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It is shown that, for the bi-harmonic equation, an optimal regularitycriterion of the vertex of typical paraboloids can be expressed in terms ofOsgood-Dini integral conditions of Petrovskii's type for the heat equationderived in 1934. Some extensions of the first Fourier coefficent method toother PDEs are discussed. A survey on boundary point regularity for ellipticand parabolic PDEs is enclosed.

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach