The Cauchy problem for time-dependent abstract parabolic equations of higher order
Consider the differential equation in a Banach space (E) \(u^{(n)}+\sum^{n-1}_{k=0}A_ k(t)u^{(k)}=f(t),\) where the operators \(A_ k(t)\) are closed, linear and have dense and constant domains. By assuming parabolicity of equation (E), which means that the operator pencil \(\mu^ n+\sum^{n-1}_{k=0}\mu^ kA_ k(t)\) has an everywhere defined bounded inverse for every \(\mu\) in a sector of the complex plane of half-amplitude greater than \(\pi\) /2 and that inverse decreases in a suitable way at infinite in this sector, and Hölder continuity of the coefficients \(A_ k(t)\) and of the r.h.s. f(t), it is proved that the Cauchy problem for (E) has a unique solution, provided the initial data satisfy conditions of Brézis-Fraenkel type [\textit{H. Brézis}, \textit{L. E. Fraenkel}, J. Funct. Anal. 29, 328-335 (1978; Zbl 0388.46026)]. The results are applicable to Petrovskij parabolic partial differential equations.
Petrovskij parabolic partial differential equations
scientific article; zbMATH DE number 4052074
scientific article