The effect of diffusion on a line on Fisher-KPP propagation In - PowerPoint PPT Presentation

The effect of diffusion on a line on Fisher-KPP propagation In honor of Jean-Michel Coron Henri Berestycki EHESS, PSL Research University, Paris, ReaDi project - ERC IHP, Paris, 22 June 2016 1/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 1 /

Fisher - KPP equation Homogeneous reaction-diffusion equations x ∈ R N , t > 0 ∂ t v − d ∆ v = f ( v ) 2/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 2 /

Fisher - KPP equation Fisher – KPP case � t > 0 , x ∈ R N ∂ t v = d ∆ v + f ( v ) x ∈ R N v | t =0 = v 0 with v 0 ≥ 0 , �≡ 0 fit . , " >p 3/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 3 /

Fisher - KPP equation Homogeneous equation – Spreading properties in KPP case Invasion: v ( x , t ) → 1 as t → ∞ , locally uniformly in x as soon as v 0 �≡ 0. 4/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 4 /

Fisher - KPP equation Homogeneous equation – Spreading properties in KPP case Invasion: v ( x , t ) → 1 as t → ∞ , locally uniformly in x as soon as v 0 �≡ 0. Asymptotic speed of propagation: ∃ w ∗ such that for any v 0 having compact support ∀ c > w ∗ sup v ( x , t ) → 0 as t → ∞ | x |≥ ct ∀ c < w ∗ sup | v ( x , t ) − 1 | → 0 as t → ∞ . | x |≤ ct 4/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 4 /

Fisher - KPP equation Homogeneous equation – Spreading properties in KPP case Invasion: v ( x , t ) → 1 as t → ∞ , locally uniformly in x as soon as v 0 �≡ 0. Asymptotic speed of propagation: ∃ w ∗ such that for any v 0 having compact support ∀ c > w ∗ sup v ( x , t ) → 0 as t → ∞ | x |≥ ct ∀ c < w ∗ sup | v ( x , t ) − 1 | → 0 as t → ∞ . | x |≤ ct w ∗ = c K = 2 � d f ′ (0) Fisher - KPP case: 4/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 4 /

Fisher - KPP equation Asymptotic position of level sets @ ÷ w*sc , a ) ocaci { ult .x1 x . : . . , 5/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 5 /

Fisher - KPP equation The effect of a “road” with fast diffusion on Fisher-KPP propagation Joint work with Jean-Michel Roquejoffre and Luca Rossi J. Math. Biology (2013) Nonlinearity (2013) Comm. Math. Phys. (2016) Nonlinear Anal. (2016) 6/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 6 /

motivation The system (B, Roquejoffre and Rossi) Ω: upper half-plane R × R + .  ∂ t u − D ∂ xx u = ν v | y =0 − µ u , x ∈ R , t ∈ R   ∂ t v − d ∆ v = f ( v ) , ( x , y ) ∈ Ω , t ∈ R  − d ∂ y v | y =0 = µ u − ν v | y =0 , x ∈ R , t ∈ R .  Note: v | y =0 := lim y ց 0 v . Birth/death rate: Logistic law (KPP type term) f : f ( v ) ≤ f ′ (0) v . f > 0 on (0 , 1) , f (0) = f (1) = 0 , 7/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 7 /

motivation Basic properties Comparison principle If ( u 1 , v 1 ) and ( u 2 , v 2 ) are solutions of the Cauchy problem with u 1 ≤ u 2 and v 1 ≤ v 2 at t = 0, then u 1 ≤ u 2 and v 1 ≤ v 2 for all t ≥ 0. “Monotone system” (kind of) 8/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 8 /

motivation Liouville-type result for stationary solutions Steady states  x ∈ R N − D ∆ x U = ν V | y =0 − µ U ,   − d ∆ V = f ( V ) , ( x , y ) ∈ Ω ,  x ∈ R N . − d ∂ y V | y =0 = µ U − ν V | y =0 ,  where Ω = { x = ( x 1 , . . . , x N , y ); y = x N +1 > 0 } . 9/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 9 /

motivation Liouville-type result for stationary solutions Steady states  x ∈ R N − D ∆ x U = ν V | y =0 − µ U ,   − d ∆ V = f ( V ) , ( x , y ) ∈ Ω ,  x ∈ R N . − d ∂ y V | y =0 = µ U − ν V | y =0 ,  where Ω = { x = ( x 1 , . . . , x N , y ); y = x N +1 > 0 } . Theorem The only bounded steady states are ( U ≡ 0 , V ≡ 0) and ( U = ν/µ, V ≡ 1) . Proof rests on a sliding method (HB - L. Nirenberg) 9/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 9 /

