Komarov, M. V. The periodic problem for the generalized Kolmogorov-Petrovskij-Piskunov equation. (English. Russian original) Zbl 0989.35071 Differ. Equ. 37, No. 1, 70-77 (2001); translation from Differ. Uravn. 37, No. 1, 66-72 (2001). The author studies the following periodic problem for the generalized Kolmogorov-Petrovskij-Piskunov equation \[ u_t+P(u) +Ku=0,\;x \in \mathbb{R}^n,\;n\geq 1,\;t\in\Psi, u|_{t=0}= \overline u(x).\tag{1} \] Here \(u (x,t)\) is a real \(2\pi\)-periodic function and \(\Psi\) a time interval, \[ P(u)= \sum^m_{l=2} a_lu^l,\;a_m\neq 0, \]\[ Ku=(2\pi)^{-n} \sum_pK_p\widehat u_p(t) e^{i (p,x)},\;\widehat\varphi_p \equiv\int_\Omega \varphi(x) e^{i(p,x)}dx \] is a linear pseudodifferential operator defined via the Fourier transform, and \(\Omega\) is an \(n\)-dimensional cube with edge length \(2\pi\). The author obtained the following results: If \(\text{Re} K_p\geq-b_0\), \(b_0>0\), \(p\in \mathbb{Z}^n\), and the initial data \(\overline u(x)\) belong to the space \(H^\delta\), \(\delta> {n\over 2}+\alpha\), \(\alpha\) the order of the operator \(K\), then there exists a unique classical solution of the periodic problem (1) on the interval \([0,T]\). If \(\text{Re} K_p\geq b>0\), \(p\in\mathbb{Z}^n\), and the initial data are sufficiently small, then the solution exists globally with respect to the time. Both of the last conditions are necessary; if one of them fails, then the solution may blow up in finite time. Reviewer: Zeng Yuesheng (Huaihua) MSC: 35K55 Nonlinear parabolic equations 35B10 Periodic solutions to PDEs Keywords:periodicity in the space variable; pseudodifferential operator; Fourier transform × Cite Format Result Cite Review PDF Full Text: DOI