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Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Cubic Nonlinearities

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Abstract

For a singularly perturbed parabolic equation

\({{\epsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\epsilon )\)

in a rectangle, a problem with boundary conditions of the first kind is considered. It is assumed that, at the corner points of the rectangle, the function \(F\) is cubic in the variable \(u\). A complete asymptotic expansion of the solution at \(\varepsilon \to 0\) is constructed, and its uniformity in a closed rectangle is substantiated.

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ACKNOWLEDGMENTS

I am grateful to V.F. Butuzov for a fruitful discussion of the results obtained.

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Correspondence to I. V. Denisov.

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Translated by E. Chernokozhin

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Denisov, I.V. Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Cubic Nonlinearities. Comput. Math. and Math. Phys. 61, 242–253 (2021). https://doi.org/10.1134/S096554252102007X

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  • DOI: https://doi.org/10.1134/S096554252102007X

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