Consider the following problem: $u_t = ku_{xx}$ on the semi-infinite strip $S = [0,\infty)\times [0,\infty)$, with the $zero$ initial condition $u(x,0) = 0,\, u(0,t)=g(t).$
I tried both the separation of variables and the fundamental solution methods, but both leads me to conclude the $u\equiv 0$, because of the zero initial condition. Usually, the initial conditions look something like: $$u(x,0) = f(x),$$ where $f(x)$ admits a Fourier Series. But in our case $f\equiv 0$ and I don't know how to circumvent this.
This is an old question from a PhD qual exam, so I highly doubt that the answer is just the identically zero function.
Best Answer
Separation of variables gives $$ \frac{T'}{kT} = -s^2 = \frac{X''}{X} \\ T(0)=0. $$ So $T(t)= e^{-ks^2 t}-1$ and $X(x) = A(s)\cos(s x)+B(s)\sin(sx)$, leading to $$ u(x,t) = \int_{0}^{\infty} (e^{-ks^2t}-1)(A(s)\cos(sx)+B(s)\sin(sx))ds. $$ $u(x,0)=0$ works out. The condition $u(0,t)=g(t)$ allows $B\equiv 0$, provided $A$ satisfies $$ g(t) = \int_{0}^{\infty}(e^{-ks^2 t}-1)A(s)ds. $$