild solutions, implying decay estimates and extending the property of strong solutions for more initial data. Moreover, we prove local and global Hölder continuity results as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation.

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confidence: 96%

A doubly nonlinear evolution problem involving the fractional p-Laplacian

Collier¹,

Hauer²

2021

Preprint

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The Dirichlet-to-Neumann operator associated with the 1-Laplacian and evolution problems

Hauer

Mazón

2022

Calc. Var.

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Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent

Deng

Tan

Xie

2023

Advances in Nonlinear Analysis

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We consider the nonlinear elliptic–parabolic boundary value problem involving the Dirichlet-to-Neumann operator of p-Laplace type at the critical Sobolev exponent. We first obtain the existence and asymptotic estimates of the global solution, and the sufficient conditions of finite time blowup of the solution by using the energy method. Second, we improve the regularity of solution by Moser-type iteration. Finally, we analyze the long-time asymptotic behavior of the global solution. Moreover, with the help of the concentration compactness principle, we present a precise description of the concentration phenomenon of the solution in the forward time infinity.

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Regularizing effect of homogeneous evolution equations with perturbation

Cited by 5 publications

References 15 publications

A doubly nonlinear evolution problem involving the fractional p-Laplacian

A doubly nonlinear evolution problem involving the fractional p-Laplacian

The Dirichlet-to-Neumann operator associated with the 1-Laplacian and evolution problems

Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent

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