Parabolic Marcinkiewicz integrals on product spaces and extrapolation

Mohammed Ali; Mohammed Al-Dolat

Displaying similar documents to “Parabolic Marcinkiewicz integrals on product spaces and extrapolation”

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...

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...

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

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The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of $p$ -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity

Kentarou Fujie, Tomomi Yokota (2014)

Mathematica Bohemica

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This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $χ (v)$ and the growth term $f (u)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0 < χ (v) \leq χ_{0} / v^{k}$ $(k \geq 1$ , $χ_{0} > 0)$ and $λ_{1} - μ_{1} u \leq f (u) \leq λ_{2} - μ_{2} u$ $(λ_{1}, λ_{2}, μ_{1}, μ_{2} > 0)$ . It is shown that if $χ_{0}$ is sufficiently small, then the system has a unique global-in-time classical solution that is uniformly bounded. This boundedness result is a generalization of a recent result by K. Fujie, M. Winkler, T. Yokota. ...

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 $L^{1}$ .

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...

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.

Characterization of the interior regularity for parabolic systems with discontinuous coefficients

Dian K. Palagachev, Lubomira G. Softova (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We deal in this Note with linear parabolic (in sense of Petrovskij) systems of order $2 b$ with discontinuous principal coefficients belonging to $V M O ⋂ L^{\infty}$ . By means of a priori estimates in Sobolev-Morrey spaces we give a precise characterization of the Morrey, BMO and Hölder regularity of the solutions and their derivatives up to order $2 b - 1$ .

Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces

Ferit Gurbuz (2016)

Open Mathematics

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In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator...

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...

The Calderón-Zygmund theorem and parabolic equations in $L P (ℝ, C^{2 + α})$ -spaces

Nicolai V. Krylov (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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A Banach-space version of the Calderón-Zygmund theorem is presented and applied to obtaining apriori estimates for solutions of second-order parabolic equations in $L_{p} (ℝ, C^{2 + α})$ -spaces.