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• # Artykuł - szczegóły

## Annales Polonici Mathematici

1996-1997 | 65 | 1 | 67-80

## Partial differential equations in Banach spaces involving nilpotent linear operators

EN

### Abstrakty

EN
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 $ℝ^{n+1}$.

EN

67-80

wydano
1996
otrzymano
1995-09-30

### Twórcy

autor
• Department of Mathematics, University of Messina, 98166 Sant'Agata-Messina, Italy
autor
• Department of Mathematics, University of Messina, 98166 Sant'Agata-Messina, Italy

### Bibliografia

• [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.
• [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] 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] 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.
• [5] F. Trèves, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966.