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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\begin{document}$$u_t=-u_{xxxx}\,\,\, {\rm in}\, Q_0\,=\{|x| < R(t), \,\,-1 < t < 0\},$$\end{document}where R(t) > 0 is a smooth function on [−1, 0) and R(t) → 0+ as t → 0−. The zero Dirichlet conditions on the lateral boundary of Q0 and bounded initial data are posed:
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\begin{document}$$u = u_x = 0\,\,\, {\rm at}\, x\,\,=\, \pm R(t), \,\, -1 \le t < 0, \quad {\rm and} \quad u(x, -1)=u_0(x).$$\end{document}The boundary point (0, 0) is then regular (in Wiener’s sense) if u(0, 0−) = 0 for any data u0, and is irregular otherwise. The proposed asymptotic blow-up approach shows that: for the backward fundamental parabolae with \documentclass[12pt]{minimal}
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\begin{document}$${R(t)=l(-t)^{1/4}}$$\end{document} , the regularity of its vertex (0, 0) depends on the constant l > 0: e.g., l = 4 is regular, while l = 5 is not;for \documentclass[12pt]{minimal}
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\begin{document}$${R(t)=(-t)^{1/4} \varphi(-{\rm ln} (-t))}$$\end{document} with \documentclass[12pt]{minimal}
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\begin{document}$${\varphi(\tau) \to +\infty}$$\end{document} as τ → + ∞, regularity/irregularity of (0, 0) can be expressed in terms of an integral Petrovskii-like (Osgood–Dini) criterion. E.g., after a special “oscillatory cut-off” of the boundary, the function\documentclass[12pt]{minimal}
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\begin{document}$$\tilde R(t) = 3^{-\frac 34} \, 2^{\frac {11}4}(-t)^{\frac 14} \left[{\rm ln} |{\rm ln}(-t)|\right]^{\frac 34}$$\end{document}belongs to the regular case, while any increase of the constant \documentclass[12pt]{minimal}
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\begin{document}$${3^{-\frac 34} \, 2^{\frac {11}4}}$$\end{document} therein leads to the irregular one. The results are based on Hermitian spectral theory of the operator \documentclass[12pt]{minimal}
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\begin{document}$${{\bf B}^*= -D_y^{(4)}- \frac 14 \,\, y D_y}$$\end{document} in \documentclass[12pt]{minimal}
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\begin{document}$${L^2_{\rho^*}(\mathbb{R})}$$\end{document} , where \documentclass[12pt]{minimal}
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\begin{document}$${\rho^*(y) = {\mathrm e}^{-a |y|^{4/3}}}$$\end{document} , \documentclass[12pt]{minimal}
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\begin{document}$${a={\rm constant} \in (0,3 \cdot 2^{-\frac {8}3})}$$\end{document} , together with typical ideas of boundary layers and blow-up matching analysis. Extensions to 2mth-order poly-harmonic equations in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{R}^{N}}$$\end{document} and other PDEs are discussed, and a partial survey on regularity/irregularity issues is presented.
