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

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: $$u_t=-u_{xxxx}\,\,\, {\rm in}\, Q_0\,=\{|x| < R(t), \,\,-1 < t < 0\},$$ 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: $$u = u_x = 0\,\,\, {\rm at}\, x\,\,=\, \pm R(t), \,\, -1 \le t < 0, \quad {\rm and} \quad u(x, -1)=u_0(x).$$ 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: $$\tilde R(t) = 3^{-\frac 34} \, 2^{\frac {11}4}(-t)^{\frac 14} \left[{\rm ln} |{\rm ln}(-t)|\right]^{\frac 34}$$ belongs to the regular case, while any increase of the constant $${3^{-\frac 34} \, 2^{\frac {11}4}}$$ therein leads to the irregular one. The results are based on Hermitian spectral theory of the operator $${{\bf B}^*= -D_y^{(4)}- \frac 14 \,\, y D_y}$$ in $${L^2_{\rho^*}(\mathbb{R})}$$ , where $${\rho^*(y) = {\mathrm e}^{-a |y|^{4/3}}}$$ , $${a={\rm constant} \in (0,3 \cdot 2^{-\frac {8}3})}$$ , together with typical ideas of boundary layers and blow-up matching analysis. Extensions to 2mth-order poly-harmonic equations in $${\mathbb{R}^{N}}$$ and other PDEs are discussed, and a partial survey on regularity/irregularity issues is presented.