Exercise 2.1. Proof of the theorem about compact subsets and maximal solutions.

UNS Nonlinear differential equations 2017-2018

2. Interval of existence of maximal solutions

Exercise 2.1. Proof of the theorem about compact subsets and maximal solutions.

We x a compact subset A ⊂ U , a solution ϕ : (α, β) → R ^d of (S) and t 0 ∈ (α, β) . We suppose that

∀t ∈ (t 0 , β), (t, ϕ(t)) ∈ A.

1. Prove that β < +∞ and that there exists l ∈ R ^d and a sequence (t n ) _n∈N in (t 0 , β) ^N such that lim _n→+∞ t n = β and

n→+∞ lim ϕ(t n ) = l.

2. Let ϕ ˜ : (α, β] → R ^d be the map whose restriction to (α, β) is ϕ and such that ϕ(β) = ˜ l . Prove that ϕ ˜ is a solution of (S) .

3. Complete the proof of the theorem.

Exercise 2.2 What can be said about the interval of existence of maximal solutions of the following dierential equations ?

(E 1 ) y ⁰ (t) = cos(e ^t y(t)) (E 2 ) y ⁰ (t) = cos(y(t))t ² + arctan(y(t)).

Exercise 2.3 Let f : R ^d → R ^d be a C ¹ map. Suppose there exists R > 0 suxh that, for any point x ∈ R ^d with kxk ≥ R , f (x) vanishes. Prove that any solution of the dierential equation y ⁰ = f (y) is bounded. What can be deduced about the interval of denition of maximal solutions ?

Exercise 2.4

1. Prove that maximal solutions of the following dierential equations are dened on R.

x ⁰ = xy ² y ⁰ = −yx ²

2. Prove that maximal solutions of the following dierential equations are dened on a neighbourhood of +∞ .

x ⁰ = −xy ² y ⁰ = −yx ²

Exercise 2.5 We denote by (S) the following dierential system x ⁰ = x ³ + y ² x

y ⁰ = y ³ + yx ² .

Let I → R

t 7→ (x(t), y(t)) be a maximal solution of (S) and let f (t) = x(t) ² + y(t) ² .

1. Find a dierential equation (E) satised by f . 2. What can we deduce about the interval I ? Exercise 2.6. Proof of the linear Gronwall lemma.

1. Let f : I → R a dierentiable function dened on an open interval I . Let t 0 ∈ I . Assume there exists A > 0 and B ≥ 0 such that

∀t ∈ [t ₀ , +∞) ∩ I, f ⁰ (t) ≤ Af(t) + B.

Prove that

∀t ∈ [t 0 , +∞) ∩ I, f(t) ≤ f (t 0 )e ^(t−t

^)A + B

A (e ^(t−t

^)A − 1).

2. Prove the linear Gronwall lemma in the case where, for any t ∈ [t 0 , +∞) ∩ I, kϕ(t)k >

0 . Hint : take ψ(t) = kϕ(t)k and use the rst question.

3. Prove the linear Gronwall lemma in the general case.

Exercise 2.7 What can be said about the interval of existence of maximal solutions of the following dierential systems ?

(S 1 )

x ⁰ = arctan(y)x + 2y + cos(t) y ⁰ = sin(y)x

(S ₂ )

( x ⁰ = −3x + 2e ^−x

y + _1+y ¹

y ⁰ = − cos(ty)x + _1+x+x ^y

+ arctan(t) .

Exercise 2.8. Linear dierential systems.

1. Prove that linear dierential systems satisfy the hypothesis of the existence and uni- queness theorem.

2. Prove the theorem about the interval of denition of maximal solutions of linear dierential systems.

Exercise 2.9. Continuity with respect to the initial solution For > 0 , let K =

(t, p) ∈ [t − , t + ] × R ^d ,

p − ϕ ^t _t

(x 0 ) ≤ . 1. Prove that there exists > 0 such that K ⊂ U .

2. In what follows, we x > 0 such that the above property is satised. Prove that there exists k > 0 such that, for any points (t, p) ∈ K and (t, p ⁰ ) ∈ K ,

kf (t, p) − f (t, p ⁰ )k ≤ k kp − p ⁰ k .

3. Fix δ > 0 such that δe ^k(t

^−t

⁾ < . Let V =

y 0 ∈ R ^d , ky 0 − x 0 k < δ .

Prove the theorem of continuity with respect to the initial condition with k > 0 and V dened as above.

Exercise 2.10 What can be said about the interval of existence of maximal solutions y _max of the dierential equation y ⁰ = (cos(ty) + 2)y ² which are dened at 0 in terms of y ₀ = y _max (0) . Exercise 2.11 Let U be an open set of R ² and f : U → R ² be a continuous function which is locally lipschitzian with respect to the phase variable. Denote by (E) the dierential equation x ⁰ = f (t, x) . Let ϕ : (α, β) → R be a subsolution of (E) and ρ : (α, β) → R be a solution of (E) . Let t 0 ∈ (α, β) .

1. Suppose that ϕ(t 0 ) ≤ ρ(t 0 ) and that, for any t ∈ [t 0 , β) , ϕ ⁰ (t) < f (t, ϕ(t)) . Suppose for a contradiction that there exists t 1 > t 0 such that ϕ(t 1 ) > ρ(t 1 ) . We let

τ = inf {t ∈ [t 0 , β), ϕ(t) > ρ(t)} . (a) Prove that ϕ ⁰ (τ) < ρ ⁰ (τ ) .

(b) Find a contradiction. What can we conclude ?

2. Now, we just suppose that ϕ(t 0 ) ≤ ρ(t 0 ) and that ϕ is a subsolution of (E) . Deduce from the rst part of the exercise that, for any t ∈ [t 0 , β) , ϕ(t) ≤ ρ(t) by using, for > 0 , f (t, x) = f (t, x) + .

Exercise 2.12 Adapt Exercise 2.11 to provide a proof of the nonlinear higher-dimensional

Gronwall lemma.