I am having some trouble understanding the proper way to take the inverse Laplace transform to obtain a solution. I am following a paper found here [1], which provides the solution to the following PDE, $$ \frac{\partial m}{\partial t} + \frac{\partial m}{\partial a} = -\lambda(a,t)m\\ m(t,0) = B(t)\\ m(0,a) = m_{0}(a) $$ If I take the laplace transform with respect to $t$, using $p$ as the transform variable, I obtain, $$ \frac{\partial \hat{m}}{\partial a} + p\hat{m}-m_{0}(a) = -\lambda(a)\hat{m} $$ I can use integrating factor $e^{\int^{a}_{0}p+\lambda(\alpha)d\alpha}$ to obtain an expression for $\hat{m}$ as, $$ \hat{m}(p,a) = e^{-pa}e^{-\int^{a}_{0}\lambda(\alpha)d\alpha}\int^{a}_{0}m_{0}(\xi)e^{p\xi}e^{\int^{\xi}_{0}\lambda(\alpha)d\alpha}d\xi+\hat{B}(p)e^{-\int^{a}_{0}\lambda(\alpha)d\alpha} $$
In the linked paper, the authors obtain the solution as, $$ m(t,a) = B(t-a)\sigma(a)H(t-a) + m_{0}(a-t)\sigma(t)H(a-t) $$
Where $\sigma(a) = e^{-\int^{a}_{0}\lambda(\alpha)d\alpha}$
Whilst I understand the general formula for inverse laplace transform in this context would be, $$ L^{-1}\{e^{-cs}F(s)\} = f(t-c)H(t-c) $$ Where H is the heaviside function. I do not understand how to apply it in this context. Would someone be able to help me out?