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

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12by using self similarity. In almost the same form as stated this lemma is proved inAppendix in [11].In the next lemma we do the first step to considering the power of summability in tequal to multiples of p.Lemma 2.6. Let T ≤∞, p ∈ (1, ∞), n =1, 2, .... For k =1, ..., n, let u k∈H 2,pp (T ) be solutions of the equationu k t = aij u k x i x j + f kwith zero initial data and with f k ∈ L p p (T ). Then∫ T0n∏||∆u k (t)|| p L pdt ≤ Nk=1n∑∫ Tk=10∏||f k (t)|| p L p||∆u j (t)|| p L pdt, (2.4)j≠kwhere N = N(n, d, p, δ).Proof. Define v k =∆u k .ForX =(x 1 , ..., x n ) ∈ R nd with x i ∈ R d ,defineV (t, X) =v 1 (t, x 1 ) · ... · v n (t, x n ).Observe thatV t (t, X) =LV (t, X)+F (t, X),
where LV = a rs (V xr1 x s 1 + ... + V x r nx s n ),13F (t, X) =∆ xi G i (t, X),G i (t, X) =f i (t, x i ) ∏ j≠iv j (t, x j ).Hence by Lemma 2.5||V || Lp ((0,T )×R nd ) ≤ N ∑ i||G i || Lp ((0,T )×R nd )and this is exactly (2.4). The lemma is proved.The second proof of Theorem 2.1.As in the first proof, we only have toconsider the case a ij ≡ δ ij and γ =0. Also obviously it suffices to prove (2.2) forf ∈ C ∞ 0 (R × Rd ). Without loss of generality we assume that f(t) =0for t ≤ 0.Let u = Rf. Thenu is a classical solution ofu t =∆u + ffor t>0 with zero initial condition and, even more than that, u(s) =0for s ≤ 0. Inaddition, it is easy to check that u ∈ L p ((0,T),H 2 p) for each T
Page 1 and 2: THE HEAT EQUATION IN L q ((0,T),L p
Page 3 and 4: Next, equation (1.1) will be consid
Page 5 and 6: Theorem 1.1. Let p, q ∈ (1, ∞),
Page 7 and 8: 7Also remember that, for f ∈ C
Page 9 and 10: where K(x, y) is a bounded operator
Page 11: In order to prove that the assumpti
Page 15 and 16: Definition 3.1. Take and fix a nonn
Page 17 and 18: Theorem 3.3. (i) We have17(H γ p,
Page 19: 194. Some smoothing properties of t
Page 23 and 24: Now we claim that Ā is a bounded o
Page 25 and 26: Proof. Clearly (6.1) becomes strong
Page 27 and 28: 27≤ N(||Mf(t)|| p H γ−2p,θ+ |

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

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