New PDF release: Linear and nonlinear diffusion problems

By Lunardi A.

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Example text

2. 4 to prove that for each f ∈ Cb ((0, T ); X), the function v = (etA ∗ f ) belongs to C 1−α ([0, T ]; DA (α, ∞)) for every α ∈ (0, 1), with norm bounded by C(α) sup0

15) has a bounded mild solution v. Then for every a < 0 and for every t ∈ [a, 0] we have v(t) = (I − P )v(t) + P v(t) t e(t−s)A (I − P )g(s)ds + P v(t) = e(t−a)A (I − P )v(a) + a t a = e(t−a)A (I − P )v(a) + e(t−s)A (I − P )g(s)ds + P v(t) − −∞ −∞ a e(a−s)A (I − P )g(s)ds + v1 (t) + P v(t) = e(t−a)A (I − P )v(a) + −∞ = e(t−a)A ((I − P )v(a) + v1 (a)) + v1 (t) + P v(t). 3, supa≤0 v1 (a) P )v(a) < ∞. Letting a → −∞ we get v(t) = v1 (t) + P v(t), t ≤ 0. 22) holds. 21). 22) belongs to Cb ((∞, 0]; X).

We recall that for 0 < θ < 1 the parabolic H¨older space C θ/2,θ ([0, T ] × RN ) is the space of the continuous functions f such that f C θ/2,θ ([0,T ]×RN ) := f ∞ + sup[f (·, x)]C θ/2 ([0,T ]) + sup [f (t, ·)]C θ (R) < ∞, x∈R t∈[0,T ] and C 1+θ/2,2+θ ([0, T ] × RN ) is the space of the functions u such that ut , and Dij u exist for all i, j = 1, . . N and belong to C θ/2,θ ([0, T ] × R). The norm is N u C 1+θ/2,2+θ ([0,T ]×RN ) := u ∞+ Di u ∞ i=1 N + ut C θ/2,θ ([0,T ]×RN ) + Dij u C θ/2,θ ([0,T ]×RN ) .

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Linear and nonlinear diffusion problems by Lunardi A.


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