Atlas: Eigenfunction expansion method for semilinear parabolic equations in circular domains by Vladimir Varlamov

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5th International ISAAC Congress
July 25-30, 2005
Department of Mathematics and Informatics, University of Catania
Catania, Sicily, Italy

Organizers
International ISAAC Board, Local organizing committee: F. Nicolosi (chairman), S. Bonafede, V. Cataldo, P. Cianci, G.R. Cirmi, S. D'Asero, G. Fiorito, L. Giusti, S. Leonardi, P.E. Ricci

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Eigenfunction expansion method for semilinear parabolic equations in circular domains
by
Vladimir Varlamov
University of Texas - Pan American, Edinburg, Texas, USA

An initial-boundary-value problem in a unit disc W is considered for the nonhomogeneous semilinear heat equation with a quadratic nonlinearity, namely

u_t=Du+u²+af

u ê
ê
t=0
=0, u ê
ê
¶W
=0,

u(r, q+2p, t)=u(r, q, t).

Here f=f(r, q, t) Î C_b(R⁺, L₂(W)) is a real function written in polar coordinates and representing the source term and a is a control parameter which should not exceed a certain constant in order not to cause the blow up. Mild solutions are defined as solutions of the corresponding integral equation. These solutions are constructed in the form of a series of eigenfuncitons of the Laplace operator in a disc. Existence and uniqueness of mild solutions in the Sobolev spaces H₀^s(W), s < 3/2, follow from the convergence of the series in this space. The results are obtained via the analysis of the nonlinear integral equations. Special functions generated by the nonlinearity are employed in order to establish the regularity of solutions. The same method can be applied for studying other semilinear parabolic equations in circular domains, e.g., the damped Boussinesq equation and the Kuramoto-Sivashinsky equation.

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Date received: April 27, 2005