Variable-coefficient parabolic initial-boundary existence and residual stability

Assume Countable Choice and the Axiom of Choice. Let $U\subset {\mathbb{R}}^d$ be nonempty, bounded and open, $T>0$, $Q=U\times (0,T)$, and $0<\alpha <1$. Use the intended divergence operator $L_{t}u={\sum }_{i,j}{D}_i(a_{ij}{D}_{j}u)+{\sum }_i{D}_i(b_{i}u)+{\sum }_i{c}_i{D}_{i}u+du,$ and its spatially expanded coefficients ${\tilde{b}}_j={\sum }_i{D}_i{a}_{ij}+b_j+c_j,\tilde{c}={\sum }_i{D}_i{b}_i+d\mathrm{.}$ Assume $a,b$ have the pointwise spatial derivatives needed for the expanded coefficients, $a=a^T\ge {\lambda }_{0}I$ with ${\lambda }_0>0$, and every spatial boundary point has an exterior tangent ball. The classical equation is understood through these expanded nondivergence coefficients. Assume $a,\tilde{b},\tilde{c},f$ have finite source order-zero weighted parabolic Hölder norms and the additional source weighted bounds $\mathrm{\mid }\delta \tilde{b}{\mathrm{\mid }}_{\alpha },\mathrm{\mid }{\delta }^2\tilde{c}{\mathrm{\mid }}_{\alpha },\mathrm{\mid }{\delta }^{2}f{\mathrm{\mid }}_{\alpha }<\mathrm{\infty }$. Here the order-zero convention is $\mathrm{\mid }v{\mathrm{\mid }}_{0,\alpha }={\sup }_{P\in Q}\mathrm{\mid }v(P)\mathrm{\mid }+{\sup }_{P\mathrm{\ne }R\in Q}{\delta }_{P,R}^{\alpha }\frac{\mathrm{\mid }v(P)-v(R)\mathrm{\mid }}{d_p(P,R{)}^{\alpha }},d_p(P,R)=(\mathrm{\mid }x-y{\mathrm{\mid }}^2+\mathrm{\mid }t-s\mathrm{\mid }{)}^{1\mathrm{/}2},$ with $\delta (P)=\min \{\mathrm{dist}(x,\mathrm{\partial }U),\sqrt{t}\}$ and ${\delta }_{P,R}=\min \{\delta (P),\delta (R)\}$. Assume the initial datum $g$ and lateral datum $\phi$ are Hölder continuous with compatible values $g(x)=\phi (x,0)$ on $\mathrm{\partial }U$. Retain the source barrier assumption that some $\theta >0$ satisfies ${\theta }^2{a}_{11}+\theta {\tilde{b}}_1\ge 1$ and its coefficient bounds ${\sum }_{i,j}\mathrm{\mid }{a}_{ij}{\mathrm{\mid }}^2\le \mathrm{\Lambda },{\lambda }_0^{-2}(\mathrm{\mid }b{\mathrm{\mid }}^2+\mathrm{\mid }c{\mathrm{\mid }}^2)+{\lambda }_0^{-1}\mathrm{\mid }d\mathrm{\mid }\le v^2\mathrm{.}$ Then the problem $-u_t+L_{t}u=f$, $u=g$ initially and $u=\phi$ laterally has a unique solution $u^{\ast }\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+\alpha ,1+\alpha \mathrm{/}2}(U\times (0,T])\cap C(\bar{U}\times [0,T])\mathrm{.}$ Write $\psi$ for the combined prescribed datum and ${\mathrm{\Gamma }}_T=(\bar{U}\times \{0\})\cup (\mathrm{\partial }U\times [0,T])$. If $\tilde{c}\le 0$, every $h\in C^{2,1}(Q)\cap C(\bar{U}\times [0,T])$ with finite residual norm satisfies $\mathrm{\parallel }h-u^{\ast }{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }h-\psi {\mathrm{\parallel }}_{C({\mathrm{\Gamma }}_T)}+T\mathrm{\parallel }f+h_t-L_{t}h{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ Consequently any sequence whose two right-hand errors tend to zero converges uniformly to $u^{\ast }$. If it exactly satisfies the initial and lateral conditions, its residuals are also locally parabolic Hölder, and the displayed divergence-to-nondivergence expansion agrees in distributions locally, its errors $e_m=h_m-u^{\ast }$ belong to $L^2(0,T;H_0^1(U))$ and satisfy $\mathrm{\parallel }{e}_m{\mathrm{\parallel }}_{L^2(0,T;H^1(U))}\to 0$. If additionally $u^{\ast }\in L^2(0,T;H^1(U))$, as holds for zero initial and lateral data, then $h_m$ also belong to that space and converge strongly there to $u^{\ast }$. Hölder prescribed data alone do not assert this additional membership of the individual functions. In particular these deterministic implications apply on every probability-one event where a PINN residual-consistency theorem supplies the stated residual and prescribed-data limits.

Shin et al., Assumption 3.4, Theorem 3.6 and Appendices F–G

The order-zero weighted Hölder requirement makes the source word “Hölder” precise through its equation (2.7), including its supremum term. If instead the original statement intended merely local Hölder continuity with only the positive-order weighted bounds, this proof does not cover that larger class: the heat equation with source $\sin (x)\mathrm{/}t$ on $(0,\pi )$ and zero initial data has no continuous zero-initial solution. Against a nonnegative compactly supported spatial test its equation would equate a time derivative of a bounded quantity plus a bounded quantity to a positive constant divided by $t$, contradicting integration near zero. The intended interpretation must be approved explicitly, not silently substituted.

The source assumes pointwise spatial differentiability. Distributional compatibility is needed only by the additional energy clause, not by classical existence or sup-norm stability. It is automatic for local spatial $C^1$ coefficients, but must be stated if that word is interpreted more broadly. The source's printed double sum places $b_{i}u$ inside the $i,j$ sum and repeats it $d$ times; this item uses the separately indexed divergence term matching its own displayed expanded coefficients.

The exact source trial-class convention weights $D_{ij}h$ by $\delta$ rather than ${\delta }^2$. It is retained as a source convention in the definition; neither existence nor this proof asserts that the solution has that stronger weighted norm. The stability conclusion applies to its classical trial members, and in fact to the larger classical class explicitly stated here.

The source barrier and divergence coefficient bounds are retained even where the existence construction does not need their full strength. The deterministic consistency clause is the PDE interface used by the separate regularized-PINN consumer; it is not a new sampling, global-minimizer, or optimizer theorem. The full Shin definitions, Assumption 3.4 and Appendix F were read. Its cited Friedman theorem was not represented as a local proof or as newly read full text.