Heat-kernels and maximal Lp − Lq−estimates: the non-autonomous case

Matthias Hieber (1) , Sylvie Monniaux (2)

1 Fachbereich Mathematik TU Darmstadt
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

In this paper, we establish maximal Lp−Lq estimates for non autonomous parabolic equations of the type u′(t) + A(t)u(t) = f(t), u(0) = 0 under suitable conditions on the kernels of the semigroups generated by the operators −A(t), t ∈ [0; T]. We apply this result on semilinear problems of the form u′(t) + A(t)u(t) = f(t; u(t)), u(0) = 0.

Mots clés: Heat-kernel estimates; Maximal L^p-L^q-regularity; Non-autonomous Cauchy problem; Singular integrals

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Matthias Hieber, Sylvie Monniaux. Heat-kernels and maximal Lp − Lq−estimates: the non-autonomous case. Journal of Fourier Analysis and Applications, 2000, 6 (5), pp.467-482. ⟨hal-00535645⟩