Dirichlet problem for parabolic equations on Hilbert spaces

Anna Talarczyk

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The $L^{p}$ solvability of the Dirichlet problems for parabolic equations

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For two general second order parabolic equations in divergence form in Lip(1,1/2) cylinders, we give a criterion for the preservation of $L^{p}$ solvability of the Dirichlet problems.

Quasilinear parabolic equations

D. E. Edmunds, L. A. Peletier (1971)

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The quasilinear parabolic kirchhoff equation

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In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.

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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.

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We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space.

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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.

New a priori estimates for nondiagonal strongly nonlinear parabolic systems

Arina Arkhipova (2008)

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We consider nondiagonal elliptic and parabolic systems of equations with quadratic nonlinearities in the gradient. We discuss a new description of regular points of solutions of such systems. For a class of strongly nonlinear parabolic systems, we estimate locally the Hölder norm of a solution. Instead of smallness of the oscillation, we assume local smallness of the Campanato seminorm of the solution under consideration. Theorems about quasireverse Hölder inequalities proved by the...

Qualitative properties of coupled parabolic systems of evolution equations

Stefano Cardanobile, Delio Mugnolo (2008)

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We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, and a heat equation with dynamic boundary conditions.

Concerning solutions of an exterior boundary-value problem for a system of non-linear parabolic equations

P. Besala (1964)

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The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are hölder continuous with respect to space variables

Davide Guidetti (1996)

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

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We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution $u$ which is bounded with values in $C^{2 + θ} (\bar{Ω})$ (0 < < 1), with $\partial_{t} u$ bounded with values in $C^{θ} (\bar{Ω})$ .

On a certain non-linear initial-boundary value problem for integro-differential equations of parabolic type

H. Ugowski (1973)

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