Global smooth solutions for a quasilinear fractional evolution equation
The authors prove the global existence of smooth solutions for quasilinear fractional evolution equations of the form \[ u_t(t, x)= \int^t_0 g_\beta(t- s)[\sigma(u_x(s, x))_x+ f(s, x)]\,ds,\tag{1} \] where \(\sigma\) is smooth and strictly increasing, and the convolution kernel is given by \[ g_\beta(t):= t^{\beta- 1}/\Gamma(\beta)\quad t> 0,\quad \beta\in (0,1). \] The proof is based on rewriting (1) as a differential equation of fractional order \(\alpha:=\beta+1\) and resorting to \(L^p (L^q)\) maximal regularity results for the corresponding linear version of (1). The paper generalizes earlier results by \textit{G. Gripenberg} [J. Differ. Equations 102, 382--390 (1993; Zbl 0780.45012)] and \textit{H. Engler} [Trans. Am. Math. Soc. 348, 267--290 (1996; Zbl 0848.45002)].
quasilinear fractional evolution equation
scientific article; zbMATH DE number 1960720