Diffusion in a Semi-Infinite Region with Nonlinear Surface Dissipation

The title problem is posed as a linear heat equation in one space dimension $(x > 0)$ and time $(t > 0)$, with a nonlinear radiative-type boundary condition on the surface $(x = 0)$. Existence and uniqueness of a nonnegative solution are shown by a simple, constructive method which leads to some useful bounds. The asymptotic behavior $(t \to \infty )$ is investigated by a formal expansion scheme. For the case in which the nonlinear boundary condition has an asymptotic power law form, a complete description of the asymptotic behavior is provided. Conservation of flux at the surface $(x = 0)$ is also determined in this case. For more general nonlinearities, some extreme cases of asymptotic behavior are examined.

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