Consider function $u:[0,T]\times \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following partial differential equation: $$ u_t(t,x)=au(t,x)-bu_x(t,x)-cu_{xx}(t,x) \quad \text{with} \quad u(T,x)=1, $$ where $a,b,c\in\mathbb{R}$.

My attempts: By Separation of variables (https://en.wikipedia.org/wiki/Separation_of_variables), let $u(t,x)=S(x)T(t)$ where $S(x)$ and $T(t)$ are functions. Then the PDE becomes: $$ \frac{T'(t)}{T(t)}=a-b\frac{S'(x)}{S(x)}-c\frac{S''(x)}{S(x)}. $$ Since both sides depends on seperate variables, then they must equaly to some constants say $-\lambda$. Then we obtain the ODEs: $$\frac{T'(t)}{T(t)}=-\lambda$$ and $$a-b\frac{S'(x)}{S(x)}-c\frac{S''(x)}{S(x)}=-\lambda.$$ Suppose $\lambda>0$, then $T(t)=e^{-\lambda (T-t)}$. Then the terminal condition implies $S(x)T(T)=S(x)=1$, which means $S(x)=1$ for all $x\in\mathbb{R}$ and $u(t,x)=e^{-\lambda (T-t)}$. However, this is a trivial solution. I wonder what approach would yield a general solution to this specific PDE?



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