motivation Long time behaviour: invasion  ∂ t u − D ∂ xx u = ν v | y =0 − µ u , x ∈ R , t ∈ R   ∂ t v − d ∆ v = f ( v ) , ( x , y ) ∈ Ω , t ∈ R  − d ∂ y v | y =0 = µ u − ν v | y =0 , x ∈ R , t ∈ R .  10/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 10 /

motivation Long time behaviour: invasion  ∂ t u − D ∂ xx u = ν v | y =0 − µ u , x ∈ R , t ∈ R   ∂ t v − d ∆ v = f ( v ) , ( x , y ) ∈ Ω , t ∈ R  − d ∂ y v | y =0 = µ u − ν v | y =0 , x ∈ R , t ∈ R .  Theorem Let ( u , v ) be a solution of the Cauchy problem with initial datum ( u 0 , v 0 ) �≡ (0 , 0) (nonnegative and bounded). Then, ( u ( x , t ) , v ( x , y , t )) → ( ν/µ, 1) , as t → ∞ , locally uniformly in ( x , y ) ∈ Ω . 10/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 10 /

asymptotic speed of propagation in the direction of the line The effect of roads on propagation 11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 /

asymptotic speed of propagation in the direction of the line The effect of roads on propagation Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w ∗ = w ∗ ( µ, d , D ) > 0 . 11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 /

asymptotic speed of propagation in the direction of the line The effect of roads on propagation Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w ∗ = w ∗ ( µ, d , D ) > 0 . That is: let ( u 0 , v 0 ) be a compactly supported initial datum (nonnegative, nontrivial). Then, locally in y: ∀ c > w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) | → 0 as t → ∞ | x |≥ ct ∀ c < w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) − ( ν/µ, 1) | → 0 as | x |≤ ct , 0 ≤ y ≤ a t → ∞ . 11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 /

asymptotic speed of propagation in the direction of the line The effect of roads on propagation Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w ∗ = w ∗ ( µ, d , D ) > 0 . That is: let ( u 0 , v 0 ) be a compactly supported initial datum (nonnegative, nontrivial). Then, locally in y: ∀ c > w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) | → 0 as t → ∞ | x |≥ ct ∀ c < w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) − ( ν/µ, 1) | → 0 as | x |≤ ct , 0 ≤ y ≤ a t → ∞ . Theorem � d f ′ (0) be the (homogenous) KPP speed of propagation. Let c K := 2 11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 /

asymptotic speed of propagation in the direction of the line The effect of roads on propagation Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w ∗ = w ∗ ( µ, d , D ) > 0 . That is: let ( u 0 , v 0 ) be a compactly supported initial datum (nonnegative, nontrivial). Then, locally in y: ∀ c > w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) | → 0 as t → ∞ | x |≥ ct ∀ c < w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) − ( ν/µ, 1) | → 0 as | x |≤ ct , 0 ≤ y ≤ a t → ∞ . Theorem � d f ′ (0) be the (homogenous) KPP speed of propagation. Let c K := 2 If D ≤ 2 d then w ∗ ( µ, ν, d , D , f ′ (0))= c K . 11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 /

asymptotic speed of propagation in the direction of the line The effect of roads on propagation Theorem There exists an asymptotic speed of propagation in the direction of the x-axis, w ∗ = w ∗ ( µ, d , D ) > 0 . That is: let ( u 0 , v 0 ) be a compactly supported initial datum (nonnegative, nontrivial). Then, locally in y: ∀ c > w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) | → 0 as t → ∞ | x |≥ ct ∀ c < w ∗ , sup | ( u ( x , t ) , v ( x , y , t )) − ( ν/µ, 1) | → 0 as | x |≤ ct , 0 ≤ y ≤ a t → ∞ . Theorem � d f ′ (0) be the (homogenous) KPP speed of propagation. Let c K := 2 If D ≤ 2 d then w ∗ ( µ, ν, d , D , f ′ (0))= c K . If D > 2 d then w ∗ ( µ, d , D ) > c K . 11/56 Henri Berestycki (Paris) Effect of a line on F-KPP propagation IHP, Paris, 22 June 2016 11 /