Mathematical models in reaction-diffusion: from Turing instabilities - PowerPoint PPT Presentation

Turing instabilities ◮ Simple idea: the new linearized system is � u 1 � d 1 � � ∆ u 1 � − a 2 � � u 1 � � � 0 − b ∂ t = + a 2 u 2 0 d 2 ∆ u 2 b − 1 u 2 ◮ Work in the spectral basis of the Laplace operator − ∆. ◮ In each direction, one has to look at the eigenvalues of the matrices � d 1 � − a 2 � � 0 − b − λ k + a 2 0 d 2 b − 1 where λ k > 0 , k ≥ 1 are the eigenvalues of − ∆. Recall that the real parts of the eigenvalues of the second matrix are negative if b < 1 + a 2 .

Turing instabilities ◮ Simple idea: the new linearized system is � u 1 � d 1 � � ∆ u 1 � − a 2 � � u 1 � � � 0 − b ∂ t = + a 2 u 2 0 d 2 ∆ u 2 b − 1 u 2 ◮ Work in the spectral basis of the Laplace operator − ∆. ◮ In each direction, one has to look at the eigenvalues of the matrices � d 1 � − a 2 � � 0 − b − λ k + a 2 0 d 2 b − 1 where λ k > 0 , k ≥ 1 are the eigenvalues of − ∆. Recall that the real parts of the eigenvalues of the second matrix are negative if b < 1 + a 2 . ◮ ...One the eigenvalues may become positive if d 2 < d 1 and � 2 � � d 2 < b < 1 + a 2 . 1 + a d 1

Turing instabilities ◮ Simple idea: the new linearized system is � u 1 � d 1 � � ∆ u 1 � − a 2 � � u 1 � � � 0 − b ∂ t = + a 2 u 2 0 d 2 ∆ u 2 b − 1 u 2 ◮ Work in the spectral basis of the Laplace operator − ∆. ◮ In each direction, one has to look at the eigenvalues of the matrices � d 1 � − a 2 � � 0 − b − λ k + a 2 0 d 2 b − 1 where λ k > 0 , k ≥ 1 are the eigenvalues of − ∆. Recall that the real parts of the eigenvalues of the second matrix are negative if b < 1 + a 2 . ◮ ...One the eigenvalues may become positive if d 2 < d 1 and � 2 � � d 2 < b < 1 + a 2 . 1 + a d 1 ◮ Instability is interesting ! It may lead to non homogeneous equilibria (depending on x ) and provide a great variety of asymptotic spatial patterns.

60 A. M. TURING ON THE was used and h was about 0.7. In the figure the set of points where f(x, y) is positive is shown black. The outlines of the black patches are somewhat less irregular than they should be due to an inadequacy in the computation procedure. I L FIGURE 2. An example of a 'dappled' pattern as resulting from a type (a) morphogen system. A marker of unit length is shown. See text, $9, 11. 10. A NUMERICAL EXAMPLE The numerous approximations and assumptions that have been made in the foregoing analysis may be rather confusing to many readers. In the present section it is proposed to consider in detail a single example of the case of most interest, (d). This will be made as specific as possible. It is unfortunately not possible to specify actual chemical reactions with the required properties, but it is thought that the reaction rates associated with the imagined reactions are not unreasonable. The detail to be specified includes (i) The number and dimensions of the cells of the ring. (ii) The diffusibilities of the morphogens. (iii) The reactions concerned. (iv) The rates at which the reactions occur. (v) Information about random disturbances. (vi) Information about the distribution, in space and time, of those morphogens which are of the nature of evocators. These will be taken in order. (i) It will be assumed that there are twenty cells in the ring, and that they have a diameter of 0.1 mm each. These cells are certainly on the large rather than the small side, but by no means impossibly so. The number of cells in the ring has been chosen rather small in order that it should not be necessary to make the approximation of continuous tissue. (ii) Two morphogens are considered. They will be called X and Y, and the same letters will be used for their concentrations. This will not lead to any real confusion. The diffusion constant for X will be assumed to be 5 x 10- cm2 s-l and that for Y to be 2.5 x crn2s-l. With cells of diameter 0.01 cni this means that X flows between neighbouring cells at the

[Rui ¡Dilao] ¡Patterns ¡in ¡butterflies ¡compared ¡with ¡patterns ¡generated ¡from ¡equation ¡(23). ¡ ¡

Alan Turing 1912-1954

Ilia Prigogine [1917-2003], Chemistry Nobel prize in 1977

Is global existence destroyed by adding diffusion? ◮ Global existence holds for the Brusselator ODE  1 = − u 1 u 2 u ′ 2 + b u 2  u 1 u 2 u ′ 2 = 2 − ( b + 1) u 2 + a u i (0) = u o i , u o i ∈ [0 , + ∞ ) , i = 1 , 2 .  ◮ Local existence holds on a maximal interval [0 , T ∗ ) by Cauchy-Lipschitz Theorem. Moreover solutions are nonnegative.

Is global existence destroyed by adding diffusion? ◮ Global existence holds for the Brusselator ODE  1 = − u 1 u 2 u ′ 2 + b u 2  u 1 u 2 u ′ 2 = 2 − ( b + 1) u 2 + a u i (0) = u o i , u o i ∈ [0 , + ∞ ) , i = 1 , 2 .  ◮ Local existence holds on a maximal interval [0 , T ∗ ) by Cauchy-Lipschitz Theorem. Moreover solutions are nonnegative. ◮ And we have an a priori estimate: ( u 1 + u 2 ) ′ = − u 2 + a ≤ a 2 ) ⇒ T ∗ = + ∞ ( u 1 + u 2 )( t ) ≤ a t + ( u o 1 + u o ⇒

