14:00h – 14:40h

Integration with values in a Neumann space for PDE’s

We It may be useful to use elements in spaces such as L^{p}([0,T] ;L^{q}_{loc}(Ω)) or H^{-1}([0,T]; W^{m,q}(Ω)-weak) … unfortunately they were not yet defined. Theses spaces L^{q}_{loc}(Ω) and W^{m,q}(Ω)-weak are Neumann spaces, that are vector spaces provided with semi-norms that made them separated and sequentially complete.

We will define D'(Ω;E), L^{p}(Ω;E) and W^{m,p}(Ω;E) where E is a Neumann space and Ω is an open set of R^{d}, and we show some applications to PDE’s.
We insist on L^{p}(Ω;E) which is composed of measures: it cannot be composed
of class of almost equal functions because the theory fails, since Egoroff theorem
does not extend if E is not metrisable.
When E is a Banach space and \hat{f} \hat{L}^{p}(Ω;E), that is the space of p-integrable
almost equal functions, then a measure \bar{f} is defined by: for all φ \in K(Ω),
<\bar{f}, φ>=\int_{Ω}\hat{f}φ
We prove that our space L^{p}(Ω;E) is {\bar{f} : \hat{f} \in \hat{L}^{p}(Ω;E)} and that \hat{f}→\bar{f} is an isomorphism. So, our L^{p}(Ω;E) is the true Lebesgue space if E is a Banach space.

14:50h – 15:30h

Decay estimates for semilinear wave equations with time-dependent time delay