= Solution
The original PDF specifies a bounded solution class $C_b^2$, rather than the $C_0^2$ transcribed in the TeX. Write $\phi(x)=\min(1,\max(0,-x))$ and fix a time $t>0$. For Brownian motion starting at $x$, consider
$$
F_s=\exp\left(-\lambda\int_0^s\phi(B_r)dr\right)u(t-s,B_s),\qquad0\leq s\leq t.
$$
The <Itô formula> and ordinary product rule give a drift proportional to $-u_t+\tfrac12u_{xx}-\lambda\phi u$, which vanishes by the equation. Therefore $F$ is a local martingale. It is bounded on this horizon because $u$ is bounded and $\phi\geq0$, so the <bounded local martingale criterion> makes it a true martingale. Its endpoint values are $F_0=u(t,x)$ and $F_t=e^{-\lambda\int_0^t\phi(B_r)dr}$, using $u(0,\cdot)=1$. Consequently
$$
\boxed{u^\lambda(t,x)=\mathbb E_x\exp\left(-\lambda\int_0^t\phi(B_s)ds\right).}
$$
This proves the required <Feynman-Kac formula with a bounded potential> directly, and also gives uniqueness within the stated bounded solution class.
Let $I=\int_0^t\phi(B_s)ds$. By <dominated convergence>, $e^{-\lambda I}\to\mathbf1_{\{I=0\}}$ in expectation. Continuity of $B$ and $\phi$ shows that $I=0$ exactly when $B_s\geq0$ throughout $[0,t]$: a negative value at any time produces a nontrivial interval with positive integrand. For $x<0$ such an interval occurs immediately, so the limiting probability is zero. For $x>0$, the <Brownian reflection principle> and the atom-free distribution of the running minimum give
$$
\mathbb P_x(\min_{s\leq t}B_s\geq0)=1-2\mathbb P_x(B_t\leq0).
$$
For $x=0$ this probability is zero: the reflection formula for levels $-\varepsilon$, followed by $\varepsilon\downarrow0$, shows that Brownian motion goes below zero before $t$ almost surely. Thus
$$
\boxed{\lim_{\lambda\to\infty}u^\lambda(t,x)=\begin{cases}1-2\mathbb P_x(B_t\leq0),&x\geq0,\\0,&x<0.\end{cases}}
$$
This is the <soft killing limit for Brownian nonnegative survival>. The assertion is pointwise for positive time, as requested; no claim of uniform convergence across the initial-time boundary is needed.
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