On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach |
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| Authors: | V A Galaktionov |
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| Institution: | 1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK
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| Abstract: | The classic problem of regularity of boundary points for higher-order partial differential equations (PDEs) is concerned.
For second-order elliptic and parabolic equations, this study was completed by Wiener’s (J. Math. Phys. Mass. Inst. Tech.
3:127–146, 1924) and Petrovskii’s (Math. Ann. 109:424–444, 1934) criteria, and was extended to more general equations including
quasilinear ones. Since the 1960–1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat’ev and Maz’ya. However, the higher-order parabolic ones, with infinitely oscillatory
kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D
bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon:
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ut=-uxxxx in Q0 ={|x| < R(t), -1 < t < 0},u_t=-u_{xxxx}\,\,\, {\rm in}\, Q_0\,=\{|x| < R(t), \,\,-1 < t < 0\}, |
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