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Let u be a smooth non-negative solution of the equation ∂u/∂t = ϕ(u) in Qρ , with 1/2 < ρ < 2. 12) Qρ where C, p, θ, N are positive constants which depend only on a and n. We shall now estimate the maximum of a solution in terms of spatial averages. 7. Let u be a smooth non-negative solution of the equation ∂u/∂t = ϕ(u), in Q∗ , with ϕ ∈ Sa . 13) u(x, t)dx where the constants C, σ depend only on n and a. Proof. Let S = Br × (−T , 0], R = Bρ × (−τ, 0] satisfy Q ⊂ S ⊂ R ⊂ Q∗ and set v = max{u, 1}.

We have 0 < σ (M) ≤ H (M) 2 and 0 < R ∗ (M) ≤ d 2 for M ∈ (0, C1 ]. Let M1 = c1 , R1 = R(M), Mk+1 = Mk − σ (Mk ) Rk+1 = min{R(Mk+1 ), kn H (Mk ) Rk }. Claim. |u(x, t)| ≤ Mk in Q(x0 ,t0 ) (Rk ). We shall use induction: (1) For k = 1, |u| ≤ c1 = M1 in QT ⊃ Q(x0 ,t0 ) (R1 ). (2) Assuming that |u| ≤ Mk on Q(Rk ) |u| ≤ Mk+1 on Q(Rk+1 ). we shall prove that 50 1 Local regularity and approximation theory Set ε = ε(Mk ) = Mk − H (Mk ). For m > 0 define z(x, t) = gε,m (u). Thus z ≤ [u − ε(Mk )]+ ≤ H (Mk ) in Q(Rk ).

2) on the unit cube Q, with 0 ≤ u ≤ 1, then max u − min u =: osc u ≤ 1 − δ. 2. 2. 5. 2 and β(1) ≤ θ . 2) in Q, with β ∈ Bθ , satisfying 0 ≤ u ≤ 1. We shall show that there exists a sequence ρk ↓ 0 for which ωk := sup oscQρk u → 0 as k → ∞. 6). To this end, set ω = sup oscQ u u∈Aθ and for u ∈ Aθ , let us define the function v= u − inf Q u . ω ˜ ˜ Then v satisfies the equation (β(v)) = β(sω + inf Q u)/ω. 5 osc v ≤ 1 − δ Qρ1 with δ = δ(η, β(ω + inf Q u)/ω) and ρ1 = ρ(η, β(ω + inf Q u)/ω). 5 Equicontinuity of solutions 41 with Cθ = sup{β(3); β ∈ Bθ } and ρ1 = ρ1 (η, Cθ /ω) < 1.

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