# The local well posedness result for the Cauchy

The local well-posedness result for the Cauchy problem (3.1) Rimonabant given as Theorem 1.3.

The proof of Theorem 1.3 is based on a truncation technique by making use of a mollifer ηLηL on [0,+∞)[0,+∞) to be defined as follows. Let L>0L>0, ηL:[0,+∞)→[0,1]ηL:[0,+∞)→[0,1] is a C∞C∞-function such thatηL(r)= 1,ifr<L,0,ifr≥L+1. Then, for k∈Nk∈N, define the operator ζkζk by the Fourier transform asF(ζkv)(ξ)=ηk( ξ )vˆ(ξ),t∈R,ξ∈Rn.

Consider the mild solution of the truncated system of (3.1):equation(3.2)u(t)=S(t)u0−iλ∫0tS(t−s)(ηL(‖u‖Ys)( x −γ? u 2)u(s))ds−i∫0tS(t−s)u(s)dW(s)−12∫0tS(t−s)u(s)F?ds.

Based on Lemma 3.1 and Strichartz estimate (1.2)–(1.3), we can obtain the well-posedness of (3.2) as follows by the contraction mapping principle.

Proposition 3.1.

Let the assumptions of Theorem 1.3hold. Then for any ρ≥qρ≥qand any T0>0T0>0such intracellular parasites for any F0F0measurable u0u0with values in Lρ(Ω;Hs(Rn))Lρ(Ω;Hs(Rn)), there exists a unique solution u of (3.2)such thatu∈ L2(Ω;YT0),fors≥γ2,Lρ(Ω;XT0),for others.