This is my first question, I hope it's not a duplicate or too easy:
I have a first order PDE for a function $\omega(x,t)$ of position $x \in \mathbb{R}$ and time $t \in [0,\infty]$: $$ \partial_t \omega = -\ln(1+e^{x}) - v \partial_x \omega + \frac{D}{2} (\partial_x \omega)^2 $$ Where $v,D>0$ are real positive constants. The initial condition are: $$ \omega(x,t=0) = -(x - \mu)^2 $$ With again $\mu>0$ real positive constant. I am interested in studying the evolution of the maximum of $\omega(x,t)$ over time. At initial time $\omega(x,t=0)$ has maximum at $x=\mu$. As time goes on how does the maximum evolve? Can I write a differential equation for its evolution? I believe that this maximum will converge to a fixed value at infinite time.
Thanks to you in advance!
Ps: ultimately my aim is finding the convergence value of the maximum at infinite time (hoping there is one). Any technique which allows to find this would do the job!
You can solve the auxiliary problem without the $-\ln(1+e^x)$ source term using separation of variables and the use Green's Functions to include the source term.
References:
M. N. Ozisik, Heat Conduction, 2nd Edition, J.Wiley & Sons, 1993. (See Chapter 6)
H.S. Carslaw & Jaeger, Conduction of Heat in Solids, Oxford Clarendon Press, 1959.