Control and inverse problems for degenerate parabolic operators - PowerPoint PPT Presentation

existence, uniqueness, regularity one space dimension a ∈ C ([ 0 , 1 ]) ∩ C 1 (] 0 , 1 ]) and a > 0 on ] 0 , 1 ] � � � u t − a ( x ) u x x = f in Q T =] 0 , 1 [ × ] 0 , T [ u ( x , 0 ) = u 0 ( x ) u ( t , 1 ) = 0 + b.c. at x = 0 u 0 ∈ L 2 ( 0 , 1 ) , f ∈ L 2 ( Q T ) Campiti, Metafune, Pallara (1998) 1 / a ∈ L 1 ( 0 , 1 ) weakly degenerate case: � 1 � � � H 1 u ∈ L 2 ( 0 , 1 ) au 2 a ( 0 , 1 ) = x dx < ∞ & u ( 0 ) = 0 = u ( 1 ) � 0 ∈ L 1 ( 0 , 1 ) strongly degenerate case: 1 / a / � 1 � � � H 1 u ∈ L 2 ( 0 , 1 ) au 2 a ( 0 , 1 ) = x dx < ∞ & u ( 1 ) = 0 � 0 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 8 / 46

existence, uniqueness, regularity well-posedness � � � � � au x ∈ H 1 ( 0 , 1 ) u ∈ H 1 D ( A ) = a ( 0 , 1 ) � � A u = au x x generates analytic semigroup in L 2 ( 0 , 1 ) u ∈ C ( 0 , T ; L 2 ( 0 , 1 )) ∩ L 2 ( 0 , T ; H 1 unique solution a ( 0 , 1 )) � � � u t − a ( x ) u x x = f in Q T =] 0 , 1 [ × ] 0 , T [ u ( x , 0 ) = u 0 ( x ) maximal regularity u 0 ∈ H 1 u ∈ H 1 ( 0 , T ; L 2 ( 0 , 1 )) ∩ L 2 ( 0 , T ; D ( A )) a ( 0 , 1 ) = ⇒ (needed to justify integration by parts) strongly degenerate case incorporates b.c. ( x → 0 ) u ∈ D ( A ) = ⇒ au x − → 0 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 9 / 46

existence, uniqueness, regularity Budyko-Sellers model � � � ( 1 − x 2 ) u x u t − x = f ( x ) g ( u ) − h ( u ) x ∈ ( − 1 , 1 ) ( 1 − x 2 ) u x | x = ± 1 = 0 ✬✩ r x = sin α � � α ✫✪ u ( t , x ) = sea-level zonally averaged temperature f ( x ) = solar input g ( u ) = co-albedo h ( u ) = outgoing infrared radiation P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 10 / 46

existence, uniqueness, regularity the simplest problem in 2d similar theory  u t − div ( A ( x ) ∇ u ) = χ ω ( x ) f ( t , x ) in Q T   n = 2 u ( x , 0 ) = u 0 ( x ) x ∈ Ω   + b. c. on Γ σ ( A ( x )) = { λ 1 ( x ) , λ 2 ( x ) } eigenvectors ε 1 ( x ) , ε 2 ( x ) � λ 1 ( x ) ∼ d Γ ( x ) α , ε 1 ( x ) ∼ − Dd Γ ( x ) = ν Γ (Π Γ ( x )) near Γ ✬ ✩ λ 2 ( x ) ≥ m > 0 ∀ x ∈ Ω Ω ε 2 ( x ) Γ q ❅ ■ q ✫ ❅ ✪ x � ε 1 ( x ) ✠ � Π Γ ( x ) C–, Rocchetti & Vancostenoble (2008) P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 11 / 46

boundary degeneracy overview Outline Examples of degenerate parabolic equations 1 Existence, uniqueness, and regularity 2 Null controllability for boundary degeneracy 3 Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems Controllability for Grushin-type operators 4 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 12 / 46

boundary degeneracy overview controlled parabolic equations ω ⊂⊂ Ω T > 0  u t − div ( A ( x ) ∇ u ) = χ ω ( x ) f ( t , x ) in Q T := Ω × ] 0 , T [   u f → u ( x , 0 ) = u 0 ( x ) x ∈ Ω   + b. c. f control χ ω characteristic function of ω � � n A ( x ) = a ij ( x ) i , j = 1 a ij = a ji ∈ C (Ω) ∩ C 1 (Ω) ✬ ✩ positive definite in Ω (not in Ω ) ✓✏ ✒✑ ω Ω ✫ ✪ also of interest boundary control Γ 1 ⊂ Γ u ( t , x ) = g ( t , x ) ( t , x ) ∈ ( 0 , T ) × Γ 1 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 13 / 46

boundary degeneracy overview the null controllability problem want to study null-controllability in time T > 0  u f ( · , T ) ≡ 0  ∀ u 0 ∈ L 2 (Ω) ∃ f ∈ L 2 ( Q T ) : � Q T | f | 2 ≤ C T � Ω | u 0 | 2  uniformly parabolic equations: ∃ m > 0 : A ( x ) ≥ m I n = ⇒ null-controllability ∀ T > 0 Fattorini and Russell (1971), Russell (1978) Lebeau and Robbiano (1995) Fursikov and Emanouilov (1996) Tataru (1997) P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 14 / 46

