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**Sample text**

30], or [20, p. 1]. In addition to the assumptions in the above theorem, we assume ∂Ω satisfies a uniform exterior cone condition and that u is Hölder continuous on the lateral boundary and on the initial manifold and zeroth order compatibility condition on ∂Ω × {t = 0}, then the above estimates extend to Ω T . The Hölder estimates for the case of oblique boundary condition can be found in [94, p. 36]. 7 Embedding Theorems The parabolic version of the embedding theorems is very similar to the elliptic version when the t derivatives is considered “half the order” of x-derivatives.

34) 2 , The right-hand side of the above inequality is unbounded as T → ∞ if p 1 + np 2 2 which gives a contradiction in this case. In the case 1 + np < p < 1 + n , we have n(p−1)2 2 > −1 + (p−1)np , so we get a contradiction by letting T → ∞ and then 2 taking t 1. 29), for t > 1, up (x, t) up1 (x, t) cp1 p|x|2 . 35) Substituting this estimate into the expression for u2 , we obtain, for t > 2, u(x, t) u2 (x, t) t 1 Rn c4 tn/2 × c4 tn/2 × = cp1 p|y|2 dydτ exp − 2τ (2πτ )np/2 t/2 tn/2 τ 1+n/2 (t − τ )n/2 1 |x|2 + |y|2 p|y|2 dy dτ − − 2(t − τ ) 2τ t/2 tn/2 1+n/2 τ (t − τ )n/2 1 2 2 |x| + |y| p|y|2 dy dτ − − t 2τ t/2 dτ τ 1 Γ (x − y, t − τ ) exp − Rn exp exp − Rn exp c5 n/2 t exp − c5 n/2 t exp − |x|2 t |x|2 t |x|2 t |x|2 t |x|2 t |x|2 log(t/2).

4, we have −f (0)(n(x0 )·γ)2 > 0. By continuity 2 ∂ u ∂γ 2 (x) ∂2 u ∂γ 2 (x0 ) = Δu(x0 )(n(x0 ) · γ)2 = > 0 on Ω ∩ Bδ (x0 ) if δ > 0 is small.