Consider the following first order linear equations: $$\partial_t u(t,x) = a(t,x)\nabla u(t,x)+b(t,x)u(t,x),u(0,x)=u_0(x).$$ If $a,b$ are real functions, this can be solved by characteristic method. But the problem I'm working on has complex valued coefficents, and $u$ is also complex valued, which I don't know how to do it. Assume $a,b$ are smooth complex valued, I want to know the existence results and local bounds for t near 0 by $a,b,u_0$.
I haven't find any references about it, and if we split it into real and imaginary parts, we obtain a linear PDE system, I checked several books including Courant and Hilbert: Methods of Mathematical Physics_ Partial Differential Equations Taylor: Partial differential equations I,II,III Only some special cases of PDE systems was mentioned.
Thanks a lot for any idea or references!