Is global existence destroyed by adding diffusion? ◮ Global existence holds for the Brusselator ODE  1 = − u 1 u 2 u ′ 2 + b u 2  u 1 u 2 u ′ 2 = 2 − ( b + 1) u 2 + a u i (0) = u o i , u o i ∈ [0 , + ∞ ) , i = 1 , 2 .  ◮ Local existence holds on a maximal interval [0 , T ∗ ) by Cauchy-Lipschitz Theorem. Moreover solutions are nonnegative. ◮ And we have an a priori estimate: ( u 1 + u 2 ) ′ = − u 2 + a ≤ a 2 ) ⇒ T ∗ = + ∞ ( u 1 + u 2 )( t ) ≤ a t + ( u o 1 + u o ⇒ ◮ Does one have global existence for the full system?  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2 + b u 2   ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 − ( b + 1) u 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  i ( · ) ≥ 0 , u o = ( u o  u i (0 , · ) = u o 1 , u o 2 ) ∈ L ∞ (Ω) 2 . 

Does global existence hold?  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2 + b u 2   ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 − ( b + 1) u 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  i ( · ) ≥ 0 , u o = ( u o  u i (0 , · ) = u o 1 , u o 2 ) ∈ L ∞ (Ω) 2 . 

Does global existence hold?  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2 + b u 2   ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 − ( b + 1) u 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  i ( · ) ≥ 0 , u o = ( u o  u i (0 , · ) = u o 1 , u o 2 ) ∈ L ∞ (Ω) 2 .  ◮ First, we can forget the linear part for the question of global existence and consider only  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2  ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 

Does global existence hold?  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2 + b u 2   ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 − ( b + 1) u 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  i ( · ) ≥ 0 , u o = ( u o  u i (0 , · ) = u o 1 , u o 2 ) ∈ L ∞ (Ω) 2 .  ◮ First, we can forget the linear part for the question of global existence and consider only  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2  ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  ◮ Using � � Ω ∆ u i = ∂ Ω ∂ ν u i = 0, we have d � � ( u 1 + u 2 )( t ) = ∂ t ( u 1 + u 2 ) = 0 dt Ω Ω � � u o 1 + u o ⇒ u 1 ( t ) + u 2 ( t ) = 2 Ω Ω

Does global existence hold?  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2 + b u 2   ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 − ( b + 1) u 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  i ( · ) ≥ 0 , u o = ( u o  u i (0 , · ) = u o 1 , u o 2 ) ∈ L ∞ (Ω) 2 .  ◮ First, we can forget the linear part for the question of global existence and consider only  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2  ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  ◮ Using � � Ω ∆ u i = ∂ Ω ∂ ν u i = 0, we have d � � ( u 1 + u 2 )( t ) = ∂ t ( u 1 + u 2 ) = 0 dt Ω Ω � � u o 1 + u o ⇒ u 1 ( t ) + u 2 ( t ) = 2 Ω Ω ◮ ⇒ L 1 (Ω)-bound, uniform in time !

Does global existence hold?  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2 + b u 2   ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 − ( b + 1) u 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  i ( · ) ≥ 0 , u o = ( u o  u i (0 , · ) = u o 1 , u o 2 ) ∈ L ∞ (Ω) 2 .  ◮ First, we can forget the linear part for the question of global existence and consider only  ∂ t u 1 − d 1 ∆ u 1 = − u 1 u 2 2  ∂ t u 2 − d 2 ∆ u 2 = u 1 u 2 2 ∂ ν u i = 0 on ∂ Ω , i = 1 , 2  ◮ Using � � Ω ∆ u i = ∂ Ω ∂ ν u i = 0, we have d � � ( u 1 + u 2 )( t ) = ∂ t ( u 1 + u 2 ) = 0 dt Ω Ω � � u o 1 + u o ⇒ u 1 ( t ) + u 2 ( t ) = 2 Ω Ω ◮ ⇒ L 1 (Ω)-bound, uniform in time ! ◮ How does this help for global existence?

Same question for the general family of systems:  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m )  ( S ) ∂ ν u i = 0   u i (0 , · ) = u 0 i ( · ) ≥ 0  d i > 0 , f i : [0 , ∞ ) m → I R locally Lipschitz continuous where ◮ (P): Positivity (nonnegativity) is preserved

Same question for the general family of systems:  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m )  ( S ) ∂ ν u i = 0   u i (0 , · ) = u 0 i ( · ) ≥ 0  d i > 0 , f i : [0 , ∞ ) m → I R locally Lipschitz continuous where ◮ (P): Positivity (nonnegativity) is preserved ◮ (M): � 1 ≤ i ≤ m f i ≤ 0

Same question for the general family of systems:  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m )  ( S ) ∂ ν u i = 0   u i (0 , · ) = u 0 i ( · ) ≥ 0  d i > 0 , f i : [0 , ∞ ) m → I R locally Lipschitz continuous where ◮ (P): Positivity (nonnegativity) is preserved ◮ (M): � 1 ≤ i ≤ m f i ≤ 0 ◮ or more generally (M’) ∀ r ∈ [0 , ∞ [ m , � 1 ≤ i ≤ m a i f i ( r ) ≤ C [1 + � 1 ≤ i ≤ m r i ] for some a i > 0

◮  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T  u i (0 , · ) = u 0  i ( · ) ≥ 0 .  ◮ (P) Preservation of Positivity: ∀ i = 1 , ..., m ∀ r = ( r 1 , ..., r m ) ∈ [0 , ∞ [ m , f i ( r 1 , ..., r i − 1 , 0 , r i +1 , ..., r m ) ≥ 0, = ” quasi-positivity ” of f = ( f i ) 1 ≤ i ≤ m . r 2 ✻ f 1 (0 , r 2 ) ≥ 0 ✁ ✕ f 2 (0 , r 2 ) ✁ ✁ f 1 ( r 1 , 0) ❆ ❑ ❆ f 2 ( r 1 , 0) ≥ 0 ✲ r 1 ❆

