Notes on Boussinesq Equation

24 2. LINEAR PROBLEM
Corollary 2.17. Under the assumptions on f and g in Proposition 2.16. The solution
u of the linear problem (2.1) satisfies
where p = 2
1−γ , q = 4 γ
‖u‖ L q (R :L p 1 (R)) ≤ c(‖f 1 ‖ 1+ γ 4 ,2 + ‖h‖ γ
4 ,2 ),
and γ ∈ [0, 1].
Proof. The result follows as a consequence of (i) and (ii) in Proposition 2.16.
From (2.27), (2.28) and Corollary 2.17 we can conclude that a solution u of the linear
problem (2.1) satisfies
u ∈ L ∞ (R : H 1 (R)) ∩ L q (R : L p 1 (R)),
where p, q and the initial data satisfy the conditions in those results.
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