Traveling-wave solutions for models of collective movements with - - PowerPoint PPT Presentation

Introduction The problem Results

Traveling-wave solutions for models of collective movements with degenerate diffusivities

Andrea Corli

Department of Mathematics and Computer Science University of Ferrara - Italy

June 13-17th 2016 11th Meeting on Nonlinear Hyperbolic PDEs and Applications On the occasion of the 60th birthday of Alberto Bressan S.I.S.S.A. Trieste Joint work with Luisa Malaguti and Lorenzo di Ruvo (University of Modena - Reggio Emilia, Italy)

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Introduction The problem Results

A tribute to Alberto

ut + A(u)ux = ǫuxx

2 / 31

Introduction The problem Results

A tribute to Alberto

ut + A(u)ux = ǫuxx

• S. Bianchini - A. Bressan: Vanishing viscosity solutions of nonlinear hyperbolic

systems, Ann. Math. 161 (2005), 223–342, dedicated to Prof. Constantine Dafermos

• n the occasion of his 60th birthday.

2 / 31

Introduction The problem Results

A tribute to Alberto

ut + A(u)ux = ǫuxx

• S. Bianchini - A. Bressan: Vanishing viscosity solutions of nonlinear hyperbolic

systems, Ann. Math. 161 (2005), 223–342, dedicated to Prof. Constantine Dafermos

• n the occasion of his 60th birthday.

. . . and if ǫ = ǫ(u)? In the case n = 1, for special solutions. . .

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Introduction The problem Results

A simple motivation

The simplest continuum (macroscopic) model for collective movements (traffic flow) is Lighthill-Whitham-Richards (1955) equation (conservation of mass) ρt +

• ρv(ρ)
• x = 0
• r

ρt + q(ρ)x = 0 (LWR) where t, x ∈ R ρ ∈ [0, ρ] is the density (of cars, pedestrians), ρ the maximal density v = v(ρ) the (assigned) speed, usually v′(ρ) < 0 and v(ρ) = 0 q = q(ρ) = ρv(ρ) the flux: usually q′′(ρ) < 0 (ok if v′′(ρ) < 0).

✲ ✻

ρ v ρ

✲ ✻

ρ q ρ Huge literature, in particular in the last twenty years: systems (different populations), networks, control.

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Introduction The problem Results

Diffusion in collective movements

Lighthill and Whitham proposed to include a linear diffusion term in (LWR) to avoid the appearance of shock waves: ρt +

• ρv(ρ)
• x = Dρxx.

The density ρ is still conserved. What is the “correct” diffusion? Several approaches: Payne (1971), Nelson (2000), Helbing (2001).

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Introduction The problem Results

Diffusion in collective movements

Lighthill and Whitham proposed to include a linear diffusion term in (LWR) to avoid the appearance of shock waves: ρt +

• ρv(ρ)
• x = Dρxx.

The density ρ is still conserved. What is the “correct” diffusion? Several approaches: Payne (1971), Nelson (2000), Helbing (2001). The simplest idea for generating diffusion (vehicular flow): drivers adjust their speed to the local density; however, there is a reaction time τ in the response to events; drivers compensate for this delay by adjusting to the density seen at some anticipation distance δ ahead of their current position.

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Introduction The problem Results

Diffusion in collective movements

Lighthill and Whitham proposed to include a linear diffusion term in (LWR) to avoid the appearance of shock waves: ρt +

• ρv(ρ)
• x = Dρxx.

The density ρ is still conserved. What is the “correct” diffusion? Several approaches: Payne (1971), Nelson (2000), Helbing (2001). The simplest idea for generating diffusion (vehicular flow): drivers adjust their speed to the local density; however, there is a reaction time τ in the response to events; drivers compensate for this delay by adjusting to the density seen at some anticipation distance δ ahead of their current position. The actual mean speed is then V (x, t)= v

• ρ(x + δ − vτ, t − τ)
• ∼ v + δv′ρx − τv′ρx − τv′ρt = v + δv′ρx + τ(v′)2ρρx

expanding in δ and τ at first order. Then ˜ q = q + ρ

• δv′(ρ) + τρ
• v′(ρ)

2 ρx and we end up with ρt +

• ρv(ρ)
• x =
• D(ρ)ρx
• x

for D(ρ) = −δρv′(ρ) − τρ2 v′(ρ) 2 .

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Introduction The problem Results

Properties of the diffusivity

Since v′(ρ) < 0: D(ρ) = −ρ

• δv′(ρ)

≤0

+ τρ

• v′(ρ)

2

• ≥0
• = Dδ(ρ)

≥0

+ Dτ(ρ)

≤0

. Usually drivers anticipate so as to more than compensate for the delay: D(ρ) ≥ 0. There are situations where the reaction time dominates anticipation: D(ρ) < 0 for large ρ. E.g.: roads providing limited sight distance ahead (fog).