◮  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T  u i (0 , · ) = u 0  i ( · ) ≥ 0 .  ◮ (P) Preservation of Positivity: ∀ i = 1 , ..., m ∀ r ∈ [0 , + ∞ [ m , f i ( r 1 , ..., r i − 1 , 0 , r i +1 , ..., r m ) ≥ 0. ◮ (M): � 1 ≤ i ≤ m f i ( r 1 , ..., r m ) ≤ 0 ⇒ ’Control of the Total Mass’: � � � � u 0 ∀ t ≥ 0 , u i ( t , x ) dx ≤ i ( x ) dx Ω Ω 1 ≤ i ≤ r 1 ≤ i ≤ r Add up, integrate on Ω, use � Ω ∆ u i = � ∂ Ω ∂ ν u i = 0: � � � � ∂ t [ u i ( t )] dx = f i ( u ) dx ≤ 0 . Ω Ω i

◮  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T  u i (0 , · ) = u 0  i ( · ) ≥ 0 .  ◮ (P) Preservation of Positivity: ∀ i = 1 , ..., m ∀ r ∈ [0 , + ∞ [ m , f i ( r 1 , ..., r i − 1 , 0 , r i +1 , ..., r m ) ≥ 0. ◮ (M): � 1 ≤ i ≤ m f i ( r 1 , ..., r m ) ≤ 0 ⇒ ’Control of the Total Mass’: � � � � u 0 ∀ t ≥ 0 , u i ( t , x ) dx ≤ i ( x ) dx Ω Ω 1 ≤ i ≤ r 1 ≤ i ≤ r Add up, integrate on Ω, use � Ω ∆ u i = � ∂ Ω ∂ ν u i = 0: � � � � ∂ t [ u i ( t )] dx = f i ( u ) dx ≤ 0 . Ω Ω i ◮ ⇒ L 1 (Ω)- a priori estimates, uniform in time.

◮  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T  u i (0 , · ) = u 0  i ( · ) ≥ 0 .  ◮ (P) Preservation of Positivity: ∀ i = 1 , ..., m ∀ r ∈ [0 , + ∞ [ m , f i ( r 1 , ..., r i − 1 , 0 , r i +1 , ..., r m ) ≥ 0. ◮ (M): � 1 ≤ i ≤ m f i ( r 1 , ..., r m ) ≤ 0 ⇒ ’Control of the Total Mass’: � � � � u 0 ∀ t ≥ 0 , u i ( t , x ) dx ≤ i ( x ) dx Ω Ω 1 ≤ i ≤ r 1 ≤ i ≤ r Add up, integrate on Ω, use � Ω ∆ u i = � ∂ Ω ∂ ν u i = 0: � � � � ∂ t [ u i ( t )] dx = f i ( u ) dx ≤ 0 . Ω Ω i ◮ ⇒ L 1 (Ω)- a priori estimates, uniform in time. ◮ Remark: L 1 -bound for all time with (M’)

QUESTION: What about Global Existence of solutions under assumption (P)+(M) ?? or more generally (P)+ (M’) ??

Several approaches and techniques ◮ L ∞ -approach: local existence ◮ An L p -approach ◮ Blow up may occur ◮ An L 1 -approach ◮ A surprising L 2 -estimate ◮ and even L 2+ ǫ ... ◮ Renormalized solutions ◮ L Log L is also involved ◮ ◮ + OPEN PROBLEMS

Local existence in L ∞ for systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ Theorem ( ` a la Cauchy-Lipschitz dans L ∞ ). Let u 0 ∈ L ∞ (Ω) + m . Then, there exist a maximum time T ∗ > 0 and u = ( u 1 , ..., u m ) unique classical nonnegative solution of (S) on [0 , T ∗ ). Moreover, < + ∞ ⇒ [ T ∗ + ∞ ] . � � sup � u ( t ) � L ∞ (Ω) + � v ( t ) � L ∞ (Ω) t ∈ [0 , T ∗ )

Local existence in L ∞ for systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ Theorem ( ` a la Cauchy-Lipschitz dans L ∞ ). Let u 0 ∈ L ∞ (Ω) + m . Then, there exist a maximum time T ∗ > 0 and u = ( u 1 , ..., u m ) unique classical nonnegative solution of (S) on [0 , T ∗ ). Moreover, < + ∞ ⇒ [ T ∗ + ∞ ] . � � sup � u ( t ) � L ∞ (Ω) + � v ( t ) � L ∞ (Ω) t ∈ [0 , T ∗ ) ◮ Corollary. If d i = d for all i = 1 , ..., m , then T ∗ = + ∞ .

Local existence in L ∞ for systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ Theorem ( ` a la Cauchy-Lipschitz dans L ∞ ). Let u 0 ∈ L ∞ (Ω) + m . Then, there exist a maximum time T ∗ > 0 and u = ( u 1 , ..., u m ) unique classical nonnegative solution of (S) on [0 , T ∗ ). Moreover, < + ∞ ⇒ [ T ∗ + ∞ ] . � � sup � u ( t ) � L ∞ (Ω) + � v ( t ) � L ∞ (Ω) t ∈ [0 , T ∗ ) ◮ Corollary. If d i = d for all i = 1 , ..., m , then T ∗ = + ∞ . ◮ Proof: ∂ t ( � i u i ) − d ∆( � i u i ) ≤ 0. � � ⇒ � u i ( t ) � L ∞ (Ω) ≤ � u i 0 � L ∞ (Ω) . i i

The L p -approach ◮ Recall the Brusselator ( β ∈ [1 , + ∞ )) ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 .

The L p -approach ◮ Recall the Brusselator ( β ∈ [1 , + ∞ )) ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 . ◮ By maximum principle � u 1 ( t ) � L ∞ (Ω) ≤ � u 0 1 � L ∞ (Ω) .

