Dynamics of nonlinear evolutionequations at resonance

Piotr Kokocki

Faculty of Mathematics and Computer ScienceNicolaus Copernicus University of Torun

1 July 2012STS Geometry in Dynamics

6th European Congress of Mathematicians

IntroductionLet Ω be an open subset Rn (n ≥ 1). Consider nonlinear heat equation

ut(x, t) = ∆u(x, t) + λu(x, t) + f(t, x, u(x, t)), t ≥ 0, x ∈ Ω

and nonlinear strongly damped wave equation

utt(x, t) = ∆u(x, t)+c∆ut(x, t)+λu(x, t)+f(t, x, u(x, t)), t ≥ 0, x ∈ Ω

where

I c > 0 and λ is a real number,

I ∆ is the Laplace operator with Dirichlet boundary conditions,

I f : [0,+∞)× Ω× R→ R is a continuous map.

We will study the dynamics for these equations, and more precisely

I the existence of T -periodic solutions (T > 0)

I the existence of orbits connecting stationary points

in the case of rezonanse at infinity, that is, λ is an eigenvalue of −∆,and f is bounded.

Main difficultiesIn the presence of resonance, the above equations may not have periodicsolutions and orbits connecting stationary points for general nonlinearityf . This fact has been explained in detail in my phd. To justify it, we put

f(t, x, s) := u0(x) for x ∈ Ω, where u0 ∈ Ker(λI + ∆) \ 0.

Therefore, if we want to look for periodic solutions and connecting orbits,we have to find additional assumptions characterizing f , which allow usto obtain the existence results.

I The classical assumptions for f that guarantee the existence ofperiodic solutions for the nonlinear heat and wave equations are:

• Landesman-Lazer conditions ,

• strong resonance conditions .

I The existence of orbits connecting stationary points for the heat andwave equations has not been investigated so far.

Landesman-Lazer conditions

Assume that there are continuous functions f+, f− : Ω→ R such that

f+(x) = lims→+∞

f(t, x, s) and f−(x) = lims→−∞

f(t, x, s).

We say that f satisfies Landesman-Lazer condition, provided∫u>0

f+(x)u(x) dx+

∫u<0

f−(x)u(x) dx > (<) 0

for u ∈ Ker(λI + ∆), u 6= 0.

Strong resonance conditions

Assume that there is a continuous function f∞ : Ω→ R such that

f∞(x) = lim|s|→+∞

f(t, x, s) · s for t ∈ [0,+∞), x ∈ Ω.

We say that f satisfies strong resonance condition, provided∫Ω

f∞(x) dx > (<) 0.

The following questions arise:

1. Do Landesman-Lazer and strong resonance conditions imply theexistence of orbits connecting the stationary points ?

2. Can we give new geometric conditions characterizing f , other thanLandesman-Lazer or strong resonance conditions ?

3. If this is the case, how these new conditions will affect on theexistence of periodic solutions and connecting orbits ?

4. Are these new geometric conditions more general than the classicalLandesman-Lazer or strong resonance conditions ?

MethodsLet A : X ⊃ D(A)→ X be a positively defined sectorial operator on aBanach space X and let Xα for α ∈ (0, 1), be a fractional space given byXα := D(Aα). The nonlinear heat and wave equation may be abstractlywritten in the form

u(t) = −Au(t) + λu(t) + F (t, u(t)), t > 0,(1)

u(t) = −Au(t)− cAu(t) + λu(t) + F (t, u(t)), t > 0,(2)

where F : [0,+∞)×Xα → X is a continuous map. It suffices to takeAu := −∆u and F (t, u) := f(t, ·, u(·)).

We say that these abstract equations are at resonance at infinity,provided Ker(λI −A) 6= 0 and F is bounded.

Our goals figure on finding periodic solutions and orbits connectingstationary points for the equations (1) and (2).

The second order equation (2) may be written as the first order equation

(3) w(t) = −Aw(t) + F(t, w(t)), t > 0,

where A : E ⊃ D(A)→ E is a linear operator on a Banach spaceE := Xα ×X given by

D(A) := (x, y) ∈ E | x+ cy ∈ D(A)A(x, y) := (−y,A(x+ cy)− λx) for (x, y) ∈ D(A),

and F : [0,+∞)×E→ E is given by

F(t, (x, y)) := (0, F (t, x)) for t ∈ [0,+∞), (x, y) ∈ E.

