Semigroups of Linear Operators and Applications to Partial Differential Equations
From the reviews: "Since E. Hille and K. Yoshida established the characterization of generators of C0 semigroups in the 1940s, semigroups of linear operators and its neighboring areas have developed into a beautiful abstract theory. Moreover, the fact that mathematically this abstract theory has many direct and important applications in partial differential equations enhances its importance as a necessary discipline in both functional analysis and differential equations. In my opinion Pazy has done an outstanding job in presenting both the abstract theory and basic applications in a clear and interesting manner. The choice and order of the material, the clarity of the proofs, and the overall presentation make this an excellent place for both researchers and students to learn about C0 semigroups." #Bulletin Applied Mathematical Sciences 4/85#1 "In spite of the other monographs on the subject, the reviewer can recommend that of Pazy as being particularly written, with a bias noticeably different from that of the other volumes. Pazy's decision to give a connected account of the applications to partial differential equations in the last two chapters was a particularly happy one, since it enables one to see what the theory can achieve much better than would the insertion of occasional examples. The chapters achieve a very nice balance between being so easy as to appear disappointing, and so sophisticated that they are incomprehensible except to the expert." #Bulletin of the London Mathematical Society#2
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adjoint analytic semigroup T(t assume AT(t Banach space bounded domain bounded linear operators bounded operators classical solution closed linear operator compact concludes the proof constant continuous function continuously differentiable Corollary D D(A definition denote densely defined dissipative elliptic operator estimate exists follows readily fractional powers given Hilbert space Hölder continuous implies inequality infinitesimal initial value problem integral equation invertible Lemma Let f Let T(t Lipschitz continuous LP(Q m-dissipative mild solution Moreover norm obtain Pazy proof is complete proof of Theorem prove resolvent set right-hand side satisfies the conditions satisfying T(t semigroup of bounded semigroup of contractions sequence solution of 1.1 strong solution strongly continuous strongly elliptic operator Theorem 3.1 u e D(A uniform operator topology uniformly bounded uniformly continuous unique solution value problem 2.1 x e D(A Y-valued solution yields Yosida