The L p -approach ◮ Recall the Brusselator ( β ∈ [1 , + ∞ )) ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 . ◮ By maximum principle � u 1 ( t ) � L ∞ (Ω) ≤ � u 0 1 � L ∞ (Ω) . ◮ We have : ∂ t u 2 − d 2 ∆ u 2 = − ( ∂ t u 1 − d 1 ∆ u 1 ).

The L p -approach ◮ Recall the Brusselator ( β ∈ [1 , + ∞ )) ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 . ◮ By maximum principle � u 1 ( t ) � L ∞ (Ω) ≤ � u 0 1 � L ∞ (Ω) . ◮ We have : ∂ t u 2 − d 2 ∆ u 2 = − ( ∂ t u 1 − d 1 ∆ u 1 ). ◮ Or : ” u 2 = ( ∂ t − d 2 ∆) − 1 ( − ∂ t + d 1 ∆) u 1 ”

The L p -approach ◮ Recall the Brusselator ( β ∈ [1 , + ∞ )) ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 . ◮ By maximum principle � u 1 ( t ) � L ∞ (Ω) ≤ � u 0 1 � L ∞ (Ω) . ◮ We have : ∂ t u 2 − d 2 ∆ u 2 = − ( ∂ t u 1 − d 1 ∆ u 1 ). ◮ Or : ” u 2 = ( ∂ t − d 2 ∆) − 1 ( − ∂ t + d 1 ∆) u 1 ” ◮ L p -Fundamental Lemma: ∂ t u 2 − d 2 ∆ u 2 ≤ a ∂ t u 1 + b ∆ u 1 , a , b ∈ I R , u 2 ≥ 0, ⇒ � u 2 � L p ( Q T ) ≤ C � u 1 � L p ( Q T ) for all 1 < p < + ∞ , all T > 0, C = C ( a , b , p , T , � u 1 (0) � p , � u 2 (0) � p ) .

The L p -approach ◮ Recall the Brusselator ( β ∈ [1 , + ∞ )) ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 . ◮ By maximum principle � u 1 ( t ) � L ∞ (Ω) ≤ � u 0 1 � L ∞ (Ω) . ◮ We have : ∂ t u 2 − d 2 ∆ u 2 = − ( ∂ t u 1 − d 1 ∆ u 1 ). ◮ Or : ” u 2 = ( ∂ t − d 2 ∆) − 1 ( − ∂ t + d 1 ∆) u 1 ” ◮ L p -Fundamental Lemma: ∂ t u 2 − d 2 ∆ u 2 ≤ a ∂ t u 1 + b ∆ u 1 , a , b ∈ I R , u 2 ≥ 0, ⇒ � u 2 � L p ( Q T ) ≤ C � u 1 � L p ( Q T ) for all 1 < p < + ∞ , all T > 0, C = C ( a , b , p , T , � u 1 (0) � p , � u 2 (0) � p ) . ◮ Lemma=Dual statement of the L p maximal regularity theory for parabolic operators. Not valid for p = + ∞ .

The L p -approach applied to the Brusselator ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 ◮ We know � u 1 ( t ) � L ∞ (Ω) ≤ � u 1 (0) � L ∞ (Ω) for all t ∈ [0 , T ∗ )

The L p -approach applied to the Brusselator ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 ◮ We know � u 1 ( t ) � L ∞ (Ω) ≤ � u 1 (0) � L ∞ (Ω) for all t ∈ [0 , T ∗ ) ◮ From the fundamental Lemma, for all p < + ∞ � u 2 � L p ( Q T ∗ ) ≤ C � u 1 � L p ( Q T ∗ ) ≤ C � u 1 � L ∞ ( Q T ∗ ) < + ∞

The L p -approach applied to the Brusselator ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 ◮ We know � u 1 ( t ) � L ∞ (Ω) ≤ � u 1 (0) � L ∞ (Ω) for all t ∈ [0 , T ∗ ) ◮ From the fundamental Lemma, for all p < + ∞ � u 2 � L p ( Q T ∗ ) ≤ C � u 1 � L p ( Q T ∗ ) ≤ C � u 1 � L ∞ ( Q T ∗ ) < + ∞ ◮ � u 1 u β 2 � L q ( Q T ∗ ) < + ∞ for all q < + ∞

The L p -approach applied to the Brusselator ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 ◮ We know � u 1 ( t ) � L ∞ (Ω) ≤ � u 1 (0) � L ∞ (Ω) for all t ∈ [0 , T ∗ ) ◮ From the fundamental Lemma, for all p < + ∞ � u 2 � L p ( Q T ∗ ) ≤ C � u 1 � L p ( Q T ∗ ) ≤ C � u 1 � L ∞ ( Q T ∗ ) < + ∞ ◮ � u 1 u β 2 � L q ( Q T ∗ ) < + ∞ for all q < + ∞ ◮ Choosing q large enough ( q > ( N + 1) / 2), this implies � u 2 � L ∞ ( Q T ∗ ) < + ∞

The L p -approach applied to the Brusselator ∂ t u 1 − d 1 ∆ u 1 = − u 1 u β  2 ( ≤ 0)  ∂ t u 2 − d 2 ∆ u 2 = u 1 u β ( S ) 2  ∂ ν u i = 0 on ∂ Ω , i = 1 , 2 ◮ We know � u 1 ( t ) � L ∞ (Ω) ≤ � u 1 (0) � L ∞ (Ω) for all t ∈ [0 , T ∗ ) ◮ From the fundamental Lemma, for all p < + ∞ � u 2 � L p ( Q T ∗ ) ≤ C � u 1 � L p ( Q T ∗ ) ≤ C � u 1 � L ∞ ( Q T ∗ ) < + ∞ ◮ � u 1 u β 2 � L q ( Q T ∗ ) < + ∞ for all q < + ∞ ◮ Choosing q large enough ( q > ( N + 1) / 2), this implies � u 2 � L ∞ ( Q T ∗ ) < + ∞ T ∗ = + ∞ ◮ ⇒