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Introduction The problem Results

Properties of the diffusivity

Since v′(ρ) < 0: D(ρ) = −ρ

• δv′(ρ)

≤0

+ τρ

• v′(ρ)

2

• ≥0
• = Dδ(ρ)

≥0

+ Dτ(ρ)

≤0

. Usually drivers anticipate so as to more than compensate for the delay: D(ρ) ≥ 0. There are situations where the reaction time dominates anticipation: D(ρ) < 0 for large ρ. E.g.: roads providing limited sight distance ahead (fog). D(0) = 0: if no density, then no diffusivity!

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Introduction The problem Results

Properties of the diffusivity

Since v′(ρ) < 0: D(ρ) = −ρ

• δv′(ρ)

≤0

+ τρ

• v′(ρ)

2

• ≥0
• = Dδ(ρ)

≥0

+ Dτ(ρ)

≤0

. Usually drivers anticipate so as to more than compensate for the delay: D(ρ) ≥ 0. There are situations where the reaction time dominates anticipation: D(ρ) < 0 for large ρ. E.g.: roads providing limited sight distance ahead (fog). D(0) = 0: if no density, then no diffusivity! Analogously one could assume: D(ρ) = 0: if no movement, then no diffusivity!

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Introduction The problem Results

Properties of the diffusivity

Since v′(ρ) < 0: D(ρ) = −ρ

• δv′(ρ)

≤0

+ τρ

• v′(ρ)

2

• ≥0
• = Dδ(ρ)

≥0

+ Dτ(ρ)

≤0

. Usually drivers anticipate so as to more than compensate for the delay: D(ρ) ≥ 0. There are situations where the reaction time dominates anticipation: D(ρ) < 0 for large ρ. E.g.: roads providing limited sight distance ahead (fog). D(0) = 0: if no density, then no diffusivity! Analogously one could assume: D(ρ) = 0: if no movement, then no diffusivity! τ = 0 for pedestrians (Bruno-Tricerri-Tosin-Venuti (2011)). There: v(ρ) = v

• 1 − e−γ
• 1

ρ − 1 ρ

• ,

D(ρ) = −δρv′(ρ) ≥ 0, ρ ∈ [0, ρ] where γ > 0 is obtained through experimental data. v and D are exponentially flat at 0 (L=leisure, C=commuters, R=rush hours).

1 2 3 4 5 6 ρ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v(ρ) Velocities [BTTV] L C R 1 2 3 4 5 6 ρ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ρ· v(ρ) Fluxes [BTTV] L C R 1 2 3 4 5 6 ρ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 D(ρ) Diffusion coefficients [BTTV] L C R

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Introduction The problem Results

Source terms

Now ρt +

• ρv(ρ)
• x =
• D(ρ)ρx
• x + g(ρ).

Source terms in modeling collective movements are often localized in space (Bagnerini-Colombo-C. (2006)): g = g(x, ρ) = χ[a,b](x)g1(ρ) with entry : g1(ρ) ≥ 0, g(ρ) = 0, g′

1(ρ) ≤ 0;

exit : g1(ρ) ≤ 0, g(0) = 0, g′

1(ρ) ≥ 0.

Entry: g1(ρ) = 0: there is no room for further entries if the maximal density is reached; g′

1(ρ) ≤ 0: the less the cars on the road, the more the cars enter;

for example: g(ρ) = L · (ρ − ρ)α, L > 0, α > 0.

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Introduction The problem Results

Source terms

Now ρt +

• ρv(ρ)
• x =
• D(ρ)ρx
• x + g(ρ).

Source terms in modeling collective movements are often localized in space (Bagnerini-Colombo-C. (2006)): g = g(x, ρ) = χ[a,b](x)g1(ρ) with entry : g1(ρ) ≥ 0, g(ρ) = 0, g′

1(ρ) ≤ 0;

exit : g1(ρ) ≤ 0, g(0) = 0, g′

1(ρ) ≥ 0.

Entry: g1(ρ) = 0: there is no room for further entries if the maximal density is reached; g′

1(ρ) ≤ 0: the less the cars on the road, the more the cars enter;

for example: g(ρ) = L · (ρ − ρ)α, L > 0, α > 0. However, source terms can be “diffused”: g = g(ρ) = g1(ρ). For instance, pedestrians moving along a long corridor (street): if the number of side entries (cross streets) is large, one drops a model with many localized entries for a model with a diffuse source term.

