Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions

Ciprian G. Gal; Mahamadi Warma

doi:10.3934/eect.2016.5.61

Article Contents

2016, Volume 5, Issue 1: 61-103. Doi: 10.3934/eect.2016.5.61

This issue Previous Article Energy decay rates for solutions of the wave equation with linear damping in exterior domain Next Article The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions

Ciprian G. Gal^1, and
Mahamadi Warma^2,

1.
Department of Mathematics, Florida International University, Miami, FL, 33199
2.
University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received: June 30, 2015

Revised: October 31, 2015

Published: February 2016

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Abstract

We investigate a class of semilinear parabolic and elliptic problems with fractional dynamic boundary conditions. We introduce two new operators, the so-called fractional Wentzell Laplacian and the fractional Steklov operator, which become essential in our study of these nonlinear problems. Besides giving a complete characterization of well-posedness and regularity of bounded solutions, we also establish the existence of finite-dimensional global attractors and also derive basic conditions for blow-up.

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Mathematics Subject Classification: Primary: 35J92, 35A15, 35B41; Secondary: 35K65.

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