 Solution

Use the negative-exponential Half-range Fourier transformFor the boundary traces $g_j(t)={\partial }_x^{j}q(0,t)$, define finite-time spectral boundary transforms $G_j(k,t)={\int }_0^t{e}^{w(k)s}{g}_j(s)ds$. Integrating the conservation law in $x$ and then time gives the global relationThe plus sign of the boundary contribution follows because the integrated flux is $-E(0,t)X(0,t,k)$.

Fourier inversion consequently gives a real-line initial integral plus a boundary integral. To put the latter in the complex plane, defineOrient $\partial D_{+}$ from the left upper infinity, through $i$, to the right upper infinity, with $D_{+}$ on the left. Each $G_j$ is entire, and in the region between the real axis and this contour, $\mathrm{Re}w\ge 0$. The causal factors $e^{-w(t-s)}$ are bounded there and $e^{ikx}$ decays for $x>0$. The Cauchy integral theorem, with the usual large-arc limiting argument, deforms the boundary integral to giveInitial smoothness, spatial decay and corner compatibility justify inversion; if necessary the integrals are first regularized before contour deformation. The unknown Neumann-type traces $g_1,g_2$ will be removed by the global relation.