Shin's parabolic operator, data, and initial-lateral boundary

Let $U\subset {\mathbb{R}}^d$ be nonempty, bounded, and open, let $T>0$, and set $Q=U\times (0,T)$ and ${\mathrm{\Gamma }}_T=(\bar{U}\times \{0\})\cup (\mathrm{\partial }U\times [0,T])\mathrm{.}$ The final-time face $U\times \{T\}$ is excluded from ${\mathrm{\Gamma }}_T$. For time-dependent coefficients define $L_t[u]={\sum }_{i,j=1}^d{D}_i(a_{ij}(x,t)D_{j}u)+{\sum }_{i=1}^d{D}_i(b_i(x,t)u)+{\sum }_{i=1}^d{c}_i(x,t)D_{i}u+d(x,t)u\mathrm{.}$ Where the coefficient derivatives exist, write its nondivergence expansion as $L_t[u]={\sum }_{i,j}{a}_{ij}{D}_i{D}_{j}u+{\sum }_i{\tilde{b}}_i{D}_{i}u+\tilde{c}u,$ where ${\tilde{b}}_i={\sum }_j{D}_j{a}_{ji}+b_i+c_i$ and $\tilde{c}={\sum }_i{D}_i{b}_i+d$. For symmetric $a$, the drift also equals ${\sum }_j{D}_j{a}_{ij}+b_i+c_i$, the form used by the symmetric source consumers. The initial-boundary value problem uses the sign convention $-u_t+L_t[u]=f\text{in }Q,u=\phi \text{on }\mathrm{\partial }U\times [0,T],u=g\text{on }\bar{U}\times \{0\},$ with compatibility $g(x)=\phi (x,0)$ on $\mathrm{\partial }U$. Its residual for a trial function $h$ is $f+h_t-L_t[h]$, and its prescribed-data error is measured on ${\mathrm{\Gamma }}_T$.

For the source notation $C_{2+\alpha }(Q)$, use the parabolic distance $d_p((x,t),(y,s))=(\mathrm{\mid }x-y{\mathrm{\mid }}^2+\mathrm{\mid }t-s\mathrm{\mid }{)}^{1\mathrm{/}2}\mathrm{.}$ Let $B=U\times \{0\}$ and $S_{\tau }=\mathrm{\partial }U\times [0,\tau ]$. For $P=(x,\tau )\in Q$, set $\delta (P)={\inf }_{R\in B\cup S_{\tau }}{d}_p(P,R)$ and ${\delta }_{P,Q}=\min \{\delta (P),\delta (Q)\}$. For an integer $m\ge 0$, define $\mathrm{\mid }v{\mathrm{\mid }}_{m,\alpha }={\sup }_{P\in Q}\delta (P{)}^m\mathrm{\mid }v(P)\mathrm{\mid }+{\sup }_{P\mathrm{\ne }Q\in Q}{\delta }_{P,Q}^{m+\alpha }\frac{\mathrm{\mid }v(P)-v(Q)\mathrm{\mid }}{d_p(P,Q{)}^{\alpha }}\mathrm{.}$ The class $C_{2+\alpha }(Q)$ consists of functions for which $\mathrm{\mid }u{\mathrm{\mid }}_{0,\alpha }+{\sum }_i\mathrm{\mid }{D}_{i}u{\mathrm{\mid }}_{1,\alpha }+{\sum }_{i,j}\mathrm{\mid }{D}_{ij}u{\mathrm{\mid }}_{1,\alpha }+\mathrm{\mid }{u}_t{\mathrm{\mid }}_{2,\alpha }<\mathrm{\infty },$ with the indicated derivatives continuous in $Q$. This fixes the source's weighted parabolic Hölder convention.

Shin et al., equations (2.1)–(2.3) and Assumption 3.4: parabolic sign convention, data, and coefficients

Published parabolic-cylinder definition: initial and lateral boundary, excluding the final-time face

The source displays ${\sum }_{i,j}{D}_i(a_{ij}{D}_{j}u+b_{i}u)$ while its expanded coefficients and energy argument count $D_i(b_{i}u)$ only once. Read literally, that displayed sum instead gives $d{\sum }_i{D}_i(b_{i}u)$ and changes both the drift and potential. The active definition above separates the two sums explicitly and agrees with the stated expanded coefficients. The original printed expression is retained here as a source discrepancy, not used as an algebraic identity.