Let u( · ;x) : [0,+∞)→ Xα be a solution for (1) starting at x ∈ Xα.

Similarly, let w( · ; (x, y)) : [0,+∞)→ E be a solution for (3) starting at(x, y) ∈ E.

Define the operators associated with the equations

u(t) = −Au(t) + λu(t) + F (t, u(t)) ∼ Φ : [0,+∞)×Xα → Xα,

u(t) = −Au(t)− cAu(t) + λu(t) + F (t, u(t)) ∼ Φ : [0,+∞)×E→ E

which are given by the formulas

Φ(t, x) := u(t;x) for t ∈ [0,+∞), x ∈ Xα,

Φ(t, (x, y)) := w(t; (x, y)) for t ∈ [0,+∞), (x, y) ∈ E.

Periodic solutions

Finding periodic solutions is reduced to finding fixed points fortranslation along trajectory operators ΦT : Xα → Xα andΦT : E→ E given by

ΦT := Φ(T, · ) and ΦT := Φ(T, · ).

In order to find fixed point of these operators we will use homotopyinvariants such as topological degree.

Orbits connecting stationary points

Here we assume that F : Xα → X is time independent map.

By a stationary point x0 ∈ Xα for equation (1) we mean a timeindependent solution, that is,

Φ(t, x0) = x0 for t ∈ [0,+∞).

By an orbit for equation (1) we mean a solution for semiflow Φ, that is,a map u : R→ Xα satisfying

Φ(s, u(t)) = u(t+ s) for s ≥ 0, t ∈ R.

We say that the orbit u : R→ Xα connects stationary points u−, u+, if

limt→−∞

u(t) = u− and limt→+∞

u(t) = u+.

The tool for finding the connecting orbits is the Conley index.

For the second order equation (2) the above definitions are analogous.

Plan of the studies

1. We find the geometrical conditions for nonlinear term F , whichguarantee the existence of periodic solutions and orbits connectingstationary points for the first and second order equations.

2. We prove that conditions from point 1. generalize the classicalLandesman-Lazer and strong resonance conditions.

3. We use the conditions from point 1. to prove the index formulas andcriteria for the existence of periodic solutions for the first and secondorder equations.

4. We use these conditions to prove the index formulas and criteria forthe existence of orbits connecting the stationary points for the firstand second order equations.

5. Applications to study heteroclinic orbits and periodic solutions ofpartial differential equations.

Geometrical assumptionsLet A : X ⊃ D(A)→ X be a positively defined sectorial operator on aBanach space X such that the following assumptions are satisfied:

(A1) operator A has a compact resolvents,

(A2) there is a Hilbert space H with a scalar product 〈 · , · 〉H and norm‖ · ‖H and linear continuous injective map (inclusion) i : X → H,

(A3) there is a self-adjoint operator A : H ⊃ D(A) → H such that

Gr (A) ⊂ Gr (A) in the sense of inclusion X ×X i×i−−→ H ×H.

Under the above assumptions (A1), (A2) and (A3):

1. Spectrum σ(A) of the operator A consists of the sequence ofeigenvalues

0 < λ1 < λ2 < . . . < λi < . . . ,

which is finite or λn → +∞ as n→ +∞.

2. Let λ = λk. There is a decomposition of X on the direct sum ofclosed spaces X := X− ⊕X0 ⊕X+ such that:

• X0 = Ker(λkI −A),

• X− is finite dimensional,

• if A± : X± ⊃ D(A±) → X± is the part of A in X±, thenλkI −A+ is positive and λkI −A− is negative.

3. Since the injection Xα ⊂ X is continuous, it follows that there is adecomposition on a sum of closed spaces Xα = Xα

− ⊕X0 ⊕Xα+,

whereXα− := Xα ∩X− and Xα

+ := Xα ∩X+.

• The part of the operator A in space Y ⊂ X is an operator AY given by

D(AY ) := x ∈ D(A) | Ax ∈ Y , AY x := Ax for x ∈ D(AY ).