Extensions of the L p -approach ◮ This approach extends to m × m systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T  u i (0 , · ) = u 0  i ( · ) ≥ 0 .  where f i have polynomial growth and  f 1 ≤ 0   f 1 + f 2 ≤ 0  ...   f 1 + f 2 + ... + f m ≤ 0 

Extensions of the L p -approach ◮ This approach extends to m × m systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T  u i (0 , · ) = u 0  i ( · ) ≥ 0 .  where f i have polynomial growth and  f 1 ≤ 0   f 1 + f 2 ≤ 0  ...   f 1 + f 2 + ... + f m ≤ 0  ◮ or more generally if for an invertible triangular matrix Q with nonnegative entries � ∀ r ∈ [0 , ∞ ) m , Q f ( r ) ≤ [1 + r i ] b , 1 ≤ i ≤ m R m , f = ( f 1 , ..., f m ) t . for some b ∈ I

Application to the case of close d i ’s  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ We may write with d := min j d j � ∂ t ( � i u i ) − d ∆( � = � i ( d i − d )∆ u i + � i u i ) i f i ≤ ∆ ( � i ( d i − d ) u i ) .

Application to the case of close d i ’s  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ We may write with d := min j d j � ∂ t ( � i u i ) − d ∆( � = � i ( d i − d )∆ u i + � i u i ) i f i ≤ ∆ ( � i ( d i − d ) u i ) . ◮ We deduce from the L p -Lemma that � � � ≤ C � � i u i � L p ( Q T ) i ( d i − d ) u i � L p ( Q T ) ≤ C max i { d i − d }� � i u i � L p ( Q T ) .

Application to the case of close d i ’s  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ We may write with d := min j d j � ∂ t ( � i u i ) − d ∆( � = � i ( d i − d )∆ u i + � i u i ) i f i ≤ ∆ ( � i ( d i − d ) u i ) . ◮ We deduce from the L p -Lemma that � � � ≤ C � � i u i � L p ( Q T ) i ( d i − d ) u i � L p ( Q T ) ≤ C max i { d i − d }� � i u i � L p ( Q T ) . ◮ If M := C max i { d i − d } < 1, we deduce u i � L p ( Q T ) ≤ (1 − M ) − 1 < + ∞ . � � i

Application to the case of close d i ’s  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) : ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ We may write with d := min j d j � ∂ t ( � i u i ) − d ∆( � = � i ( d i − d )∆ u i + � i u i ) i f i ≤ ∆ ( � i ( d i − d ) u i ) . ◮ We deduce from the L p -Lemma that � � � ≤ C � � i u i � L p ( Q T ) i ( d i − d ) u i � L p ( Q T ) ≤ C max i { d i − d }� � i u i � L p ( Q T ) . ◮ If M := C max i { d i − d } < 1, we deduce u i � L p ( Q T ) ≤ (1 − M ) − 1 < + ∞ . � � i ◮ Whence global existence if f i at most polynomial.

Extensions and limits of the L p -approach ◮ L p -approach does not apply to � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2

Extensions and limits of the L p -approach ◮ L p -approach does not apply to � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2 ◮ neither to the system � ∂ t u 1 − d 1 ∆ u 1 = u 3 1 u 2 2 − u 2 1 u 3 2 ∂ t u 2 − d 2 ∆ u 2 = u 2 1 u 3 2 − u 3 1 u 2 2

Extensions and limits of the L p -approach ◮ L p -approach does not apply to � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2 ◮ neither to the system � ∂ t u 1 − d 1 ∆ u 1 = u 3 1 u 2 2 − u 2 1 u 3 2 ∂ t u 2 − d 2 ∆ u 2 = u 2 1 u 3 2 − u 3 1 u 2 2 ◮ and even not to the ”better” system with λ ∈ (0 , 1) ∂ t u 1 − d 1 ∆ u 1 = λ u 3 1 u 2 2 − u 2 1 u 3 � 2 [=: f 1 ( u )] ∂ t u 2 − d 2 ∆ u 2 = u 2 1 u 3 2 − u 3 1 u 2 2 [=: f 2 ( u )] where : f 1 ( u ) + f 2 ( u ) ≤ 0, and also : f 1 ( u ) + λ f 2 ( u ) ≤ 0

Finite time L ∞ -blow up may appear! � ∂ t u 1 − d 1 ∆ u 1 = f 1 ( u 1 , u 2 ) ∂ t u 2 − d 2 ∆ u 2 = f 2 ( u 1 , u 2 ) ◮ Theorem: (D. Schmitt, MP) One can find polynomial nonlinearities f , g satisfying (P) and ( M ) f + g ≤ 0 , and also : ∃ λ ∈ [0 , 1[ , f + λ g ≤ 0 , for which T ∗ < + ∞ with t → T ∗ � u ( t ) � L ∞ (Ω) = lim lim t → T ∗ � v ( t ) � L ∞ (Ω) = + ∞ .

