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

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: u(t) = -u(xxxx) in Q(0) = {vertical bar x vertical bar < R(t), -1 < t < 0}, 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 Q(0) and bounded initial data are posed: u = u(x) = 0 at x = +/- R(t), -1 <= t < 0, and u(x, -1) = u(0)(x). The boundary point (0, 0) is then regular (in Wiener's sense) if u(0, 0(-)) = 0 for any data u(0), and is irregular otherwise. The proposed asymptotic blow-up approach shows that: (i) for the backward fundamental parabolae with R(t) = l(-t)(1/4), the regularity of its vertex (0, 0) depends on the constant l > 0: e.g., l = 4 is regular, while l = 5 is not; (ii) for R(t) = (-t)(1/4)phi(-ln(-t)) with phi(tau) -> +infinity as tau -> +infinity, 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 (R) over tilde = 3(-3/4)2(11/4)(-t)(1/4) [ln vertical bar ln(-t)vertical bar](3/4) belongs to the regular case, while any increase of the constant 3(-3/4)2(11/4) therein leads to the irregular one. The results are based on Hermitian spectral theory of the operator B* = -D-y((4)) - 1/4 yD(y) in L-rho*(2)(R), where rho*(y) = e(-a vertical bar y vertical bar 4/3), a = constant is an element of (0, 3.2(-8/3)), together with typical ideas of boundary layers and blow-up matching analysis. Extensions to 2mth-order poly-harmonic equations in RN and other PDEs are discussed, and a partial survey on regularity/irregularity issues is presented.