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This article is cited in 4 scientific papers (total in 4 papers)
The KPP-problem and $\log t$-front shift for higher-order semilinear parabolic equations
V. A. Galaktionov Department of Mathematical Sciences, University of Bath, Bath, UK
Abstract:
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.
Received in November 2012
Citation:
V. A. Galaktionov, “The KPP-problem and $\log t$-front shift for higher-order semilinear parabolic equations”, Function theory and equations of mathematical physics, Collected papers. In commemoration of the 90th anniversary of Lev Dmitrievich Kudryavtsev's birth, Trudy Mat. Inst. Steklova, 283, MAIK Nauka/Interperiodica, Moscow, 2013, 49–79; Proc. Steklov Inst. Math., 283 (2013), 44–74
Linking options:
https://www.mathnet.ru/eng/tm3500https://doi.org/10.1134/S0371968513040055 https://www.mathnet.ru/eng/tm/v283/p49
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