Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 4

On a semigroup generated by the heat equation with a nonlinear Neumann boundary condition

Pages: 299 – 319

DOI: http://dx.doi.org/10.4310/DPDE.2014.v11.n4.a1

Author

Ruediger Landes (Department of Mathematics, University of Oklahoma, Norman, Ok., U.S.A.)

Abstract

Recently, a nonlinear boundary value problem of the heat equation has been introduced to describe critical heated boiling regimes. Speetjens et al have conjectured that the solution gives rise to semigroup with an attractor being the unstable manifold of its fixed point set. We verify that the solution has indeed that property. But we point out that the semigroup should be considered acting on $L^2$ and not on $H^1$ as suggested in the conjecture.

Keywords

semigroup, heat equation, nonlinear boundary conditions, longtime behavior of boiling regimes

2010 Mathematics Subject Classification

Primary 14E20, 54C40. Secondary 20C20, 46E25.

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