Analytic Semigroups and Optimal Regularity in Parabolic Problems
Springer Science & Business Media, 1995. jan. 1. - 424 oldal
This book gives a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and of how such a theory may be used in parabolic PDE's. It takes into account the developments of the theory during the last fifteen years, and it is focused on classical solutions, with continuous or Holder continuous derivatives. On one hand, working in spaces of continuous functions rather than in Lebesgue spaces seems to be appropriate in view of the number of parabolic problems arising in applied mathematics, where continuity has physical meaning; on the other hand it allows one to consider any type of nonlinearities (even of nonlocal type), even involving the highest order derivatives of the solution, avoiding the limitations on the growth of the nonlinear terms required by the LP approach. Moreover, the continuous space theory is, at present, sufficiently well established. For the Hilbert space approach we refer to J. L. LIONS - E. MAGENES , M. S. AGRANOVICH - M. l. VISHIK , and for the LP approach to V. A. SOLONNIKOV , P. GRISVARD , G. DI BLASIO , G. DORE - A. VENNI  and the subsequent papers , , . Many books about abstract evolution equations and semigroups contain some chapters on analytic semigroups. See, e. g. , E. HILLE - R. S. PHILLIPS ' S. G. KREIN , K. YOSIDA , A. PAZY , H. TANABE , PH.
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spaces of continuous
Analytic semigroups and intermediate spaces
Generation of analytic semigroups by elliptic operators
Linear parabolic problems
Linear nonautonomous equations
A G C asymptotically stable Au(t Banach space belongs to C([0 boundary condition bounded functions bounded with values C D(A C(Rn C2m boundary classical solution coefficients consider constant continuous and bounded continuous functions continuous with values continuously differentiable continuously embedded Corollary Da(a DA(a,oo derivatives Dirichlet boundary condition eigenvalue elliptic elliptic operator equations equivalent etAx exists fixed point fully nonlinear Holder continuous Holder spaces holds implies initial datum interpolation spaces interval l,oo Lemma Let f Let us prove linear operator Lipschitz continuous mild solution Moreover null solution open set priori estimates proof of Theorem regularity properties Reiteration Theorem replaced resolvent set second order Section sectorial operator semigroup set in Rn solution of 4.0.1 spectral set stability statement follows strict solution strong solution Subsection T-periodic thanks to Proposition uniformly C2 uniformly continuous unique solution values in D(A