The KPP-problem and $\log t$-front shift for higher-order semilinear parabolic equations

The seminal paper by Kolmogorov, Petrovskii, and Piskunov (KPP) of 1937 on the travelling wave propagation in the reaction–diffusion equation $u_t=u_{xx}+u(1-u)$ in $\mathbb R\times\mathbb R_+$ with $u_0(x)=H(-x)\equiv1$ for $x<0$ and $0$ for $x\ge0$ (here $H(\cdot)$ is the Heaviside function) opened a new era in the general theory of nonlinear PDEs and various applications. This paper became an encyclopedia of deep mathematical techniques and tools for nonlinear parabolic equations, which, in the last seventy years, were further developed in hundreds of papers and in dozens of monographs. The KPP paper established the fundamental fact that, in the above equation, there occurs a travelling wave $f(x-\lambda _0t)$, with the minimal speed $\lambda_0=2$, and, in the moving frame with the front shift $x_f(t)$ ($u(x_f(t),t)\equiv1/2$), there is uniform convergence $u(x_f(t)+y,t)\to f(y)$ as $t\to+\infty$, where $x_f(t)=2t(1+o(1))$. In 1983, by a probabilistic approach, Bramson proved that there exists an unbounded $\log t$-shift of the wave front in the indicated PDE problem and $x_f(t)=2t-(3/2)\log t(1+o(1))$ as $t\to+\infty$. Our goal is to reveal some aspects of KPP-type problems for higher-order semilinear parabolic PDEs, including the bi-harmonic equation and the tri-harmonic one, $u_t=-u_{xxxx}+u(1-u)$ and $u_t=u_{xxxxxx}+u(1-u)$. Two main questions to study are (i) existence of travelling waves via any analytical/numerical methods and (ii) a formal derivation of the $\log t$-shifting of moving fronts.