Finite time L ∞ -blow up may appear! � ∂ t u 1 − d 1 ∆ u 1 = f 1 ( u 1 , u 2 ) ∂ t u 2 − d 2 ∆ u 2 = f 2 ( u 1 , u 2 ) ◮ Theorem: (D. Schmitt, MP) One can find polynomial nonlinearities f , g satisfying (P) and ( M ) f + g ≤ 0 , and also : ∃ λ ∈ [0 , 1[ , f + λ g ≤ 0 , for which T ∗ < + ∞ with t → T ∗ � u ( t ) � L ∞ (Ω) = lim lim t → T ∗ � v ( t ) � L ∞ (Ω) = + ∞ . ◮ Blow up may appear even in space dimension N = 1 (with high degree polynomial nonlinearities)

Finite time L ∞ -blow up may appear! � ∂ t u 1 − d 1 ∆ u 1 = f 1 ( u 1 , u 2 ) ∂ t u 2 − d 2 ∆ u 2 = f 2 ( u 1 , u 2 ) ◮ Theorem: (D. Schmitt, MP) One can find polynomial nonlinearities f , g satisfying (P) and ( M ) f + g ≤ 0 , and also : ∃ λ ∈ [0 , 1[ , f + λ g ≤ 0 , for which T ∗ < + ∞ with t → T ∗ � u ( t ) � L ∞ (Ω) = lim lim t → T ∗ � v ( t ) � L ∞ (Ω) = + ∞ . ◮ Blow up may appear even in space dimension N = 1 (with high degree polynomial nonlinearities) ◮ Blow up may appear with any superquadratic growth 2 + ǫ for the f i (with high dimension N )

To proceed: Look for weak solutions which are allowed to go out of L ∞ (Ω) from time to time or even often (”Incomplete blow up”). We ask the nonlinearities to be at least in L 1 ( Q T ): f i ( u ) ∈ L 1 ( Q T ) and the solution is understood in the sense of distributions or of Duhamel integral formula.

An L 1 -approach  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m )  ( S ) ∂ ν u i = 0   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ L 1 -Theorem. Assume the two conditions (P)+ (M’) hold. Assume moreover that the following a priori estimate holds: � ∀ i = 1 , ..., m , | f i ( u ) | ≤ C ( T ) < + ∞ , ∀ T ∈ (0 , + ∞ ) . Q T Then, there exists a global weak solution for System (S), even for all u 0 ∈ L 1 (Ω) + m !

An L 1 -approach  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m )  ( S ) ∂ ν u i = 0   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ L 1 -Theorem. Assume the two conditions (P)+ (M’) hold. Assume moreover that the following a priori estimate holds: � ∀ i = 1 , ..., m , | f i ( u ) | ≤ C ( T ) < + ∞ , ∀ T ∈ (0 , + ∞ ) . Q T Then, there exists a global weak solution for System (S), even for all u 0 ∈ L 1 (Ω) + m ! ◮ Proof involves truncations T k ( r ) := inf { r , k } , r ∈ [0 , ∞ ) and L 1 -type estimates like � �� � � d i |∇ u i | 2 ≤ k u 0 | f i ( u ) | + . i [0 ≤ u i ≤ k ] Ω

L 1 -Theorem applies to many situations � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2 ◮ Easy L 1 ( Q T )-estimate of the nonlinearity : � � � u 1 e u 2 2 = u 0 u 1 ( T ) + 1 . Ω Ω Q T

L 1 -Theorem applies to many situations � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2 ◮ Easy L 1 ( Q T )-estimate of the nonlinearity : � � � u 1 e u 2 2 = u 0 u 1 ( T ) + 1 . Ω Ω Q T ◮ ⇒ Global existence of weak solutions

L 1 -Theorem applies to many situations � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2 ◮ Easy L 1 ( Q T )-estimate of the nonlinearity : � � � u 1 e u 2 2 = u 0 u 1 ( T ) + 1 . Ω Ω Q T ◮ ⇒ Global existence of weak solutions 2 ∈ L 1 (Ω) + !! ◮ Existence holds for any u 0 1 , u 0

L 1 -Theorem applies to many situations � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2 ◮ Easy L 1 ( Q T )-estimate of the nonlinearity : � � � u 1 e u 2 2 = u 0 u 1 ( T ) + 1 . Ω Ω Q T ◮ ⇒ Global existence of weak solutions 2 ∈ L 1 (Ω) + !! ◮ Existence holds for any u 0 1 , u 0 ◮ Recall that the equation ∂ t u 2 − ∆ u 2 = e u 2 2 , u 2 (0) = u 0 2 , does not have even local solutions in general when 2 ∈ L 1 (Ω) + only. u 0

L 1 -Theorem applies to many situations � ∂ t u 1 − d 1 ∆ u 1 = − u 1 e u 2 2 ∂ t u 2 − d 2 ∆ u 2 = u 1 e u 2 2 ◮ Easy L 1 ( Q T )-estimate of the nonlinearity : � � � u 1 e u 2 2 = u 0 u 1 ( T ) + 1 . Ω Ω Q T ◮ ⇒ Global existence of weak solutions 2 ∈ L 1 (Ω) + !! ◮ Existence holds for any u 0 1 , u 0 ◮ Recall that the equation ∂ t u 2 − ∆ u 2 = e u 2 2 , u 2 (0) = u 0 2 , does not have even local solutions in general when 2 ∈ L 1 (Ω) + only. u 0 ◮ OPEN PROBLEM: are the solutions classical ?

L 1 -Theorem applies to many situations ◮ Let λ ∈ ( −∞ , 1) and � ∂ t u 1 − d 1 ∆ u 1 = λ u 3 1 u 2 2 − u 2 1 u 3 2 in Q T , ∂ t u 2 − d 2 ∆ u 2 = u 2 1 u 3 2 − u 3 1 u 2 2 in Q T

L 1 -Theorem applies to many situations ◮ Let λ ∈ ( −∞ , 1) and � ∂ t u 1 − d 1 ∆ u 1 = λ u 3 1 u 2 2 − u 2 1 u 3 2 in Q T , ∂ t u 2 − d 2 ∆ u 2 = u 2 1 u 3 2 − u 3 1 u 2 2 in Q T ◮ λ ∈ ( −∞ , 0] ⇒ OK ! (triangular structure).

