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2009•Korean Journal of MathematicsRequires access

LERAY-SCHAUDER DEGREE THEORY APPLIED TO THE PERTURBED PARABOLIC PROBLEM

Tacksun Jung, Q-Heung Choi

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Abstract

We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the flnite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

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What this paper is about

We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the flnite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

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Available abstract

We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the flnite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

Key concepts: Mathematics, Degree (music), Mathematical analysis, Eigenvalues and eigenvectors, Hilbert space, Boundary value problem, Laplace operator, Dirichlet boundary condition

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