1. Introduction We are going to investigate the equation ut(t, x ... - ICMS

10Therefore we may and will assume that a ij ≡ δ ij . Also assuming γ =0does notrestrict generality.Now we are ready to use Theorem 2.4. From Sec. 4.3 of [8] we know that A isuniquely defined and is bounded as an operator acting in L p (R,L p ). We are going tocheck that A satisfies the assumptions of Theorem 2.4 with F = G = L p .Observe the simple fact that, for t>0, k =1, 2, ..., andf ∈ L p ,wehave∂T t f/∂t =∆T t f and ||∂ k T t f/∂t k || Lp ≤ Nt −k ||f|| Lp ,where N depends only on d and k. Fort>0 introduce the operatorK(t) =∆T t : L p → L pwith norm bounded by Nt −1 ,whereN is independent of t. Fort ≤ 0, letK(t) =0.Since a ij = δ ij ,wehaveRf(t, x) =∫ t−∞T t−s f(s, ·)(x) ds.In addition, if t is at a distance from the support of f, then differentiating the aboveformula presents no difficulties and we findAf(t, x) =∫ t−∞∆T t−s f(s, ·) dx =∫RK(t − s)f(s)(x) ds.