Boundary characteristic point regularity for semilinear reaction-diffusion equations: towards an ode criterion |
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Authors: | V A Galaktionov V Maz’ya |
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Institution: | 1.Department of Mathematical Sciences,University of Bath,Bath,UK;2.Department of Mathematical Sciences,M&O Building University of Liverpool,Liverpool,UK;3.Department of Mathematics,Link?ping University,Link?ping,Sweden |
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Abstract: | The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet
conditions is considered. The Petrovskii ( 2?{loglog} ) \left( {2\sqrt {{\log \log }} } \right) criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations
of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after
a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical
system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous
matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type.
Extensions to regularity problems of backward paraboloid vertices in
\mathbbRN {\mathbb{R}^N} are discussed. Bibliography: 54 titles. Illustrations: 1 figure. |
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