L 1 -Theorem applies to many situations ◮ Let λ ∈ ( −∞ , 1) and � ∂ t u 1 − d 1 ∆ u 1 = λ u 3 1 u 2 2 − u 2 1 u 3 2 in Q T , ∂ t u 2 − d 2 ∆ u 2 = u 2 1 u 3 2 − u 3 1 u 2 2 in Q T ◮ λ ∈ ( −∞ , 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0 , 1) ? � � � u 3 1 u 2 u 0 1 + u 0 ( u 1 + u 2 )( T ) = − (1 − λ ) 2 + 2 , Ω Q T Ω � � � u 2 1 u 3 u 0 1 + λ u 0 ( u 1 + λ u 2 ) = − (1 − λ ) 2 + 2 . Ω Q T Ω

L 1 -Theorem applies to many situations ◮ Let λ ∈ ( −∞ , 1) and � ∂ t u 1 − d 1 ∆ u 1 = λ u 3 1 u 2 2 − u 2 1 u 3 2 in Q T , ∂ t u 2 − d 2 ∆ u 2 = u 2 1 u 3 2 − u 3 1 u 2 2 in Q T ◮ λ ∈ ( −∞ , 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0 , 1) ? � � � u 3 1 u 2 u 0 1 + u 0 ( u 1 + u 2 )( T ) = − (1 − λ ) 2 + 2 , Ω Q T Ω � � � u 2 1 u 3 u 0 1 + λ u 0 ( u 1 + λ u 2 ) = − (1 − λ ) 2 + 2 . Ω Q T Ω Q T u 3 1 u 2 Q T u 2 1 u 3 ◮ ⇒ � � 2 , 2 ≤ C ⇒ Global weak solutions

L 1 -Theorem applies to many situations More generally, the same method applies if there exists an invertible matrix Q with nonnegative entries such that ∀ r ∈ [0 , ∞ ) m , Q f ( r ) ≤ [1 + � r i ] b , 1 ≤ i ≤ m R m , f = ( f 1 , ..., f m ) t . for some b ∈ I In other words: there are m independent inequalities between the f i ’s (not necessarily triangular)

L 1 -Theorem does not solve everything ◮ Let λ ∈ ( −∞ , 1) and � ∂ t u − d 1 ∆ u = λ u 3 v 2 − u 2 v 3 in Q T , ∂ t v − d 2 ∆ v = u 2 v 3 − u 3 v 2 in Q T ◮ λ ∈ ( −∞ , 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0 , 1) ⇒ Global weak solutions

L 1 -Theorem does not solve everything ◮ Let λ ∈ ( −∞ , 1) and � ∂ t u − d 1 ∆ u = λ u 3 v 2 − u 2 v 3 in Q T , ∂ t v − d 2 ∆ v = u 2 v 3 − u 3 v 2 in Q T ◮ λ ∈ ( −∞ , 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0 , 1) ⇒ Global weak solutions ◮ OPEN PROBLEMS ◮ For λ ∈ (0 , 1): is the solution classical ???(i.e. bounded)

L 1 -Theorem does not solve everything ◮ Let λ ∈ ( −∞ , 1) and � ∂ t u − d 1 ∆ u = λ u 3 v 2 − u 2 v 3 in Q T , ∂ t v − d 2 ∆ v = u 2 v 3 − u 3 v 2 in Q T ◮ λ ∈ ( −∞ , 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0 , 1) ⇒ Global weak solutions ◮ OPEN PROBLEMS ◮ For λ ∈ (0 , 1): is the solution classical ???(i.e. bounded) ◮ What happens for λ = 1???

A surprising L 2 -estimate for these systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ L 2 -Theorem. Assume (P)+(M’) . Then, the following a priori estimate holds for the solutions of (S): � m � 2 m � � � u 2 � u 0 ∀ T > 0 , i ≤ C , C = C ( T , ( d i )) . i Ω Q T i =1 i =1

A surprising L 2 -estimate for these systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T   u i (0 , · ) = u 0 i ( · ) ≥ 0 .  ◮ L 2 -Theorem. Assume (P)+(M’) . Then, the following a priori estimate holds for the solutions of (S): � m � 2 m � � � u 2 � u 0 ∀ T > 0 , i ≤ C , C = C ( T , ( d i )) . i Ω Q T i =1 i =1 ◮ The proof uses only the sum of the equations � � ∂ t ( u i ) − ∆( d i u i ) ≤ 0 . i i

Idea of the proof of the L 2 -estimate ◮ �� � �� � ∂ t u i − ∆ d i u i ≤ 0 . i i

Idea of the proof of the L 2 -estimate ◮ �� � �� � ∂ t u i − ∆ d i u i ≤ 0 . i i ◮ � i d i u i � ∂ t W − ∆ ( a W ) ≤ 0 , W = u i a = � i u i i

Idea of the proof of the L 2 -estimate ◮ �� � �� � ∂ t u i − ∆ d i u i ≤ 0 . i i ◮ � i d i u i � ∂ t W − ∆ ( a W ) ≤ 0 , W = u i a = � i u i i ◮ � i d i u i 0 ≤ min d i ≤ a = ≤ max d i < + ∞ � i u i i i

Idea of the proof of the L 2 -estimate ◮ �� � �� � ∂ t u i − ∆ d i u i ≤ 0 . i i ◮ � i d i u i � ∂ t W − ∆ ( a W ) ≤ 0 , W = u i a = � i u i i ◮ � i d i u i 0 ≤ min d i ≤ a = ≤ max d i < + ∞ � i u i i i ◮ The operator W → ∂ t W − ∆( aW ) is not of divergence form and a is not continuous, but bounded from above and from below so that the operator is parabolic and this implies � W � L 2 ( Q T ) ≤ C � W 0 � L 2 (Ω) .

