Partial differential equations in Banach spaces involving nilpotent linear operators
Tom 65 / 1996
Annales Polonici Mathematici 65 (1996), 67-80
DOI: 10.4064/ap-65-1-67-80
Streszczenie
Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D^{k}_{t}u + ∑_{j=0}^{k-1}∑_{|α|≤m} A_{j,α}(D^{j}_{t} D^{α}_{x}u) = f$ in $ℝ^{n+1}$, ⎨ ⎩ $D^{j}_{t} u(0,x) = φ_j(x)$ in $ℝ^n$, j=0,...,k-1, where each $A_{j,α}$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j,α}$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u ∈ C^∞(ℝ^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of $ℝ^{n+1}$.
