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Sobolev imbeddings and integrated semigroups. (English) Zbl 0762.47013
Semigroup theory and evolution equations, Proc. 2nd Int. Conf., Delft/Neth. 1989, Lect. Notes Pure Appl. Math. 135, 29-40 (1991).
In the reviewing paper, the author considers the Cauchy problem \[ u'(t) = Au(t), \quad t \geq 0, \quad u(0) = x, \] with Schrödinger operator \( A = i\Delta_ p\), \(\Delta_ p \) being the Laplacian on \(L^ p(\Omega )\), where \( \Omega \subset \mathbb{R} ^ n \) is a bounded set.
To this aim, he deals with the concept of \(k\)-times integrated semigroup, see [the author, Israel J. Math. 59, 327-352 (1987; Zbl 0637.44001), and F. Neubrander, Pac. J. Math. 135, 111-155 (1988; Zbl 0675.47030)]. Let the operator \(A\) be an operator on a Banach space \(E\) , and let \(k \in \mathbb{N}\). One says that \(A\) generates a \(k\)-times integrable semigroup \(S\), if there exists \(w \in \mathbb{R}\) and a (uniquely determined) strongly continuous function \( S: [0, \infty ) \rightarrow \mathcal L (E)\) satisfying \( \sup_{t\geq 0} | e^{-wt} S (t)| < \infty \) , such that \(( w,\infty ) \subset \rho (A) \) (the resolvent set), and \[ R (\lambda , A) := (\lambda - A )^{-1} = \lambda \int_{0}^{\infty } e^{-\lambda t}S(t) \text d t \text {\;\;\;for all\;\;\;} \lambda > 0 . \] (In the case that \(A\) generates a \( C_ 0 \) - semigroup \(T\), one can write \[ S(T) = \int_{0}^{t} [(t-s)^{k-1}/(k-1)!]T(S) \text{d} S ; \] and the operator which generates the \(k\)-times integrated semigroup, generates, for all \(l \geq k ,\) the \(k\)-times integrated semigroup as well. Hence, the best possible \(k\) is an expression for the regularity of the Cauchy problem defined by \(A\).)
Using the technique of interpolation of semigroups and Sobolev imbedding theorems, the author shows that for \( \Omega \in C^ 2 (\Omega )\) \(i\Delta \) (with Dirichlet or Neumann boundary conditions) generates a \(k\)-times integrable semigroup on \( L^{\infty }(\Omega )\) (\(1 < p <\infty\)) if \( k \geq \frac{N}{2}| \frac{1}{2}-\frac{1}{p} | \) and in \( C_ 0(\Omega)\), \(C(\overline \Omega)\), \(L^ \infty (\Omega )\), \(L^ 1(\Omega )\) if \(k > \frac{N}{4}\).
The method still works if the Laplacian is replaced by a strictly elliptic operator with smooth coefficients.
For the entire collection see [Zbl 0741.00048].
Reviewer: P.Doktor (Praha)

47D06 One-parameter semigroups and linear evolution equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems