Evolution inclusions governed by the difference of two subdifferentials in reflexive Banach spaces
Let \(V\) be a real reflexive Banach space and let \(V^*\) be its dual. The authors investigate the local and global existence for the Cauchy problem associated with the evolution equation \(f(t)\in u^{\prime}(t) + \partial \phi (u(t)) - \partial \psi (u(t))\) in \(V^*\), \(0<t<T\). For previous related results in Hilbert space setting, they refer the reader to \textit{Y. Koi} and \textit{J. Watanabe}, [Proc. Japan Acad. 52, 413--416 (1976; Zbl 0361.35037); \textit{H. Ishii}, J. Diff. Equations 26, 291--319 (1977; Zbl 0339.34062); \textit{M. Otani}, J. Fac. Sci. Univ. Tokyo, Sec. IA Math. 24, 575--605 (1977; Zbl 0386.47040); \textit{M. Otani}, in: M. Farkas (Ed.), Qualitative Theory of Differential Equations, Vol. I, II, Colloquia Math. Societatis Janos Bolyai, 30, 795--809 (1981; Zbl 0506.35075); \textit{M. Otani}, J. Diff. Equations 46, 268--299 (1982; Zbl 0495.35042)]. The nonlinear heat equation \(u_t - {\Delta}_pu -| u| ^{q-2}u = f(x,t)\) with a Dirichlet boundary condition is studied as an application. Here \({\Delta}_p\) denotes the \(p\)-Laplacian.
scientific article; zbMATH DE number 2136119