✲

ρ

✻

g ρ entries

✲

ρ

✻

g ρ exits

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Introduction The problem Results

Some references

Degenerate diffusivities are common in physics (porous media, heat conduction), neurophysics and biophysics, chemical physics, biology (chemotaxis), population genetics, tumor growth, and mathematical ecology, jointly with source terms vanishing at some point (Kalashnikov (1987)). Diffusivities changing sign: in traffic, Kerner-Osipov (1994); in biology, Maini-Malaguti (2006) (diffusion-aggregation phenomena, also with source terms). For traveling waves (degenerate diffusivities): Gilding-Kersner (2004). Engler (1985) gave a general and interesting transformation that reduces the degenerate diffusivity to a constant one. Not used in the following, but there are some common features.

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Introduction The problem Results

A general framework

Consider the scalar parabolic (advection-reaction-diffusion) equation ρt + f(ρ)x =

• D(ρ)ρx
• x + g(ρ),

(x, t) ∈ R × [0, +∞), (E) where for some ρ > 0: f ∈ C1[0, ρ], f(0) = 0; denote f′ = h; g ∈ C[0, ρ]; D ∈ C1[0, ρ].

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Introduction The problem Results

A general framework

Consider the scalar parabolic (advection-reaction-diffusion) equation ρt + f(ρ)x =

• D(ρ)ρx
• x + g(ρ),

(x, t) ∈ R × [0, +∞), (E) where for some ρ > 0: f ∈ C1[0, ρ], f(0) = 0; denote f′ = h; g ∈ C[0, ρ]; D ∈ C1[0, ρ]. For the moment, assume that the diffusivity D satisfies one of: (D0) D(ρ) > 0 for ρ ∈ [0, ρ]; (D1) D(ρ) > 0 for ρ ∈ (0, ρ] and D(0) = 0, ˙ D(0) > 0; (D2) D(ρ) > 0 for ρ ∈ (0, ρ] and D(0) = ˙ D(0) = 0. We refer to condition (D) when we indifferently assume one of the above.

✲

ρ

✻

D (D0) (D1) (D2) ρ The nonlinear heat equation (or the porous media equation with 1 < m < 2), does not enter in this framework: D(0) = 0 but ˙ D(0) = ∞.

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Introduction The problem Results

A general framework (2)

About g: (g) g(ρ) > 0 for ρ ∈ [0, ρ) and g(ρ) = 0.

✲ ✻

ρ g ρ Other cases ( ˙ D(0) = ∞, D(ρ) = 0, g modeling exits) in the last part of the talk.

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Introduction The problem Results

Outline and aim: semi-wavefronts for (E)

Travelling-wave solution (TW) of (E) is ρ(x, t) = ϕ(x − ct) for some profile ϕ(ξ) and c ∈ R. Then ϕ = ϕ(ξ) (formally) satisfies in some I ⊆ R

• D(ϕ)ϕ ′ ′ +
• c − h(ϕ)
• ϕ ′ + g(ϕ) = 0.

Note: we cannot integrate because of g.

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Introduction The problem Results

Outline and aim: semi-wavefronts for (E)

Travelling-wave solution (TW) of (E) is ρ(x, t) = ϕ(x − ct) for some profile ϕ(ξ) and c ∈ R. Then ϕ = ϕ(ξ) (formally) satisfies in some I ⊆ R

• D(ϕ)ϕ ′ ′ +
• c − h(ϕ)
• ϕ ′ + g(ϕ) = 0.

Note: we cannot integrate because of g. A monotonic TW between two stationary states of (E) is a wavefront solution; for such g, equation (E) usually supports wavefront solutions. Difficulty: the system has only one equilibrium point: ρ = ρ! Only semi-wavefronts (SWFs) can be constructed: I = (−∞, ̟) or I = (̟, ∞).

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Introduction The problem Results

Outline and aim: semi-wavefronts for (E)

Travelling-wave solution (TW) of (E) is ρ(x, t) = ϕ(x − ct) for some profile ϕ(ξ) and c ∈ R. Then ϕ = ϕ(ξ) (formally) satisfies in some I ⊆ R

• D(ϕ)ϕ ′ ′ +
• c − h(ϕ)
• ϕ ′ + g(ϕ) = 0.

Note: we cannot integrate because of g. A monotonic TW between two stationary states of (E) is a wavefront solution; for such g, equation (E) usually supports wavefront solutions. Difficulty: the system has only one equilibrium point: ρ = ρ! Only semi-wavefronts (SWFs) can be constructed: I = (−∞, ̟) or I = (̟, ∞). We would construct SWFs to (E), extending previous results by Gilding and Kersner; characterize the slope of the profiles when they reach 0; characterize the source terms g providing non-strictly monotonic profiles. The interest of SWFs is lesser than TWs: ρ is only defined in half-planes x − ct ≷ ̟. However, ̟ is arbitrary.

