Nonhomogeneous quasilinear partial differential equation

Question

I have such quasilinear PDE: $$\begin{cases}u_t+bDu+au=f, \ \ \ \Bbb R^n\times (0, \infty) \\ u=g,\ \ \ \Bbb R^n\times {t=0}\end{cases}$$

Which I tried to solve but without any results, so any help is appreciated!

Answer 1

Using $u=e^{-at}v$ we transform the equation to $v_t+bDv=e^{at}f$ with $v(0, x)=g(x)$. Note that $au$ term has been eliminated. Now using the method of characteristics we have $\frac{dv}{dt}=e^{at}f$ on $\frac{dx}{dt}=b$. From here we get the family of characteristics $x=bt+\xi$ (straight lines passing through $(0, \xi)$), where $\xi$ is a parameter. Integrating $\frac{dv}{dt}=e^{at}f$ we obtain $v(t, x)=\int_0^te^{a\tau}f(\tau, b\tau+\xi)d\tau+g(\xi)$ with $v(0,\xi)=g(\xi)$ and $\xi=x-bt$. This yields $v(t, x)=\int_0^te^{a\tau}f(\tau, b(\tau-t)+x)d\tau+g(x-bt)$ or $u(t, x)=\int_0^te^{a(\tau-t)}f(\tau, b(\tau-t)+x)d\tau+e^{-at}g(x-bt)$.