On the Cauchy Problem for the b-Family Equations with a Strong Dispersive Term

Theorem 1.1. Given u₀ ∈ H^s(ℝ), s > 3/2, then there exist a T and a unique solution u to (1.3) such that

Lemma 2.1. Assume that u₀ ∈ H^s(ℝ), s > 2. If b = 1/2, then the solution of (1.2) will exist globally in time. If b > 1/2, then the solution blows up if and only if u_x becomes unbounded from below in finite time. If b < 1/2, the solution blows up in finite time if and only if u_x becomes unbounded from above in finite time.

Proof. By direct computation, one has

Hence, Applying y on (1.2) and integration by parts, we obtain

If b = 1/2, then (d/dt)∫_ℝy²dx = 0. Hence,

Equation (2.4) implies the corresponding solution exists globally.

If b > 1/2, due to the Gronwall inequality, it is clear that, from (2.3), u_x is bounded from below on [0, T) and then the H₂-norm of the solution is also bounded on [0, T). On the other hand,

Therefore where we use (2.2). Hence, (2.6) tells us if H₂-norm of the solution is bounded, then the L_∞-norm of the first derivative is bounded.

By the same argument, we can get the similar result for b < 1/2.

This completes the proof.

Theorem 2.2. Let b ≥ 2. Suppose that u₀ ∈ H²(ℝ) and there exists an x₀ ∈ ℝ such that $$,

Then the corresponding solution u(x, t) to (1.2) with u₀ as the initial datum blows up in finite time.

Proof. Suppose that the solution exists globally. Due to (2.11) and the initial condition (2.12), we have y(q(x₀, t), t) = 0, and

for all t ≥ 0. Since u(x, t) = G*y(x, t), one can write u(x, t) and u_x(x, t) as Consequently, for all t > 0.

For any fixed t, if x ≤ q(x₀, t), then

Similarly, for x ≥ q(x₀, t), we also have Combining (2.16) and (2.17) together, we get that for any fixed t, for all x ∈ ℝ.

Differentiating (1.3), we get

Differentiating u_x(q(x₀, t), t) with respect to t, where q is the diffeomorphism defined in (2.7), where we use (2.18) and the following inequality: $$.

Suppose not; that is, there exists a t₀ such that $$ on [0, t) and $$. Now, let

Firstly, differentiating I(t), we have Secondly, by the same argument, we get Hence, it follows from (2.22), (2.23), and the continuity property of ODEs that Moreover, due to (2.22) and (2.23) again, we have the following equation for $$: Now, substituting (2.20) into (2.25), it yields Before completing the proof, we need the following technical lemma.

Let $$; then, (2.26) is an equation of type (2.27) with C₀ = 1/2. The proof is complete by applying Lemma 2.3.

Remark 2.4. Mckean got the necessary and sufficient condition for the Camassa-Holm equation in [7]. It is worth pointing out that Zhou and his collaborators [13] gave a new proof to Mckean′s theorem. However, the necessary and sufficient condition for (1.2) is still a challenging problem for us at present.

Theorem 3.1. Supposing that u₀ ∈ H³, $$ is one sign. Then the corresponding solution to (1.2) exists globally.

Proof. We can assume that y₀ ≥ 0. It is sufficient to prove u_x(x, t) has a lower and supper bound for all t. In fact,

so This completes the proof.

Theorem 4.1. Assume that for some T > 0 and s > 5/2, u ∈ C([0, T]; H^s(ℝ)) is a strong solution of (1.2) and that u₀(x) = u(x, 0) satisfies that for some θ ∈ (0,1),

Then uniformly in the time interval [0, T].

Proof. First, we will introduce the weight function to get the desired result. This function φ_N(x) with N ∈ ℤ⁺ is independent on t as follows:

which implies that From (1.3), we can get Multiplying (4.5) by (uφ_N) ^2p−1 with p ∈ ℤ⁺ and integrating the result in the x-variable, we get from which we can deduce that Denoting $$ and by Gronwall′s inequality, we obtain Taking the limits in (4.8), we get Next differentiating (1.3) in the x-variable produces the equation Using the weight function, we can rewrite (4.10) as Multiplying (4.11) by (u_xφ_N) ^2p−1 with p ∈ ℤ⁺ and integrating the result in the x-variable, it follows that For the second term on the right side of (4.12), we know Using the above estimate and the Hölder inequality, we deduce that Thanks to Gronwall′s inequality, it holds that Taking the limits in (4.15), we have Combining (4.9) and (4.16) together, it follows that A simple calculation shows that there exists c₀ > 0, depending only on θ ∈ (0,1), such that for any N ∈ ℤ⁺, Thus, for any appropriate function f and g, one sees that Similarly, we can get Thus, inserting the above estimates into (4.17), there exists a constant $$ such that Hence, for any t ∈ ℤ⁺ and any t ∈ [0, T], we have Finally, taking the limit as N goes to infinity in (4.22), we find that for any t ∈ [0, T] which completes the proof of the theorem.

Theorem 5.1. Let 0 ≤ b ≤ 3. Assume that for some T ≥ 0 and s ≥ 5/2, u ∈ C([0, T]; H^s(ℝ)) is a strong solution of (1.2). If u₀(x) = u(x, 0) has compact support [a, c], then for t ∈ (0, T], one has

where f₊(t) and f₋(t) denote continuous nonvanishing functions, with f₊(t) > 0 and f₋(t) < 0 for t ∈ (0, T]. Furthermore, f₊(t) is strictly increasing function, while f₋(t) is strictly decreasing function.

Proof. Since u₀ has compact support in x in the interval [a, c], from (2.11), so does y(x, t) in the interval [q(a, t), q(c, t)] in its lifespan. Hence the following functions are well defined:

with Then for x > q(c, t), we have Similarly, when x < q(a, t), we get Hence, as consequences of (5.4) and (5.5), we have On the other hand, It is easy to get Substituting identity (5.8) into dE(t)/dt, we obtain where we use (5.6).

Therefore, in the lifespan of the solution, we have

By the same argument, one can check that the following identity for F(t) is true: In order to complete the proof, it is sufficient to let f₊(t) = (1/2)E(t) and f₋(t) = (1/2)F(t).

Remark 5.2. The main result in [18] is that any nontrivial classical solution of the b-family equation with dispersive term will not have compact support if its initial data has this property. But Theorem 4.1 means that no matter what the profile of the compactly supported initial datum u₀(x) is (no matter whether it is positive or negative), for any t > 0 in its lifespan, the solution u(x, t) is positive at infinity and negative at negative infinity. So Theorem 4.1 is an improvement of that in [18].