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Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia Chinnì, Paolo Cubiotti (1996)

Annales Polonici Mathematici

Let E be a Banach space. We consider a Cauchy problem of the type ⎧ D t k u + j = 0 k - 1 | α | m A j , α ( D t j D x α u ) = f in n + 1 , ⎨ ⎩ D t j u ( 0 , x ) = φ j ( x ) in 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 ( n + 1 , E ) whose derivatives are equi-bounded on each bounded subset of n + 1 .

Partial differential operators depending analytically on a parameter

Frank Mantlik (1991)

Annales de l'institut Fourier

Let P ( λ , D ) = | α | m a α ( λ ) D α be a differential operator with constant coefficients a α depending analytically on a parameter λ . Assume that the family { P( λ ,D) } is of constant strength. We investigate the equation P ( λ , D ) 𝔣 λ g λ where 𝔤 λ is a given analytic function of λ with values in some space of distributions and the solution 𝔣 λ is required to depend analytically on λ , too. As a special case we obtain a regular fundamental solution of P( λ ,D) which depends analytically on λ . This result answers a question of L. Hörmander.

Partial exact controllability of a nonlinear system.

A. K. Nandakumaran, R. K. George (1995)

Revista Matemática de la Universidad Complutense de Madrid

In this article, we prove the partial exact controllability of a nonlinear system. We use semigroup formulation together with fixed point approach to study the nonlinear system.

Partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth

Mario Marino, Antonino Maugeri (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Sfruttando i risultati di [1], si prova che le derivate spaziali D α u di ordine | α | con | α | < m - 1 delle soluzioni in Q di un sistema parabolico quasilineare di ordine 2 m con andamenti strettamente controllati, sono parzialmente hölderiane in Q con esponente di hölderianità decrescente al crescere di | α | .

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