Assumptions for the first order equations:

(G1)

for every ball B ⊂ Xα

+ ⊕Xα− there is R > 0 such that

〈F (t, x+ y), x〉H > 0 for (t, y, x) ∈ [0, T ]×B ×X0, ‖x‖H ≥ R,

(G2)

for every ball B ⊂ Xα

+ ⊕Xα− there is R > 0 such that

〈F (t, x+ y), x〉H < 0 for (t, y, x) ∈ [0, T ]×B ×X0, ‖x‖H ≥ R,

Assumptions for the second order equations:

(G3)

for every balls B1 ⊂ Xα

+ ⊕Xα− and B2 ⊂ X0 there is R > 0

such that 〈F (t, x+ y), x〉H > −〈F (t, x+ y), z〉Hfor (t, y, z) ∈ [0, T ]×B1 ×B2, x ∈ X0 such that ‖x‖H ≥ R.

(G4)

for every balls B1 ⊂ Xα

+ ⊕Xα− and B2 ⊂ X0 there is R > 0

such that 〈F (t, x+ y), x〉H < −〈F (t, x+ y), z〉Hfor (t, y, z) ∈ [0, T ]×B1 ×B2, x ∈ X0 such that ‖x‖H ≥ R.

Orbits connecting stationary pointsIn order to find the orbits connecting stationary points:

1. we prove index formulas expressing the Conley index for semiflowsΦ and Φ on sufficiently large ball in the terms of geometricalassumptions (G1)–(G4),

2. we apply theorems concerning irreducible invariant sets proved byRybakowski (1980).

From now on we assume that λ = λk for some k ≥ 1, is an eigenvalue ofthe operator A.

Furthermore, for any l ≥ 0 we define

dl :=

0 for l = 0,∑li=1 dim Ker(λiI −A) for l ≥ 1.

Then one can check that dk−1 = dimX− for k ≥ 1.

Theorem 1 (Index formula for first order equations)There is a closed isolated neighborhood N ⊂ Xα, admissible with respectto the semiflow Φ such that 0 ∈ intN and, for K := Inv(N,Φ), thefollowing statements hold:

(i) if condition (G1) is satisfied, then h(Φ,K) = Σdk ,

(ii) if condition (G2) is satisfied, then h(Φ,K) = Σdk−1 .

Theorem 2 (Index formula for second order equations)There is a closed isolated neighborhood N ⊂ E, admissible with respectto the semiflow Φ such that 0 ∈ intN and, for K := Inv(N,Φ), thefollowing statements hold:

(i) if condition (G3) is satisfied, then h(Φ,K) = Σdk ,

(ii) if condition (G4) is satisfied, then h(Φ,K) = Σdk−1 .

• If (X,x0) is a topological space and x0 ∈ X, then the reduciblesuspension is the topological space defined by

ΣX := (X × [0, 1])/(X × 0 ∪X × 1 ∪ x0 × [0, 1]).

Then Σ0 := S0 and Σk := Σ(Σk−1) for k ≥ 1.

• An invariant set in N with respect to Φ is given by

Inv(N,Φ) := x ∈ N | there is an orbit u : R→ Xα for ϕ

such that u(0) = x and u(R) ⊂ N.

• A closed invariant set K ⊂ X is called isolated, if there is a closed setN ⊂ X such that

K = Inv(N,Φ) ⊂ intN.

Such a set N is referred to as isolated neighborhood for K.

Additionally, we assume that:

(A4) F (0) = 0 and F is differentiable at 0 and there is ν ∈ R such thatDF (0)[x] = νx for x ∈ Xα.

Theorem 3 (Criterion for the first order equation)Equation (1) admits a relatively compact nonzero orbit u : R→ Xα suchthat

limt→−∞

u(t) = 0 or limt→+∞

u(t) = 0,

provided one of the following conditions holds:

(i) condition (G1) is satisfied and λl < λ+ ν < λl+1 where λl 6= λ;

(ii) condition (G1) is satisfied and λ+ ν < λ1;

(iii) condition (G2) is satisfied, λl−1 < λ + ν < λl and λ 6= λl, wherel ≥ 2;

(iv) condition (G2) is satisfied, λ+ ν < λ1 and λ 6= λ1.

Theorem 4 (Criterion for the second order equation)Equation (2) admits a relatively compact nonzero orbit w : R→ E suchthat

limt→−∞

w(t) = 0 or limt→+∞

w(t) = 0,

provided one of the following conditions holds:

(i) condition (G3) is satisfied and λl < λ+ ν < λl+1 where λl 6= λ;

(ii) condition (G3) is satisfied and λ+ ν < λ1;

(iii) condition (G4) is satisfied, λl−1 < λ + ν < λl and λ 6= λl, wherel ≥ 2;

(iv) condition (G4) is satisfied, λ+ ν < λ1 and λ 6= λ1.

