Leray-Volevich conditions for systems of abstract evolution equations of Nirenberg/Nishida type (Q1340406)

The author treats the Cauchy problem for a system of abstract evolution equations \(u'(t)= F(u, t)\) for \(t\geq 0\), \(u(0)= 0\), \(u= (u_1,\dots, u_n)\), in a scale of Banach spaces \((B_s, |u|_s)\), \(0< s< s_0\), derived from a scale \((H_s, |\cdot |_s)\), which means that \(H_s\subset H_{s'}\) and \(|\cdot|_{s'}\leq |\cdot |_s\) for \(0< s'< s< s_0\) and \(B_s= H^n_s\), \(|u|^2_s= \sum^n_1 |u_i|^2_s\). It is assumed that \(F= (F_1,\dots, F_n)\) maps a set \(\{u\in B_s: |u|_s< R\}\times [0, \eta]\) continuously into \(B_{s'}\) \((0< s'< s< s_0< \eta)\) and satisfies \[ |F_i(u, t)- F_i(v, t)|_s\leq \sum^n_{j= 1} C_{ij} {|u_j- v_j|_s\over (s- s')^{p_i j}},\quad |F_i(0, t)|_s\leq K/(s_0- s)^{q_i}. \] Under the Leray-Volevich condition \(p_{ij}\leq q_i- q_j+ 1\) there exists for some \(b> 0\), \(s_\infty< s_0\) a unique solution \(u(t)\in C^1([0, b(s_\infty- s)), B_s)\) \((0< s< s_\infty)\) with `conical evolution', which means that \[ \sup_{s, t, i} |u_i(t)|_s (b(s_\infty- s)- t)^{q_i}< \infty. \] A similar theorem for `cylindrical evolution', where the solution belongs to \(C^1([0, T], B_s)\) for all \(s\), holds under the stronger assumption \(p_{ij}\leq q_i- q_j\). These results belong to the abstract Cauchy-Kowalewsky theory; concrete examples are given.