Convergence to a traveling wave solution for the Burgers-FKPP equation

We study the large time behavior of solutions for the Burgers-FKPP equation. When the coefficient $\beta$ of the Burgers nonlinearity increases, it leads to a phase transition from pulled fronts to pushed fronts. By introducing a novel nonlinear, nonlocal transformation, we capture the criticality of phase transitions at $\beta=2$. With that, we can show the convergence of a solution to a single traveling wave in the Burgers-KPP equation for all $\beta$. Furthermore, we discuss the spreading speeds of solutions for the Keller-Segel-FKPP equation, which involves a nonlocal drift term describing chemotaxis. We will show how our new approach can improve the spreading speed results.