Maximal regularity for evolution equations in weighted L p -spaces

Abstract

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy probleṁif and only if it has the property of maximal L p -regularity. Moreover, it is also shown that the derivation operator D = d/dt admits an H ∞ -calculus in weighted L p -spaces.Introduction. Let X be a Banach space and let A be a closed linear operator on X with domain D(A). We consider the abstract Cauchy probleṁ

Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

“…According to the above analysis, we can pass to the limit as n → +∞ in (38)- (39) and we obtain that…”

“…Existence of regular bounded solutions on (0, +∞) may be found for example in [27,17,29,28,20,19,38,11,10,23,42,15,18,3,4] and in several other articles whose references may be found in the survey [33] or in the book [39]. However, it is well-known that the solutions may blow up in L ∞ (Ω) in finite time as proved in [35,36] where explicit finite time blow up in L ∞ (Ω) are given.…”

On quasilinear parabolic evolution equations in weighted L p -spaces

“…To be precise, this means that for each f ∈ L p,µ (R + ; X 0 ) there exists a unique solution u ∈ H of the problemu + A 0 u = f, t > 0, u(0) = 0. Thanks to [21,Theorem 2.4] the characterization A 0 ∈ MR p,µ (X 1 , X 0 ) ⇔ A 0 ∈ MR p (X 1 , X 0 ) for a closed linear operator A 0 in X 0 holds true, provided µ ∈ (1/p, 1], p ∈ (1, ∞). Here we use the notation A 0 ∈ MR p (X 1 , X 0 ) for the 'classical' case µ = 1.…”

“…[11]. Concerning nontrivial initial data, it was shown in [21,Theorem 3.2] that if A 0 ∈ MR p (X 1 , X 0 ), then the initial value probleṁ u + A 0 u = f, t > 0, u(0) = u 0 .…”

On R-boundedness of Unions of Sets of Operators

“…During the past few years a theory of L p -multipliers for operator valued functions has been developed by means of the notion of R-boundedness of sets of operators, see for example [1,2,3,4,5,6,7,8,10,12,13,17,18,19]. This theory has been applied to study maximal regularity of certain abstract evolution equations.…”