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Introduction The problem Results

Definitions

Definition (Weak traveling waves) Let I ⊆ R and ϕ: I → [0, ρ] such that ϕ ∈ C(I), D(ϕ)ϕ ′ ∈ L1

loc(I) and

• I
• D
• ϕ(ξ)
• ϕ′(ξ) − f
• ϕ(ξ)
• + cϕ(ξ)
• ψ′(ξ) − g
• ϕ(ξ)
• ψ(ξ)
• dξ = 0,

for every ψ ∈ C∞

0 (I). The function ρ(x, t) = ϕ(x − ct) is said a traveling-wave

solution with wave speed c and wave profile ϕ. The TW solution is global if I = R, classical if ϕ is differentiable.

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Introduction The problem Results

Definitions

Definition (Weak traveling waves) Let I ⊆ R and ϕ: I → [0, ρ] such that ϕ ∈ C(I), D(ϕ)ϕ ′ ∈ L1

loc(I) and

• I
• D
• ϕ(ξ)
• ϕ′(ξ) − f
• ϕ(ξ)
• + cϕ(ξ)
• ψ′(ξ) − g
• ϕ(ξ)
• ψ(ξ)
• dξ = 0,

for every ψ ∈ C∞

0 (I). The function ρ(x, t) = ϕ(x − ct) is said a traveling-wave

solution with wave speed c and wave profile ϕ. The TW solution is global if I = R, classical if ϕ is differentiable. Definition (Semi-wavefronts) Let ρ be a TW with ϕ : (̟, +∞) → R and ̟ ∈ R; let ℓ+ ∈ [0, ρ] such that g(ℓ+) = 0. If ϕ is monotonic, non-constant and ϕ(ξ) → ℓ+ as ξ → +∞, then ρ is said a semi-wavefront solution to ℓ+. Similarly, ϕ is said a SWF from ℓ−. In both cases, a SWF is strict if it is not extendible to a global traveling-wave solution.

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Introduction The problem Results

A picture for semi-wavefronts

✲

ξ

✻

ϕ ρ ϕ1 ̟1

❥

ϕ2 ̟2

✯

Figure: A SWF ϕ1 from ρ, a SWF ϕ2 to ρ with ϕ2(̟2) = 0.

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Introduction The problem Results

A starter: two known results in the case g = 0

Consider the case g = 0 (recall h = f′): ρt + f(ρ)x =

• D(ρ)ρx
• x .

(E0)

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Introduction The problem Results

A starter: two known results in the case g = 0

Consider the case g = 0 (recall h = f′): ρt + f(ρ)x =

• D(ρ)ρx
• x .

(E0) Theorem (Gilding and Kersner) Assume condition (D). (i) If c < h(ρ) (c > h(ρ)), then equation (E0) has a classical SWF from ρ (resp., to ρ) with wave speed c; (ii) if c = h(ρ), the same result holds if and only if for some 0 < δ ≤ ρ we have ρ

ρ−s

• h(σ) − h(ρ)
• dσ > 0 (resp., < 0) ,

for all 0 < s < δ; (*) (iii) if c > h(ρ) (c < h(ρ)), then equation (E0) has no classical SWF from ρ (resp., to ρ) with wave speed c.

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Introduction The problem Results

A starter: two known results in the case g = 0

Consider the case g = 0 (recall h = f′): ρt + f(ρ)x =

• D(ρ)ρx
• x .

(E0) Theorem (Gilding and Kersner) Assume condition (D). (i) If c < h(ρ) (c > h(ρ)), then equation (E0) has a classical SWF from ρ (resp., to ρ) with wave speed c; (ii) if c = h(ρ), the same result holds if and only if for some 0 < δ ≤ ρ we have ρ

ρ−s

• h(σ) − h(ρ)
• dσ > 0 (resp., < 0) ,

for all 0 < s < δ; (*) (iii) if c > h(ρ) (c < h(ρ)), then equation (E0) has no classical SWF from ρ (resp., to ρ) with wave speed c. Clearly any ˜ ρ is an equilibrium for (E0); if ˜ ρ = ρ the statement is more complicate. Condition (∗) holds if f is strictly concave (resp., convex) in a neighborhood of ρ. . .

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Introduction The problem Results

A picture for condition (*)

. . . indeed, condition ρ

ρ−s

• h(σ) − h(ρ)
• dσ > 0 (resp., < 0) ,

for all 0 < s < δ; (*) can be written as f(ρ) − f(ρ − s) s > f′(ρ).

✲

ρ

✻

f ρ

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

ρ − s . . . . . . . . . . . . . . . .

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Introduction The problem Results

A picture

✲

ξ

✻

ϕ ρ ϕ1 c < h(ρ) c > h(ρ) ̟1

❥ ✛

ϕ2 ̟2

✯ ✲

Figure: A SWF ϕ1 from ρ, a SWF ϕ2 to ρ with ϕ2(̟2) = 0. Arrows for the case h(ρ) = 0.

