Unlike in your earlier question, the function $u(t,x)$ is analytic in $\xi$, so there are no complications arising from a nonzero real part of $\xi$.

You can simply invert the Fourier transform of $\exp\bigl[-t\bigl((\omega+i\xi)^2+k^2\bigr)\bigr]$ for any complex $\xi$, to arrive at $$u(t,x_4,r) = e^{\xi x_4} e^{-(x_4^2+r^2)/4t}\frac{1}{4t^2}.$$ This decays at large $x_4$ or large $r$ for any $t>0$, irrespective of whether $\xi$ is real or imaginary.

The integral $\int_0^\infty u(t,x_4,r)\,dt$ does not decay when ${\rm Re}\,\xi\neq 0$, but there is no reason it should.

Unlike in your earlier question, the function $u(t,x)$ is analytic in $\xi$, so there are no complications arising from a nonzero real part of $\xi$.

You can simply invert the Fourier transform of $\exp\bigl[-t\bigl((\omega+i\xi)^2+k^2\bigr)\bigr]$ for any complex $\xi$, to arrive at $$u(t,x_4,r) = e^{\xi x_4} e^{-(x_4^2+r^2)/4t}\frac{1}{4t^2}.$$ This decays at large $x_4$ or large $r$ for any $t>0$, irrespective of whether $\xi$ is real or imaginary.

The integral $\int_0^\infty u(t,x_4,r)\,dt$ does not decay as a function of $x_4$ when ${\rm Re}\,\xi\neq 0$, but there is no reason it should.