Let $f \in C^{0}([0, \infty))$ satisfy $f(0)=0$. Find an integral representation for the solution of the problem: $$ \left\{\begin{array}{ll} u_{t}-u_{x x}=0, & \text { in }(0, \infty) \times(0, \infty) \\ u(x, 0)=f(x), & \text { for } x \in[0, \infty) \\ u(0, t)=0, & \text { for } t \in[0, \infty) \end{array}\right. $$
I have no idea how to start. I know the representation integrals for the homogeneous problem (containing the heat kernel) and Duhamel's principle for the non-homogenous problem. But is there any connection to inifinte propagation speed? if $f$ starts from zero doesn't it say that $u$ is identically zero? would be grateful for a clue.