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I am working with this system of first order PDEs:

\begin{equation} \left\{ \begin{aligned} %Suscettibili &\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,t)S(a,t)\\ %Infection detention: CIN0 - DNA POSITIVE &\frac{\partial{I}(a,t)}{\partial{t}} + \frac{\partial{I}(a,t)}{\partial{a}}= \lambda(a,t)S(a,t)-{\eta}I(a,t)+rL(a,t)\\ %CIN1: persistence- clearance %&\frac{\partial{P}(a,t)}{\partial{t}} + \frac{\partial{P}(a,t)}{\partial{a}}={\eta}I_{f}(a,t)-o(a)P(a,t)\\ &\frac{\partial{L}(a,t)}{\partial{t}} + \frac{\partial{L}(a,t)}{\partial{a}}= {\eta}I(a,t)-rL(a,t)\\ \end{aligned} \right. \end{equation}

where $\eta$, $r$ are constant parameters and with conditions: $$S(0,t)=1,\quad S(a,0)=S^{0},\quad I(0,t)=0, \quad I(a,0)=I^{0}$$ $$\quad L(0,t)=0, \quad L(a,0)=L^{0}$$.

I want to solve it with the method of characteristic but actually I have no clue because it it a system. From the first equation, integrating along the characteristic line t-a = constant I get $$ S(a,t)=e^{-\int_{0}^{a}\lambda(t-a+\sigma,\sigma)d\sigma} $$ But, then how can I procede for solving the other two equations? Any suggestions?

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    $\begingroup$ Adding the second and third equations allows to get $I+L=f$. After that substitute $L=f-I$ in the third equation. $\endgroup$
    – Andrew
    Commented Jun 11, 2016 at 9:33
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    $\begingroup$ There is no generally applicable method of characterstics for first order systems, as far as I know. After all, any PDE system could be reduced to a first order system by introducing more variables and equations. However, in your case, after a change of variables $a=s+b$, $t=s$, your PDE becomes an ODE in $s$ with a parameter $b$. I can't say in advance whether you'll find any "explicit solutions" this way, but at least the usual ODE theory (existence, uniqueness, continuous/smooth dependence on parameters) gives an abstract starting point. $\endgroup$
    – Igor Khavkine
    Commented Jun 11, 2016 at 13:47
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    $\begingroup$ Writing $D = \partial_t + \partial_a$, $X = (S, I, L)$, and $A$ the matrix $$ \begin{pmatrix} - \lambda \\ \lambda & -\eta & r \\ & \eta & - r\end{pmatrix} $$ your equation is the linear system of ODE (as Igor said) $$ DX = AX $$ where the matrix $A$ has variable coefficients. You can still explicitly integrate to get $X = X_0 \exp \int A$. $\endgroup$
    – Willie Wong
    Commented Jun 13, 2016 at 16:37

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