I am cross-posting this question https://physics.stackexchange.com/questions/590549/possible-applications-for-this-semi-linear-first-order-pde
First of all I should say that I work in probability and my knowledge about PDEs is quite small, so this question could make little sense, please let me know if something is not well stated.
While dealing with approximations methods for SDEs a I've noticed a particular connection between an SDE and a deterministic PDE of the form:
$$\partial_t u+\sigma(t)\partial_x u=b(t,x,u)$$
where $b$ is a Lipschitz continuous function in $u$.
I've tried searching online for application of this kind of equation but unfortunately I wasn't able to find anything concrete.
In the book by Moussiaux, Zaitsev and Polyanin "Handbook of first order PDEs" they discuss methods for solving this kind of equations but they don't provide examples of applications.
I suspect this could be connected somehow to the transport equations, but I am not entirely sure.
Do you know of some references for applications of this particular equation?
Best Answer
Your equation is quasilinear, but sometimes it can be connected with a semilinear equation $$ v_s +v_x = c(s,x,v). $$ Specifically, define $\Sigma(t) = \int_0^t \sigma (\tau)d\tau$. If $\Sigma$ is an invertible function, for example if $\sigma$ is increasing, then the change of variables $$ s = \Sigma(t), \qquad u(x,t) = v(x,s) $$ gives $$ u_t+\sigma u_x = \sigma(v_s+v_x) = b(\Sigma^{-1}(s),x,v) $$ so dividing by $\sigma$ you have $v_x+v_x = c(s,x,v)$ with $$c(s,x,v) = \frac{b(\Sigma^{-1}(s),x,v)}{\Sigma'(\Sigma^{-1}(s))}$$
Edit: for an example application, suppose $w(x,t)$ is the depth of a layer of particles that are being transported to the right by some process, or perhaps the density of animals at longitude $x$ during an eastward migration. Consider a conservation law of the form $$ \frac{d}{dt}\int_a^{a+h} w(x,t)dx = kw(a,t)-kw(a+h,t) +\int_a^{a+h}c(w,x,t)dx $$ where some function $c(w,x,t)$ represents a source of particles/length, or a birth rate of animals, and $kw$ is the flux of particles from left to right, $k$ being a fraction of the local density that are able to migrate past.
Thus the conservation law says that the time rate of change of the population in the interval $[a,a+h]$ is given by flux in (imigration), minus flux out (emigration), plus the birth rate.
Divide by $h$ and take the limit as $h$ tends to zero; you get by the Fundamental Theorem of Calculus that $$ w_t = -kw_x+c(w,x,t). $$