Idea of the proof of the L 2 -estimate ◮ �� � �� � ∂ t u i − ∆ d i u i ≤ 0 . i i ◮ � i d i u i � ∂ t W − ∆ ( a W ) ≤ 0 , W = u i a = � i u i i ◮ � i d i u i 0 ≤ min d i ≤ a = ≤ max d i < + ∞ � i u i i i ◮ The operator W → ∂ t W − ∆( aW ) is not of divergence form and a is not continuous, but bounded from above and from below so that the operator is parabolic and this implies � W � L 2 ( Q T ) ≤ C � W 0 � L 2 (Ω) . ◮ Seen on the dual operator ψ → − ( ∂ t ψ + a ∆ ψ ) which is regularizing and satisfies L 2 -maximal regularity.

Extensions of the L 2 -estimate for such systems ◮ Variable coefficients d i = d i ( t , x ) , nonlinear diffusions − ∆ d i ( u i )

Extensions of the L 2 -estimate for such systems ◮ Variable coefficients d i = d i ( t , x ) , nonlinear diffusions − ∆ d i ( u i ) ◮ W 0 ∈ L 1 (Ω) only !: The L 2 -estimate may be localized for ∂ t W − ∆( aW ) ≤ 0 , d ≤ a ≤ d . � W � L 2 (( τ, T ) × Ω) ≤ C ( d , d , T ) � W 0 � L 1 (Ω) . τ N / 4

Extensions of the L 2 -estimate for such systems ◮ Variable coefficients d i = d i ( t , x ) , nonlinear diffusions − ∆ d i ( u i ) ◮ W 0 ∈ L 1 (Ω) only !: The L 2 -estimate may be localized for ∂ t W − ∆( aW ) ≤ 0 , d ≤ a ≤ d . � W � L 2 (( τ, T ) × Ω) ≤ C ( d , d , T ) � W 0 � L 1 (Ω) . τ N / 4 ◮ [J.A. Ca˜ nizo, L. Desvillettes, K. Fellner]: there exists ǫ ( N ) > 0 such that � W � L 2+ ǫ ( Q T ) ≤ C � W 0 � L 2+ ǫ (Ω) .

Applications to quadratic systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T  i ( · ) ≥ 0 , u 0 ∈ L 1 (Ω) m .  u i (0 , · ) = u 0  ◮ Corollary of the L 1 and L 2 Theorems. Assume (P)+(M’) and the f i are at most quadratic, i.e. � ∀ 1 ≤ i ≤ m , ∀ r ∈ [0 , ∞ ) 2 , | f i ( r ) | ≤ C [1 + r 2 j ] . Then, ( S ) has a global weak solution.

Applications to quadratic systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T  i ( · ) ≥ 0 , u 0 ∈ L 1 (Ω) m .  u i (0 , · ) = u 0  ◮ Corollary of the L 1 and L 2 Theorems. Assume (P)+(M’) and the f i are at most quadratic, i.e. � ∀ 1 ≤ i ≤ m , ∀ r ∈ [0 , ∞ ) 2 , | f i ( r ) | ≤ C [1 + r 2 j ] . Then, ( S ) has a global weak solution. ◮ Proof if u 0 i ∈ L 2 (Ω) . By L 2 -estimate + at most quadratic growth of f i , we have the a priori estimate for the solutions of ( S ) � ∀ 1 ≤ i ≤ m , | f i ( u ) | ≤ C ( T ) , Q T Then, we apply the L 1 -theorem.

Applications to quadratic systems  ∀ i = 1 , ..., m   ∂ t u i − d i ∆ u i = f i ( u 1 , u 2 , ..., u m ) in Q T  ( S ) ∂ ν u i = 0 on Σ T  i ( · ) ≥ 0 , u 0 ∈ L 1 (Ω) m .  u i (0 , · ) = u 0  ◮ Corollary of the L 1 and L 2 Theorems. Assume (P)+(M’) and the f i are at most quadratic, i.e. � ∀ 1 ≤ i ≤ m , ∀ r ∈ [0 , ∞ ) 2 , | f i ( r ) | ≤ C [1 + r 2 j ] . Then, ( S ) has a global weak solution. ◮ Proof if u 0 i ∈ L 2 (Ω) . By L 2 -estimate + at most quadratic growth of f i , we have the a priori estimate for the solutions of ( S ) � ∀ 1 ≤ i ≤ m , | f i ( u ) | ≤ C ( T ) , Q T Then, we apply the L 1 -theorem. ◮ Proof when u 0 i ∈ L 1 (Ω): use the local L 2 -estimates and work a little more near t = 0.

Application to the Lotka-Volterra system ◮ Applies to the (quadratic) Lotka-Volterra system: for all u 0 ∈ L 1 (Ω) + m , there exists a global weak solution to the system: For all i = 1 , ..., m ,    �  u i =: f i ( u ) , ∂ t u i − d i ∆ u i = e i u i + p ij u j 1 ≤ j ≤ m where e i ∈ I R , p ij ∈ I R + ”Dissipation”.

Application to the Lotka-Volterra system ◮ Applies to the (quadratic) Lotka-Volterra system: for all u 0 ∈ L 1 (Ω) + m , there exists a global weak solution to the system: For all i = 1 , ..., m ,    �  u i =: f i ( u ) , ∂ t u i − d i ∆ u i = e i u i + p ij u j 1 ≤ j ≤ m where e i ∈ I R , p ij ∈ I R + ”Dissipation”. ◮ ”Dissipation”: we assume that, for some ( a i ) ∈ (0 , + ∞ ) m m � ∀ ξ ∈ [0 , ∞ ) m , a i p ij ξ i ξ j ≤ 0 i , j =1 � � ⇒ a i f i ( u ) ≤ a i e i u i (whence (M’) ) i i

Applications to chemical quadratic systems ◮ u i = u i ( t , x ) = concentration of U i k + U 1 + U 3 U 2 + U 4 ⇋ k − for i = 1 , 2 , 3 , 4 , ∂ t u i − d i ∆ u i = ( − 1) i ( k + u 1 u 3 − k − u 2 u 4 ) .