Uniform convergence of a $C^2$ sequence with its first and second derivatives on a rectangle passes $C^2$ regularity and termwise differentiation to the limit

Fix $x\in [0,L]$ and apply [F1] to the sequence $t\mapsto u_k(t,x)$ on $[0,T]$ with $c=0$: $u_k(0,x)\to u(0,x)$ and ${\mathrm{\partial }}_t{u}_k(\cdot ,x)\to v(\cdot ,x)$ uniformly on $[0,T]$ (the convergence of ${\mathrm{\partial }}_t{u}_k$ is uniform on all of $R$, hence on the slice). We obtain: $t\mapsto u(t,x)$ is differentiable on $[0,T]$ with ${\mathrm{\partial }}_{t}u(t,x)=v(t,x)$, and $u_k(\cdot ,x)\to u(\cdot ,x)$ uniformly on $[0,T]$. Applying the same statement to ${\mathrm{\partial }}_t{u}_k(\cdot ,x)$ with its $t$-derivative ${\mathrm{\partial }}_t^2{u}_k(\cdot ,x)\to v_2(\cdot ,x)$, $t\mapsto v(t,x)$ is differentiable with ${\mathrm{\partial }}_{t}v(t,x)=v_2(t,x)$.

Fix $t\in [0,T]$ and apply [F1] in the variable $x$ to $x\mapsto u_k(t,x)$ and to $x\mapsto {\mathrm{\partial }}_x{u}_k(t,x)$: $x\mapsto u(t,x)$ has ${\mathrm{\partial }}_{x}u(t,x)=w(t,x)$, and $x\mapsto w(t,x)$ has ${\mathrm{\partial }}_{x}w(t,x)=w_2(t,x)$. Applying [F1] in $x$ to $x\mapsto {\mathrm{\partial }}_t{u}_k(t,x)$ is legitimate because its $x$-derivative equals ${\mathrm{\partial }}_t{\mathrm{\partial }}_x{u}_k$ by [F4]; the limits are $v(t,x)$ and $z(t,x)$, so ${\mathrm{\partial }}_{x}v(t,x)=z(t,x)$. Applying [F1] in $t$ to the sequence $t\mapsto {\mathrm{\partial }}_x{u}_k(t,x)$, whose values converge to $w(t,x)$ and whose derivatives converge uniformly to $z(t,x)$, gives ${\mathrm{\partial }}_{t}w(t,x)=z(t,x)$.

The functions $v,w,v_2,z,w_2$ are continuous on $R$ as uniform limits of continuous functions [F2]. Fix $(t_0,x_0)\in R$. By continuity, $v$ and $w$ are bounded on a relative neighbourhood of $(t_0,x_0)$. The integral-evaluation theorem The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, with continuous slice derivatives integrable by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion and reversed endpoints handled by The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$, and steps 1.1-1.2 give, for $(t,x)$ in that neighbourhood, $u(t,x)-u(t_0,x_0)={\int }_{x_0}^{x}w(t_0,s)ds+{\int }_{t_0}^{t}v(s,x)ds\mathrm{.}$ Thus $\mathrm{\mid }u(t,x)-u(t_0,x_0)\mathrm{\mid }\le M_w\mathrm{\mid }x-x_0\mathrm{\mid }+M_v\mathrm{\mid }t-t_0\mathrm{\mid }$ for local bounds $M_w,M_v$, proving continuity of $u$ on $R$, including its boundary. Steps 1.1-1.2 identify the first and second partial derivatives of $u$ with these continuous functions; in particular ${\mathrm{\partial }}_t{\mathrm{\partial }}_{x}u={\mathrm{\partial }}_x{\mathrm{\partial }}_{t}u=z$. By [F3], $u$ is $C^2$ on the interior of $R$ with the stated derivatives, all extending continuously to $R$.