Noyaux de la chaleur et estimations mixtes Lp — Lq optimales: le cas non autonome
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Consider the non-autonomous initial value problem u′(t) + A(t)u(t) = f(t), u(0) = 0, where - A(t) is for each t 
[0, T], the generator of a bounded analytic semigroup on L2 (Ω). We prove maximal Lp - Lq a priori estimates for the solution of the above equation provided the semigroups Tt are associated to kernels which satisfies an upper Gaussian bound and {A(t), t [0, T]} fulfills a Acquistapace-Terreni commutator condition.
[0, T], the generator of a bounded analytic semigroup on L2 (Ω). We prove maximal Lp - Lq a priori estimates for the solution of the above equation provided the semigroups Tt are associated to kernels which satisfies an upper Gaussian bound and {A(t), t [0, T]} fulfills a Acquistapace-Terreni commutator condition.