Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold

Assumption (A)

There exists a C² vector field H on 𝓜 such that

where δ > 0 is a constant and Ω ⊂ 𝓜 is an open set with smooth boundary.

Moreover, a(x) satisfies

where a₀ > 0 is a constant.

Remark 2.1

The vector field given by assumption (A) is called escape vector field and it was introduced by Yao [33] for the controllability of the wave equation with variable coefficients, which is also a useful condition for the controllability and the stabilization of the quasilinear wave equation (see [17, 34, 36]). Existence of escape vector field depends on the sectional curvature of the Riemannian manifold (𝓜, g). There are a number of methods and examples in [35] to find out escape vector field. The explicit expression of Dr∂∂r in Riemannian manifold (ℝⁿ, g) is given by [25, 26].

Assumption (B)

(Unique continuation) Let Ω ⊂ 𝓜 be an open set with smooth boundary and ω ⊂ Ω be an open subset. Assume that ω satisfies the geometric control condition:

Remark 2.2

Let H = Dφ, where H is given by (2.2) and φ is a strictly convex function, then assumption (B) follows from Proposition B.3 in [24]. It can also be proved in particular cases by Carleman estimates in Euclidean space, see [22, 23, 31].

Theorem 2.1

Let assumption (A) and assumption (B) hold true. Given constant E₀ > 0. Assume that ∥u₀∥_L²(𝓜) ≤ E₀. Then there exist positive constants C₁ and C₂, which are only dependent on E₀, such that

Lemma 3.1

Suppose that u(x, t) solves the following equation:

Let 𝓗 be a C¹ vector field defined on 𝓜. Then

Moreover, assume that the real function P ∈ C¹(𝓜). Then

Proof

Multiplying the Schrödinger equation in (3.1) by 𝓗(ū) and then integrating over 𝓜 × (0, T), we deduce that

and

The equality (3.2) follows from Green’s formula.

In addition, we multiply the Schrödinger equation in (3.1) by Pū and integrate over 𝓜 × (0, T). It gives that

and

The equality (3.3) follows from Green’s formula. □

Lemma 3.2

Let Ω ⊂ 𝓜 be a bounded domain. Assume that there exists a C¹ vector field H on 𝓜 such that

where δ > 0 is a constant.

Then, for any x ∈ Ω and any unit-speed geodesic y (t) starting at x, if

we have

Proof

Note that

Then

Hence

□

Lemma 3.3

Let u(x, t) solve the system (1.3). Then

for any T > 0.

Proof

After multiplying the Schrödinger equation in (1.3) by 2ū and integrating over 𝓜 × (0, T), the equality (3.16) holds.

Multiplying the Schrödinger equation in (1.3) by 2ū_t and integrating over 𝓜 × (0, T), we obtain

Let P = a(x) in (3.3). Substituting (3.3) into (3.18), we have

□

Lemma 4.1

Let assumption (A) hold true. Then, there exists t₀ > 0, for any x ∈ Ω and any unit-speed geodesic y (t) starting at x, there exists t < t₀ such that

Lemma 4.2

(Unique continuation) Let assumption (B) hold true. Let Ω ⊂ 𝓜 be an open set with smooth boundary and ω ⊂ Ω be an open subset. Assume that ω satisfies the geometric control condition:

(GCC) There exists constant T₀ > 0 such that for any x ∈ Ω and any unit-speed geodesic y (t) of (𝓜, g) starting at x, there exists t < T₀ such that y (t) ⊂ ω.

Accordingly, for every T > 0, the only solution in C([0, T], H¹(Ω)) to the system

is the trivial one u ≡ 0.

Lemma 4.3

(Unique continuation) Let assumption (B) hold true. Let Ω ⊂ 𝓜 be an open set with smooth boundary and ω ⊂ Ω be an open subset. Assume that ω satisfies the geometric control condition:

(GCC) There exists constant T₀ > 0 such that for any x ∈ Ω and any unit-speed geodesic y (t) of (𝓜, g) starting at x, there exists t < T₀ such that y (t) ⊂ ω.

Therefore, for every T > 0, the only solution in C([0, T], H¹(Ω)) to the system

is the trivial one u ≡ 0.

Proof

Let

Note that

and

Hence

It follows from assumption (B) that u ≡ 0. □

Lemma 4.4

For any ϵ > 0, there exists C_ϵ > 0 such that

where a(x) is given by (1.3).

Proof

We prove (4.8) by contradiction. If (4.8) doesn’t hold true, then there exist constant ε₀ > 0 and {xk}k=1∞⊂M such that

Therefore, there exists x₀ and a subset of {xk}k=1∞ , still denoted by {xk}k=1∞ , such that

Note that

and

It follows from (4.9) that

and

With (4.11) and (4.13), we obtain

which contradicts (4.14). □

Lemma 4.5

Let assumption (A) hold true. Let u(x, t) solve the system (1.3). Then

for sufficiently large T.

Proof

Note that a(x) ∈ C¹(𝓜), then there exists an open set Ω₀ ∈ 𝓜, such that

Let b(x) ∈ C^∞(𝓜) be a nonnegative function satisfying

Let

Note that

It follows from (3.2) that

Let P=divgH2 in (3.3). Substituting (3.3) into (4.23), we obtain

Then

Therefore

Hence

With (3.16) and (3.17), we deduce that

and

Substituting (4.28) and (4.29) into (4.27), for T > 5C, we have

Therefore

The estimate (4.16) holds true. □

Lemma 4.6

Let assumption (A) and assumption (B) hold true. Let T be sufficiently large. Then for any ∥u₀∥_L^2(𝓜) ≤ E₀, there exists positive constant C(E₀, T) such that

Proof

We apply compactness-uniqueness arguments to prove the conclusion. It follows from (4.16) that

Then, if the estimate (4.16) doesn’t hold true, there exist {uk}k=1∞ satisfying (1.3) and

Thus,

where

It follows from (3.16), (3.17) and (4.36) that there exists C(T) > 0 such that

which implies

Therefore, there exists û and a subset of {uk}k=1∞ , still denoted by {uk}k=1∞ , such that

and

Note that

Lebesgue’s dominated convergence theorem yields

1. ∫0T∫M|u^|2dxgdt>0. (4.44)

   Note that

   H1M↪L2pM. (4.45)

   Therefore, it follows from (4.39) that

   {|uk|p−1uk}areboundedin  L2(M×(0,T)). (4.46)

   Hence, there exists a subset of {uk}k=1∞ , still denoted by {uk}k=1∞ , such that

   |uk|p−1uk→|u^|p−1u^weaklyinL2(M×(0,T)). (4.47)

   It follows from (4.34) and(4.35) that

   a(x)u^=0(x,t)∈M×(0,T). (4.48)

   Therefore, with (4.40) and (4.47), we obtain

   iu^0t+Δgu^−|u^|p−1u^=0(x,t)∈M×(0,T),a(x)u^=0(x,t)∈M×(0,T). (4.49)

   With (4.1) and Lemma 4.3, we have

   u^≡0onM×(0,T), (4.50)

   which contradicts (4.44).

2. u^≡0onM×(0,T). (4.51)

Denote

where

Then v_k satisfies

and

It follows from (4.35) that

Therefore, it follows from (4.33) that