Atlas: Applications of Banach-valued Besov spaces to Evolution Equations in Banach spaces by Tosinobu Muramatu

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5th International ISAAC Congress
July 25-30, 2005
Department of Mathematics and Informatics, University of Catania
Catania, Sicily, Italy

Organizers
International ISAAC Board, Local organizing committee: F. Nicolosi (chairman), S. Bonafede, V. Cataldo, P. Cianci, G.R. Cirmi, S. D'Asero, G. Fiorito, L. Giusti, S. Leonardi, P.E. Ricci

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Applications of Banach-valued Besov spaces to Evolution Equations in Banach spaces
by
Tosinobu Muramatu
Dept.Math.Tokyo University of Science

It is well known that Besov spaces are very usefull in studying many problems. I report here their applications to the evolution equation: (EE) du/dt+A(t)u=f(t), a < t < b, where -A(t) is the generator of a semigroup of linear operators in a Banach space X. We consider only the case where -A(t) is the generator of an analytic semogroup.

Case 1. A(t)=A is independent of t. Crandall-Pazy, 1969, proved that F(t):=ò_a^te^-(t-s)Af(s)ds ( e^-tA denotes the semigroup generated by -A) is strongly differentiable and satisfies (EE) if the modulus of continuity w(h:f) of f is integrable near 0 with the measure dh/h. Furthermoere, Baillon, 1980, showed that if F is differantiable for every continuous function f then X has a spacial property or A is bounded. We prove that F is strongly differantiable and satisfies (EE) if f belongs to B⁰_{¥, 1}(I;X)_locÇL¹(I;X), I:=(a, b) ( J.Math.Soc. Japan, 1990).

Case 2. The domain D(A(t)) of A(t) is independent of t, which we write by Y. Tanabe, 1960, has constructed the evolutin operator U(t, s) to (EE) when A(t) is Hölder continuous L(Y, X)-valued function. We have improved his result, that is, we have constructed it under the assumption that the modulas of continuity w(h) of A(t) as an L(Y, X)-valued function is integrable near 0 with dh/h( Osaka J. Math. 2001).We also showed that F(t):=ò_a^tU(t, s)f(s)ds is strongly differentiable and satisfies (EE) if f satisfies the same conditon as in Case I.

Case 3. The domain D(A(t)^1/m) is independent of t, where m is some positive integer m greater than 1. We put Y=D(A(t)^1/m). Assuming that A(t)^1/m is Hölder continuous with a exponent q greater than 1-1/m as an L(Y, X)-valued function, T. Kato, 1961, has constructed the evolution operator. We recently improved his result. Our assumption is `A(t) belongs to B_{¥, 1}^1-1/m(I;L(Y, X))'. We also have the same result for F as in Case 2.

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Date received: May 26, 2005