The sign of the velocity c is the effect of diffusion, which spreads the wave and makes the function t → ρ(x, t) decrease. If profiles reach 0, they have to stop at 0 because f and D are only defined in [0, ρ]. Since 0 is an equilibrium, in some cases they can be extended to R with 0 (not strict). Then ρ(x, t) = ϕ(x − ct) is defined in R2 and vanishes if x < ̟1 + ct (x > ̟2 + ct): degenerate diffusivities lead to the “finite propagation speed”.

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Introduction The problem Results

A second known result

In the case f is strictly concave, more precise results on WF solutions can be given: Theorem (Gilding and Kersner) Consider (E0) under assumption (D), f strictly concave; fix ρ± ∈ [0, ρ], ρ− = ρ+. Then, wavefront solutions connecting ρ− with ρ+ exist if and only if ρ− < ρ+. Moreover, in all cases we have ϕ(ξ) ∈ (0, ρ) for every ξ ∈ R, except in the case ρ− = 0 and under either (D1) or (D2).

✲

ξ

✻

ϕ ρ ρ+ ρ− ϕ1 ϕ2 In the case ρ− = 0 and (D1) or (D2) lim

ξ→̟− ϕ ′(ξ) =

λ > 0 if (D1) holds, ∞ if (D2) hold.

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Introduction The problem Results

An application

Consider the model of crowd dynamics ρt +

• ρv(ρ)
• x =
• D(ρ)ρx
• x ,

where f(ρ) = ρv(ρ) is strictly concave and D(ρ) = −δρv′(ρ); then h(ρ) = − γv

ρ . If

(D) holds, we deduce that the equation has (i) a semi-wavefront solution from ρ (to ρ) for every c ≤ h(ρ) (resp., c > h(ρ)); (ii) no semi-wavefront solution from ρ (to ρ) if c > h(ρ) (resp., c ≤ h(ρ)).

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Introduction The problem Results

An application

Consider the model of crowd dynamics ρt +

• ρv(ρ)
• x =
• D(ρ)ρx
• x ,

where f(ρ) = ρv(ρ) is strictly concave and D(ρ) = −δρv′(ρ); then h(ρ) = − γv

ρ . If

(D) holds, we deduce that the equation has (i) a semi-wavefront solution from ρ (to ρ) for every c ≤ h(ρ) (resp., c > h(ρ)); (ii) no semi-wavefront solution from ρ (to ρ) if c > h(ρ) (resp., c ≤ h(ρ)). Lemma (Slope at 0) Let c ≤ h(ρ) and ϕ be a classical semi-wavefront profile from ρ. Then, there exists ̟ ∈ R such that ϕ(ξ) → 0 as ξ → ̟−; moreover, lim

ξ→̟− ϕ ′(ξ) =

λ < 0 if (D0) holds, −∞ if (D1) or (D2) hold.

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Introduction The problem Results

Main results: Semi-wavefront solutions to (E)

We come back to equation (E): ρt + f(ρ)x =

• D(ρ)ρx
• x + g(ρ).

Theorem (Semi-wavefront solutions, C.-Malaguti) Consider equation (E) under assumptions (D) and (g). Then, the following holds. (i) For every c ∈ R, equation (E) has a classical strict SWF solution from ρ (to ρ). (ii) Consider a SWF from ρ; then there exists c∗ ∈ R such that in case (D0): lim

ξ→̟− ϕ ′(ξ) ∈ (−∞, 0),

in case (D1): lim

ξ→̟− ϕ ′(ξ) =

             −∞ if c < c∗, r−(c∗) ˙ D(0) if c = c∗, r+(c) ˙ D(0) if c > c∗, in case (D2): lim

ξ→̟− ϕ ′(ξ) =

     −∞ if c ≤ c∗, − g(0) c − h(0) if c > c∗. (iii) If ϕ1, ϕ2 correspond to speeds c1 < c2 and ̟1 = ̟2 =: ̟, then ϕ2(ξ) < ϕ1(ξ), for ξ ∈ (−∞, ̟) with ϕ2(ξ) < ρ.

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Introduction The problem Results

Semi-wavefront solutions: pictures and remarks

✲

ξ

✻

ϕ ρ ϕ1 ̟1

❘

ϕ2 ̟2

✯ ✛

g = 0

✲ g = 0 ✲

D = 0

✛

D = 0

Figure: A strictly decreasing SWF ϕ1 from ρ, a non-strictly increasing SWF ϕ2 to ρ.

