The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi; Giuseppe Mingione

Displaying similar documents to “The Wolff gradient bound for degenerate parabolic equations”

Estimates of Green functions and their applications for parabolic operators with singular potentials

Lotfi Riahi (2003)

Colloquium Mathematicae

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We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a $C^{1, 1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.

On admissibility for parabolic equations in ℝⁿ

Martino Prizzi (2003)

Fundamenta Mathematicae

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We consider the parabolic equation (P) $u_{t} - Δ u = F (x, u)$ , (t,x) ∈ ℝ₊ × ℝⁿ, and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-admissible in the sense of Rybakowski. If F(x,u) is asymptotically linear, under appropriate non-resonance conditions, we use Conley’s index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained...

Existence results for some nonlinear parabolic equations with degenerate coercivity and singular lower-order terms

Rabah Mecheter, Fares Mokhtari (2023)

Mathematica Bohemica

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In this paper, we study the existence results for some parabolic equations with degenerate coercivity, singular lower order term depending on the gradient, and positive initial data in $L^{1}$ .

Boundary estimates for certain degenerate and singular parabolic equations

Benny Avelin, Ugo Gianazza, Sandro Salsa (2016)

Journal of the European Mathematical Society

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We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$ -Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_{T}$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of...

Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices

Arina A. Arkhipova, Jana Stará (2019)

Commentationes Mathematicae Universitatis Carolinae

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Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A (t)$ -caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.

$B^{q}$ for parabolic measures

Caroline Sweezy (1998)

Studia Mathematica

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If Ω is a Lip(1,1/2) domain, μ a doubling measure on $\partial_{p} Ω, \partial / \partial t - L_{i}$ , i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures $ω_{0}$ , $ω_{1}$ have the property that $ω_{0} \in B^{q} (μ)$ implies $ω_{1}$ is absolutely continuous with respect to $ω_{0}$ whenever a certain Carleson-type condition holds on the difference function of the coefficients of $L_{1}$ and $L_{0}$ . Also $ω_{0} \in B^{q} (μ)$ implies $ω_{1} \in B^{q} (μ)$ whenever both measures are center-doubling measures. This is B. Dahlberg’s result for elliptic measures...

Parabolic Marcinkiewicz integrals on product spaces and extrapolation

Mohammed Ali, Mohammed Al-Dolat (2016)

Open Mathematics

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In this paper, we study the the parabolic Marcinkiewicz integral [...] MΩ,hρ1,ρ2 $ℳ_{Ω, h}^{ρ_{1,} ρ_{2}}$ on product domains Rn × Rm (n, m ≥ 2). Lp estimates of such operators are obtained under weak conditions on the kernels. These estimates allow us to use an extrapolation argument to obtain some new and improved results on parabolic Marcinkiewicz integral operators.

Extension of a result of Ladyženskaja and Uralceva concerning second-order parabolic equations of arbitrary order

Wolf von Wahl (1983)

Annales Polonici Mathematici

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Solvability problem for strong-nonlinear nondiagonal parabolic system

Arina A. Arkhipova (2002)

Mathematica Bohemica

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A class of $q$ -nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities $q \in (1, 2)$ , $q = 2$ , $q > 2$ , are analyzed.

Estimates of weak solutions to nondiagonal quasilinear parabolic systems

Dmitry Portnyagin (2005)

Annales Polonici Mathematici

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$L^{\infty}$ -estimates of weak solutions are established for a quasilinear non-diagonal parabolic system with a special structure whose leading terms are modelled by p-Laplacians. A generalization of the weak maximum principle to systems of equations is employed.

Some theorems on the estimate and existence of solutions of integro-differential equations of parabolic type

H. Ugowski (1972)

Annales Polonici Mathematici

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Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

Pernilla Johnsen, Tatiana Lobkova (2018)

Applications of Mathematics

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This paper is devoted to the study of the linear parabolic problem $ε \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x / ε, t / ε^{3}) \nabla u_{ε} (x, t)) = f (x, t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $ε$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic...

Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions

Changchun Liu (2013)

Annales Polonici Mathematici

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We consider an initial-boundary problem for a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures. Using Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions for the problem in two space dimensions.

Controllability of a parabolic system with a diffuse interface

Jérôme Le Rousseau, Matthieu Léautaud, Luc Robbiano (2013)

Journal of the European Mathematical Society

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We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain or on a Riemannian manifold, where the transmission conditions involve an additional parabolic operator on the interface. This system is an idealization of a three-layer model in which the central layer has a small thickness $δ$ . We prove a Carleman estimate in the neighborhood of the interface for an associated elliptic operator by means of partial estimates in several microlocal...

On the long-time behaviour of solutions of the p-Laplacian parabolic system

Paweł Goldstein (2008)

Colloquium Mathematicae

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Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be $¹_{l o c}$ , and in the variable exponent case, L² and $W^{1, p (x)}$ -weak.

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Yūki Naito (2006)

Banach Center Publications

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We consider a nonlinear parabolic system modelling chemotaxis $u_{t} = \nabla \cdot (\nabla u - u \nabla v)$ , $v_{t} = Δ v + u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

Full regularity of bounded solutions to nondiagonal parabolic systems of two equations

Dmitry Portnyagin (2008)

Applicationes Mathematicae

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Hölder continuity and, basing on this, full regularity and global existence of weak solutions is studied for a general nondiagonal parabolic system of nonlinear differential equations with the matrix of coefficients satisfying special structure conditions and depending on the unknowns. A technique based on estimating a certain function of unknowns is employed to this end.

A note on the paper of Y. Naito

Piotr Biler (2006)

Banach Center Publications

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This note contains some remarks on the paper of Y. Naito concerning the parabolic system of chemotaxis and published in this volume.