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Preliminary results: Basic theory of elliptic operators
Published in Takao Nambu, Theory of Stabilization for Linear Boundary Control Systems, 2017
by choosing a suitable sector Σ¯={λ−b∈ℂ;θ0⩽|argλ|⩽π}, 0 < ∀θ0 < π/2, ∃b ∈ ℝ1. An operator L whose resolvent satisfies (4.1) in some sector Σ¯ with angle more than π is called a sectorial operator. It is well known that a sectorial operator generates an analytic semigroup. We briefly review some properties of a sectorial operator in this section. The class of these operators are apparently narrower than those generating C0-semigroups, since, for example, the operator in wave equations with no damping term has the spectrum lying in the imaginary axis in the complex plane, and thus is not sectorial. Let () e−tL=−12πi∫∂Σ¯e−tλ(λ−L)−1dλ,t>0,
Speed of wave propagation for a nonlocal reaction–diffusion equation
Published in Applicable Analysis, 2020
Let be the Banach space of all bounded and uniformly continuous function from to with conventional supremum norm . Set Then is a closed cone in , and is a Banach space under the partial ordering induced by . It is well known that the differential operator generates an analytic semigroup on . The standard parabolic maximum principle (see Corollary 7.2.3 of [12]) implies that the semigroup is strongly positive in the sense that for all t>0 where Int() is the interior of .
Well-posedness of a mathematical model of diabetic atherosclerosis with advanced glycation end-products
Published in Applicable Analysis, 2022
To prove the main theorem, we are going to use similar steps to that in [52]. We first rewrite the system (61)–(96) as a quasi-linear abstract evolution equation in Banach spaces and then apply the analytic semigroup theory [43].