Periodic solutionsTo find periodic solutions we will use index formulas expressingtopological degree for the operators ΦT and ΦT with respect tosufficiently large ball, in the terms of conditions (G1)–(G4).

I ΦT is completely continuous – Leray-Schauder degree degLS,

I ΦT is condensing – Sadovskii degree degC.

Let N be a Banach space.

• By measure of noncompactness we mean map βN : B(N)→ R given by

βN (Ω) := infr > 0 | Ω ⊂kr⋃i=1

B(xi, r), where xi ∈ N for i = 1, . . . , kr.

• A map F is completely continuous, provided F (Ω) is relatively compact for everybounded Ω ⊂ N .

• A map F is condensing, provided there is k ∈ [0, 1) such that

βN (F (Ω)) ≤ kβN (Ω)

for every bounded Ω ⊂ N .

Theorem 5 (Index formula for the first order equations)The following statements are true.

(i) If condition (G1) is satisfied, then there is R > 0 such that ΦT (x) 6=x for x ∈ Xα with norm ‖x‖α ≥ R and

degLS(I − ΦT , B(0, R)) = (−1)dk .

(ii) If condition (G2) is satisfied, then there is R > 0 such that ΦT (x) 6=x for x ∈ Xα with norm ‖x‖α ≥ R and

degLS(I − ΦT , B(0, R)) = (−1)dk−1 .

Theorem 6 (Index formula for the second order equations)The following statements are true.

(i) If condition (G3) is satisfied, then there is R > 0 such thatΦT (x, y) 6= (x, y) for (x, y) ∈ E with norm ‖(x, y)‖E ≥ R and

degC(I −ΦT , B(0, R)) = (−1)dk .

(ii) If condition (G4) is satisfied, then there is R > 0 such thatΦT (x, y) 6= (x, y) for (x, y) ∈ E with norm ‖(x, y)‖E ≥ R and

degC(I −ΦT , B(0, R)) = (−1)dk−1 .

Application to partial differential equationsLet Ω ⊂ Rn for n ≥ 1, be an open bounded set with C∞ boundary.Assume that:

I ∆ is Laplace operator with Dirichlet conditions,

I f : [0,+∞)× Ω× R× Rn → R is a continuous, bounded map,which is lipschitz on bounded sets.

Let X := Lp(Ω) for p ≥ 1. Define Ap : X ⊃ D(Ap)→ X as linearoperator given by

D(Ap) := W 2,p0 (Ω) and Apu := −∆u for u ∈ D(Ap).

Then Ap is a positive definite sectorial operator.

I Assume that α ∈ (3/4, 1) and p ≥ 2n.

Embedding theorems for fractional spaces imply that the inclusionXα ⊂ C(Ω) is continuous.

Therefore we can define the Niemycki operator F : [0,+∞)×Xα → Xgiven for every u ∈ Xα by the formula

F (t, u)(x) := f(t, x, u(x)) for t ∈ [0,+∞) and x ∈ Ω.

1. One can prove that A2 is symmetric and Ap has compact resolvents.Hence assumption (A1) is satisfied.

2. Take H := L2(Ω) with standard inner product and norm. Since Ω isbounded and p ≥ 2, we derive that i : Lp(Ω) → L2(Ω) is acontinuous embedding. In consequence assumption (A2) is satisfied.

3. Using again the boundedness of Ω, one can prove that for theoperator A := A2, the inclusion Ap ⊂ A is satisfied in the sense ofthe map i× i. Therefore (A3) holds.

Theorem 7 (Properties of Niemycki operator)Assume that f+, f− : Ω→ R are continuous functions such that

f+(x) = lims→+∞

f(t, x, s) and f−(x) = lims→−∞

f(t, x, s)

for x ∈ Ω, uniformly for t ∈ [0,+∞).

(i) If the following condition is satisfied

(LL1)

∫u>0

f+(x)u(x) dx+

∫u<0

f−(x)u(x) dx > 0

for u ∈ Ker(λI −Ap) \ 0, then conditions (G1) and (G3) hold.

(ii) If the following condition is satisfied

(LL2)

∫u>0

f+(x)u(x) dx+

∫u<0

f−(x)u(x) dx < 0

for u ∈ Ker(λI −Ap) \ 0, then conditions (G2) and (G4) hold.