Therefore: we have SWF of speed c for every value of c. If g ≡ 0 this is true only for some values of c: if h ≡ 0, then SWFs from ρ move to the left, SWFs to ρ move to the right. If D ≡ 0, g makes the function t → ρ(x, t) increase; as a consequence, SWFs from ρ move to the right and SWFs to ρ move to the left. In presence of both diffusion and source term, these opposite behaviors tune up and lead to the existence of SWFs for every c.

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Introduction The problem Results

Characterization of strictly monotonic solutions

The wave profiles can reach the equilibrium ρ for a finite ξ0. If g = 0 then they assume identically the value ρ for ξ < ξ0 (or ξ > ξ0).

✲

ξ

✻

ϕ ρ

❝

ϕ ϕ ̟3

s

ϕ (i) (ii) ̟1

❘

20 / 31

Introduction The problem Results

Characterization of strictly monotonic solutions

The wave profiles can reach the equilibrium ρ for a finite ξ0. If g = 0 then they assume identically the value ρ for ξ < ξ0 (or ξ > ξ0).

✲

ξ

✻

ϕ ρ

❝

ϕ ϕ ̟3

s

ϕ (i) (ii) ̟1

❘

Theorem (Characterization of strictly monotonic solutions, C.-Malaguti) Consider equation (E) under assumptions (D) and (g); let L > 0 and ρ1 ∈ [0, ρ). (i) If g(ρ) ≤ L(ρ − ρ) for ρ ∈ [ρ1, ρ], then ϕ satisfies ϕ(ξ) < ρ for every ξ in its domain. (ii) If there is α ∈ (0, 1) such that g(ρ) ≥ L(ρ − ρ)α for ρ ∈ [ρ1, ρ], then every ϕ satisfies ϕ(ξ) ≡ ρ on (−∞, ξ] (or on [ξ, +∞)), for some ξ in its domain. As a consequence of D(ρ) = 0, then non-strictly monotone profiles are C1 if they reach ρ for finite ξ: all profiles are classical.

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Introduction The problem Results

Technical tools

We use a singular order reduction of equation (E). Since g ≥ 0, every SWF ϕ is strictly monotone (where 0 ≤ ϕ(ξ) < ρ); let ξ = ξ(ϕ) the inverse function. If z(ϕ) := D(ϕ)ϕ ′ ξ(ϕ)

• ,

ϕ ∈ (0, ρ), then z satisfies the singular equation ˙ z(ϕ) = h(ϕ) − c − D(ϕ)g(ϕ) z(ϕ) , ϕ ∈ (0, ρ). Notice: D(ϕ)g(ϕ) vanishes at 0 (because of D) and at ρ (because of g).

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Introduction The problem Results

Technical tools

We use a singular order reduction of equation (E). Since g ≥ 0, every SWF ϕ is strictly monotone (where 0 ≤ ϕ(ξ) < ρ); let ξ = ξ(ϕ) the inverse function. If z(ϕ) := D(ϕ)ϕ ′ ξ(ϕ)

• ,

ϕ ∈ (0, ρ), then z satisfies the singular equation ˙ z(ϕ) = h(ϕ) − c − D(ϕ)g(ϕ) z(ϕ) , ϕ ∈ (0, ρ). Notice: D(ϕ)g(ϕ) vanishes at 0 (because of D) and at ρ (because of g). Theorem Assume (D) and (g). The existence of a strict SWF from ρ of (E) with speed c is equivalent to the solvability of the problem      ˙ z = h(ϕ) − c − D(ϕ)g(ϕ)

z

, z(ϕ) < 0, ϕ ∈ (0, ρ), z(ρ) = 0. (Ez) Solutions z are meant in the sense z ∈ C0[0, ρ] ∩ C1(0, ρ).

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Introduction The problem Results

A picture for z’s

     ˙ z = h(ϕ) − c − D(ϕ)g(ϕ)

z

, z(ϕ) < 0, ϕ ∈ (0, ρ), z(ρ) = 0. (Ez) Notice z(0) ≤ 0. Recall z(0) =

• D(ϕ)ϕ′

(ξ = 0), so ϕ′(̟) = z(0)

D(0) when ϕ reaches 0.

✲

ϕ

✻

z ρ

❜

z(0)

❜

z(0) Proof: Given ϕ, show ϕ′(ξ) < 0 when 0 < ϕ(ξ) < ρ and ϕ′(ξ) → 0, as ξ → −∞, D

• ϕ(ξ)
• ϕ′(ξ) → ℓ as ξ → ̟−.

Then show that z solves (Ez). Conversely, given z, show that ϕ is strict, i.e., ̟ ∈ R: consider different cases according the values of c; use lower- and upper-solutions, comparison arguments, approximation procedures.

