# Partial differential equations in Banach spaces involving nilpotent linear operators

• Volume: 65, Issue: 1, page 67-80
• ISSN: 0066-2216

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## Abstract

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Let E be a Banach space. We consider a Cauchy problem of the type ⎧ ${D}_{t}^{k}u+{\sum }_{j=0}^{k-1}{\sum }_{|\alpha |\le m}{A}_{j,\alpha }\left({D}_{t}^{j}{D}_{x}^{\alpha }u\right)=f$ in ${ℝ}^{n+1}$, ⎨ ⎩ ${D}_{t}^{j}u\left(0,x\right)={\phi }_{j}\left(x\right)$ in ${ℝ}^{n}$, j=0,...,k-1, where each ${A}_{j,\alpha }$ is a given continuous linear operator from E into itself. We prove that if the operators ${A}_{j,\alpha }$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in {C}^{\infty }\left({ℝ}^{n+1},E\right)$ whose derivatives are equi-bounded on each bounded subset of ${ℝ}^{n+1}$.

## How to cite

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Antonia Chinnì, and Paolo Cubiotti. "Partial differential equations in Banach spaces involving nilpotent linear operators." Annales Polonici Mathematici 65.1 (1996): 67-80. <http://eudml.org/doc/269972>.

@article{AntoniaChinnì1996,
abstract = {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 $ℝ^\{n+1\}$.},
author = {Antonia Chinnì, Paolo Cubiotti},
journal = {Annales Polonici Mathematici},
keywords = {partial differential equations in Banach spaces; nilpotent operators; nilpotent linear operator; well-posedness; Cauchy problem},
language = {eng},
number = {1},
pages = {67-80},
title = {Partial differential equations in Banach spaces involving nilpotent linear operators},
url = {http://eudml.org/doc/269972},
volume = {65},
year = {1996},
}

TY - JOUR
AU - Antonia Chinnì
AU - Paolo Cubiotti
TI - Partial differential equations in Banach spaces involving nilpotent linear operators
JO - Annales Polonici Mathematici
PY - 1996
VL - 65
IS - 1
SP - 67
EP - 80
AB - 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 $ℝ^{n+1}$.
LA - eng
KW - partial differential equations in Banach spaces; nilpotent operators; nilpotent linear operator; well-posedness; Cauchy problem
UR - http://eudml.org/doc/269972
ER -

## References

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1. [1] L. Cattabriga, On the surjectivity of differential polynomials on Gevrey spaces, Rend. Sem. Mat. Univ. Politec. Torino (1983), special issue on Linear partial and pseudodifferential operators'', 81-89. Zbl0561.35008
2. [2] L. Cattabriga and E. De Giorgi, Una dimostrazione diretta dell'esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti, Boll. Un. Mat. Ital. (4) 4 (1971), 1015-1027.
3. [3] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 271-355.
4. [4] B. Ricceri, On the well-posedness of the Cauchy problem for a class of linear partial differential equations of infinite order in Banach spaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991), 623-640. Zbl0810.35169
5. [5] F. Trèves, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966. Zbl0164.40602

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