Theorem 8 (Properties of Niemycki operator)Assume that f∞ : Ω→ R is a continuous function such that

f∞(x) = lim|s|→+∞

f(t, x, s) · s

for x ∈ Ω, uniformly for t ∈ [0,+∞).

(i) The following inequality

(SR1)

∫Ω

f∞(x) dx > 0,

implies that conditions (G1) and (G3) are satisfied.

(i) The following inequality

(SR2)

∫Ω

f∞(x) dx < 0.

implies that conditions (G2) and (G4) are satisfied.

We consider the following equations

ut(t, x) = ∆u(t, x) + λu(t, x) + f(t, x, u(t, x))(4)

utt(t, x) = ∆u(t, x) + ∆ut(t, x) + λu(t, x) + f(t, x, u(t, x)).(5)

Theorem 9 (Existence of periodic solutions)

1. Assume that f+, f− : Ω→ R are continuous functions such that

f+(x) = lims→+∞

f(t, x, s), f−(x) = lims→−∞

f(t, x, s) for x ∈ Ω.

If one of the conditions (LL1) or (LL2)is satisfied, then equations(4) and (5) admit T -periodic solutions.

2. Assume that there is a continuous function f∞ : Ω→ R such that

f∞(x) = lim|s|→+∞

f(t, x, s) · s for x ∈ Ω.

If one of the conditions (SR1) or (SR2) is satisfied, then equations(4) and (5) admit T -periodic solution.

Assume that f : Ω× R→ R is a map of class C1 such that

I f(x, 0) = 0 for x ∈ Ω

I there is ν ∈ R such that ν = Dsf(x, 0) for x ∈ Ω.

Theorem 10 (Existence of heteroclinic orbits)Assume that f+, f− : Ω→ R are continuous functions such that

f+(x) = lims→+∞

f(x, s), f−(x) = lims→−∞

f(x, s) for x ∈ Ω.

Equations (4) and (5) admit heteroclinic orbit, provided one of thefollowing conditions is satisfied:

(i) condition (LL1) is satisfied and λl < λ+ ν < λl+1 gdzie λl 6= λ;

(ii) condition (LL1) is satisfied and λ+ ν < λ1;

(iii) condition (LL2) is satisfied, λl−1 < λ+ ν < λl and λ 6= λl, wherel ≥ 2;

(iv) condition (LL2) is satisfied, λ+ ν < λ1 and λ 6= λ1.

Theorem 11 (Existence of heteroclinic orbits)Assume that f∞ : Ω→ R is a continuous function such that

f∞(x) = lim|s|→+∞

f(x, s) · s for x ∈ Ω.

Equations (4) and (5) admit heteroclinic orbit, provided one of thefollowing conditions holds:

(i) condition (SR1) is satisfied and λl < λ+ ν < λl+1 where λl 6= λ;

(ii) condition (SR1) is satisfied and λ+ ν < λ1;

(iii) condition (SR2) is satisfied and λl−1 < λ + ν < λl where λ 6= λl,l ≥ 2;

(iv) condition (SR2) is satisfied and λ+ ν < λ1.

Literatura

I Rybakowski K. P., The homotopy index and partial differentialequations, Universitext, Springer-Verlag 1987;

I Rybakowski K. P., Nontrivial solutions of elliptic boundary valueproblems with resonance at zero, Ann. Mat. Pura Appl. 139 (1985)237–277;

I Rybakowski K. P., Irreducible invariant sets and asymptotically linearfunctional-differential equations, Boll. Un. Mat. Ital. B (6) 3(1984), no. 2, 245–271;

I Landesman E. M., Lazer A. C., Nonlinear perturbations of linearelliptic boundary value problems at resonance, J. Math. Mech. 19(1970) 609–623.

I Bartolo P., Benci V., oraz Fortunato D., Abstract critical pointtheorems and applications to some nonlinear problems with “strong”resonance at infinity, Nonlinear Anal. 7 (1983), no. 9, 981–1012.

Lapunov functions V : Xα → R and V : E→ R for equations (4) and(5), respectively are given by

V (u) :=1

2

(‖∆1/2u‖2L2(Ω) − ‖u‖

2L2(Ω)

)−∫

Ω

f(x, u(x)) dx,

V (u, v) :=1

2

(‖∆1/2u‖2L2(Ω) + ‖v‖2L2(Ω) − ‖u‖

2L2(Ω)

)−∫

Ω

f(x, u(x)) dx.