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Introduction The problem Results

Solvability of the first-order problem

Theorem Assume (D) and (g). Then, problem (Ez) is uniquely solvable for every c ∈ R. In case (D0) we have z(0) < 0 for every c. In cases (D1) and (D2) there exists c∗ ∈ R satisfying the estimate 2

• ˙

D(0)g(0) + h(0) ≤ c∗ ≤ 2

• sup

s∈(0,ρ)

D(s)g(s) s + max

ρ∈[0,ρ] h(ρ),

(*) such that (Ez) with z(0) = 0 is solvable if and only if c ≥ c∗.

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Introduction The problem Results

Solvability of the first-order problem

Theorem Assume (D) and (g). Then, problem (Ez) is uniquely solvable for every c ∈ R. In case (D0) we have z(0) < 0 for every c. In cases (D1) and (D2) there exists c∗ ∈ R satisfying the estimate 2

• ˙

D(0)g(0) + h(0) ≤ c∗ ≤ 2

• sup

s∈(0,ρ)

D(s)g(s) s + max

ρ∈[0,ρ] h(ρ),

(*) such that (Ez) with z(0) = 0 is solvable if and only if c ≥ c∗. In cases (D1) and (D2) if c < c∗, then z(0) < 0. In case (D2) the inequalities reduce to h(0) ≤ c∗ ≤ 2

• sup

s∈(0,ρ)

D(s)g(s) s + max

ρ∈[0,ρ] h(ρ).

Proof: use again lower- and upper-solutions and approximation procedures.

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Introduction The problem Results

Composite traveling waves?

Global solutions? Fix a wave speed c and ̟ ∈ R. We have: a SWF ϕ1 from ρ with speed c; a SWF ϕ2 to ρ with the same speed c. by a shift, ϕ1(̟) = ϕ2(̟) = 0. Such profiles are unique. Define (guess) ϕ(ξ) = ϕ1(ξ) if ξ ≤ ̟, ϕ2(ξ) if ξ > ̟, ξ ∈ R. Pasting would be possible only at 0 and when D(0) = 0.

✲

ξ

✻

ϕ ρ ϕ1 ϕ2 ϕ(ξ) ̟

✲

ξ

✻

ϕ ρ ϕ1 ϕ2 ϕ(ξ) ̟

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Introduction The problem Results

No composite traveling waves!

However, pasting is not possible. A motivation: We are given ϕ1 from ρ; let c∗

1 be the threshold for that equation.

Construct ϕ2 to ∞ as ϕ2(ξ) = ψ(−ξ), where ψ(ξ) solves the equation with reversed flux −h and speed −c:

• D(ψ)ψ ′ ′ −
• c − h(ψ)
• ψ ′ + g(ψ) = 0.

Let c∗

2 the threshold for ϕ2.

Compute for ζ a test function, assuming ζ(̟) = 1, ̟

̟−ǫ

• D(ϕ)ϕ′ − f(ϕ) + cϕ
• ζ′ − g(ϕ)ζ
• dξ =

if c ≥ c∗

1,

z1(0) < 0 if c < c∗

1,

̟+ǫ

̟

• D(ϕ)ϕ′ − f(ϕ) + cϕ
• ζ′ − g(ϕ)ζ
• dξ =

−z2(0) < 0 if c > −c∗

2,

if c ≤ −c∗

2.

Only the cases 0 = 0 may match, for c∗

1 ≤ c ≤ −c∗ 2; however c∗ 1 < −c∗ 2 never

holds but for trivial cases.

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Introduction The problem Results

A naive interpretation

Consider the initial-boundary value problem      ρt + f(ρ)x =

• D(ρ)ρx
• x + g(ρ)

x < 0, t > 0, ρ(0, t) = b(t) t > 0, ρ(x, 0) = a(x) x < 0, with 0 ≤ a(x), b(t) ≤ ρ: a pedestrian motion in the half-line x < 0, with initial datum a; pedestrians enter through the x axis or through the boundary x = 0. Choose c > 0 and a SWF ψ from ρ; shift ψ so that it is defined in (−∞, 0]. Consider ρ(x, t) = ψ(x − ct), x < 0, t > 0. Since c > 0, then ρ is well defined and solves the IBVP for a(x) := ψ(x, 0), b(t) = ψ(−ct), x < 0, t > 0. If g(ρ) ≥ L(ρ − ρ)α, with 0 < α < 1, the street can be filled in finite time (even if D(ρ) = 0 we have classical profiles). If g ≤ 0 and D(ρ) = 0, sharp profiles may appear.

