naa a22 4500 510811299 CHVBK 20180411083443.0 cr unu---uuuuu 180411e20131201xx s 000 0 eng 10.1134/S0081543813080051 doi (NATIONALLICENCE)springer-10.1134/S0081543813080051 Galaktionov V. Department of Mathematical Sciences, University of Bath, BA2 7AY, Bath, UK aut The KPP-problem and log t -front shift for higher-order semilinear parabolic equations [Elektronische Daten] [V. Galaktionov] 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 ℝ × ℝ+ with u 0(x) = H(−x) ≡ 1 for x < 0 and 0 for x ≥ 0 (here H(·) 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 − λ 0 t), with the minimal speed λ 0 = 2, and, in the moving frame with the front shift x f (t) (u(x f (t),t) ≡ 1/2), there is uniform convergence u(x f (t) + y, t) → f(y) as t → +∞, 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 → +∞. 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. Pleiades Publishing, Ltd., 2013 Proceedings of the Steklov Institute of Mathematics Springer US; http://www.springer-ny.com 283/1(2013-12-01), 44-74 0081-5438 283:1<44 2013 283 11501 https://doi.org/10.1134/S0081543813080051 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1134/S0081543813080051 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Galaktionov V. Department of Mathematical Sciences, University of Bath, BA2 7AY, Bath, UK aut NATIONALLICENCE 773 0- Proceedings of the Steklov Institute of Mathematics Springer US; http://www.springer-ny.com 283/1(2013-12-01), 44-74 0081-5438 283:1<44 2013 283 11501 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer