Notes on Boussinesq Equation

CHAPTER 5
Asymptotic Behavior of Solutions
In this chapter we study some aspects related to the asymptotic behavior of solutions
to the IVP (1.1). In particular, we study the decay of solutions with time. We will show
that the decay of solutions of the nonlinear problem is the same inherited from the linear
problem. The decay result will allow us to show the existence of solutions to the linear
problem that approximate to solutions of the nonlinear problem. This is called nonlinear
scattering. In the last section of this chapter we present a blow-up result for solutions of
the IVP (1.1)
5.1. Decay
One interesting question concerning solutions of evolution equations is the behavior of
these regarding the time. In this direction we have the following result.
Theorem 5.1. Let f ∈ H 1 (R) ∩ L q γ (R), g = h ′ , h ∈ L 2 (R) ∩ L q (R), and α > 4−3γ−γ2
γ
.
2
If ||f||| + ||g|| ≡ ‖f‖ 1,2 + ‖f‖γ
2 ,q + ‖h‖ 2 + ‖h‖ q < δ small. Then there exists c > 0 such that
the solution u of the IVP (1.1) satisfies
where p = 2
1−γ , q = 2
1+2γ
‖u‖ p ≤ c (1 + t) −γ/2 , t > 0,
and γ ∈ (0, 1/2).
Proof. The solution of the IVP (1.1) is written as
∫ t
u(t) = V 1 (t)f(x) + V 2 (t)g(x) −
V 1 (t)· and V 2 (t)· are defined as in (2.26) and (2.32), respectively.
From (5.1) it follows that
∫ t
‖u(t)‖ p ≤ ‖V 1 (t)f‖ p + ‖V 2 (t)g‖ p +
then use of Proposition 2.13 and Lemma 2.15 leads to
∫ t
‖u(t)‖ p ≤ c(1 + t) −γ/2 (||f|| + ||g||) + c
0
43
0
0
V 2 (t − τ)(|u| α−1 u) xx (τ) dτ, (5.1)
‖V 2 (t − τ)(|u| α−1 u) xx (τ)‖ p dτ,
(t − τ) −γ/2 ‖|u| α−1 u(τ)‖ 2 dτ. (5.2)
1+γ