✲

ξ

✻

ϕ ρ ψ ψ

✲ g ≥ 0

ϕ

✲ g ≤ 0

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Introduction The problem Results

Related results: the case D(ρ) = 0

If D(ρ) = 0, non-strictly monotone profiles are C1 if they reach ρ for finite ξ: ϕ′(ξ) = z(ρ) D(ρ) = 0. If D(ρ) = 0, then both D and g vanish at ρ; sharp profiles can appear. Definition Let ϕ : I ⊆ R → R a profile; ϕ is sharp at ℓ if g(ℓ) = 0 and there exists ξ0 ∈ I such that ϕ(ξ0) = ℓ, with ϕ classical in I \ {ξ0} and not differentiable at ξ0. In the previous case D(ρ) > 0, every profile was classical.

✲

ξ

✻

ϕ ρ ϕ2 ϕ2 ̟2

s

. . . . . . . . . . . . ξ2 ϕ3 ϕ3 ̟3

s

. . . . . . . . . . . . ξ3 ϕ1 ̟1

❘

ϕ5 ̟5

✯

ϕ4 ϕ4 ̟4

✒

. . . . . . . . . . . . ξ4

Figure: A strictly decreasing SWF ϕ1 from ρ; a strictly increasing SWF ϕ5 to ρ. Non-strictly decreasing, sharp (at ρ) SWF ϕ2 and ϕ3 from ρ; a non-strictly increasing, classical SWF ϕ4 to ρ. While ϕ4 is smooth at ξ4, ϕ2 and ϕ3 are not at ξ2 and ξ3, respectively.

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Introduction The problem Results

Related results: the case D(ρ) = 0

Theorem (Existence of sharp SWF, C.-di Ruvo-Malaguti) Assume D(ρ) = 0 and (g). Then, for every wave speed c ∈ R, we have SWFs from ρ and to ρ. Moreover, they are strict and ϕ is sharp if c < h(ρ), classical if c > h(ρ), from ρ, while ϕ is classical if c < h(ρ), sharp if c > h(ρ), to ρ, In the case c = h(ρ), the profiles are classical if ˙ D(ρ) < 0 while they can be either classical or sharp if ˙ D(ρ) = 0.

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Introduction The problem Results

Related results: the case D(ρ) = 0

Theorem (Existence of sharp SWF, C.-di Ruvo-Malaguti) Assume D(ρ) = 0 and (g). Then, for every wave speed c ∈ R, we have SWFs from ρ and to ρ. Moreover, they are strict and ϕ is sharp if c < h(ρ), classical if c > h(ρ), from ρ, while ϕ is classical if c < h(ρ), sharp if c > h(ρ), to ρ, In the case c = h(ρ), the profiles are classical if ˙ D(ρ) < 0 while they can be either classical or sharp if ˙ D(ρ) = 0. Roughly speaking: slow profiles from ρ are sharp, fast profiles are smooth. In the remaining case c = h(ρ) and ˙ D(ρ) = 0, profiles are either classical or sharp according to the order of vanishing of h(ρ) − c at ρ (and not only on that of D and g). Proof: refine the previous analysis.

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Introduction The problem Results

Related results: the case ˙ D(0) = ∞

We assume D ∈ C[0, ρ] ∩ C1(0, ρ) and one of the following conditions: (ˆ D0) D(ρ) > 0 for ρ ∈ [0, ρ] and ˙ D(0) = ±∞. (ˆ D1) D(ρ) > 0 for ρ ∈ (0, ρ] and D(0) = 0, ˙ D(0) = ∞.

✲ ρ ✻

D (ˆ D0) (ˆ D1) (ˆ D0) ρ Under (ˆ D1) we formally deduce c∗ = ∞, which would suggest the solvability for any c and, moreover, z(0+) < 0. This is indeed the case. Theorem (C.-Malaguti) Assume (ˆ D) and (g); then, the singular IVP for z is uniquely solvable for every c ∈ R and z(0+) < 0 for every c. In turn, equation (E) has a strict classical semi-wavefront solution from ρ for every c; solutions are unique up to shifts and their wave profiles are

• f class C2 in (−∞, ̟). Moreover,

in case (ˆ D0): lim

ξ→̟− ϕ ′(ξ) ∈ (−∞, 0),

in case (ˆ D1): lim

ξ→̟− ϕ ′(ξ) = −∞.

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Introduction The problem Results

Conclusions

We considered the scalar equation ρt + f(ρ)x =

• D(ρ)ρx
• x + g(ρ),

where D(0) = 0 (possibly D(ρ) = 0) and g(ρ) = 0. This equation: never admits wavefronts; has semi-wavefronts for every c ∈ R; According to the order of vanishing of D at 0 and to the growth of g we characterized the slope of the profiles when they reach 0; when profiles are strictly monotone. We used an original reduction to a first-order singular equation.

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Introduction The problem Results

Thank you for your attention. . .

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Introduction The problem Results

Thank you for your attention. . . . . . and thanks to Alberto for his work, patience, altruism in sharing his ideas and, last but not least, his friendship!

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