boundary degeneracy overview the roadmap to null controllability show equivalence with observability inequality � adjoint v t + div ( A ( x ) ∇ v ) = 0 in Q T problem v = 0 on Γ × ] 0 , T [ � T � � v 2 ( x , 0 ) dx ≤ C T v 2 ( x , t ) dxdt = ⇒ Ω 0 ω prove observability by Carleman estimates τ >> 1 � T �� e 2 s φ ( x , t ) dxdt ≤ C v 2 dxdt τ 3 θ 3 ( t ) v 2 � �� Q T 0 ω + τθ ( t ) | Dv | 2 + ··· � e r ψ ( x ) − e 2 r � ψ � ∞ � φ ( x , t ) = θ ( t ) any D ψ ( x ) � = 0 in Ω \ ω P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 15 / 46

boundary degeneracy overview difficulties in degenerate case observability ( ⇒ null controllability) may fail (for violent degeneracies) φ in Carleman must be adapted to degeneracy Hardy’s inequality essential � � w 2 dx ≤ C α Γ |∇ w | 2 dx d α − 2 d α ( α � = 1 ) Γ Ω Ω P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 16 / 46

boundary degeneracy one space dimension Outline Examples of degenerate parabolic equations 1 Existence, uniqueness, and regularity 2 Null controllability for boundary degeneracy 3 Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems Controllability for Grushin-type operators 4 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 17 / 46

boundary degeneracy one space dimension the simplest example of degeneracy a ( x ) = x α ω =] a , b [ ⊂⊂ ( 0 , 1 ) ( α > 0 ) � � x α u x u t − x = χ ω f , u ( 0 , x ) = u 0 ( x ) Theorem (C – Martinez – Vancostenoble, 2008)  false α ≥ 2 ( → regional null controllability )   � n. c. any b.c. 0 ≤ α < 1 weak true 0 ≤ α < 2   Neumann b.c. 1 ≤ α < 2 strong r r r r regional T r r r 0 1 ω P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 18 / 46

boundary degeneracy one space dimension Carleman estimate 0 < α < 2 � � x α w x w t + x = f in ( 0 , 1 ) × ( 0 , T ) + b. c. let ϕ ( t , x ) = θ ( t ) ψ ( x ) where � � 4 ψ ( x ) = x 2 − α − 2 1 θ ( t ) = t ( T − t ) ( 2 − α ) 2 Theorem (C – Martinez – Vancostenoble, 2008) There exists τ 0 , C > 0 such that ∀ τ ≥ τ 0 �� w 2 x + τ 3 θ 3 x 2 − α w 2 � e 2 τϕ dxdt τθ + τθ x α w 2 t Q T � T �� | f | 2 e 2 τϕ dxdt + C w 2 ( x , t ) dxdt ≤ C Q T 0 ω P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 19 / 46

boundary degeneracy one space dimension more general 1-d problems divergence form � � u t − a ( x ) u x x = χ ω f Martinez – Vancostenoble (2006) � � Alabau – C – Fragnelli (2006) u t − a ( x ) u x x + g ( u ) = χ ω f � x θ u x � x + x σ b ( x , t ) u x = χ ω f Flores – de Teresa (2010) u t − non-divergence form C – Fragnelli – Rocchetti (2007, 2008) u t − a ( x ) u xx − b ( x ) u x = χ ω f degenerate/singular problems Vancostenoble – Zuazua (2008), Vancostenoble (2009) λ x θ u x � � u t − x − x σ u = χ ω f systems C – de Teresa (2009) cascade 2 × 2 Maniar et al. (2011) general 2 × 2 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 20 / 46

boundary degeneracy higher space dimension Outline Examples of degenerate parabolic equations 1 Existence, uniqueness, and regularity 2 Null controllability for boundary degeneracy 3 Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems Controllability for Grushin-type operators 4 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 21 / 46

boundary degeneracy higher space dimension extension to n ≥ 2  u t − div ( A ( x ) ∇ u ) = χ ω ( x ) f ( t , x ) in Q T   n = 2 u ( x , 0 ) = u 0 ( x ) x ∈ Ω   + b. c. on Γ σ ( A ( x )) = { λ 1 ( x ) , λ 2 ( x ) } eigenvectors ε 1 ( x ) , ε 2 ( x ) � λ 1 ( x ) ∼ d Γ ( x ) α , ε 1 ( x ) ∼ − Dd Γ ( x ) = ν Γ (Π Γ ( x )) near Γ λ 2 ( x ) ≥ m > 0 ∀ x ∈ Ω ✬ ✩ Ω ε 2 ( x ) Γ q ❅ ■ q ✫ ❅ ✪ x � ε 1 ( x ) ✠ � Π Γ ( x ) P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 22 / 46

boundary degeneracy higher space dimension null controllability ( n = 2 )  u t − div ( A ( x ) ∇ u ) = χ ω ( x ) f ( t , x ) in Q T σ ( A ( x )) = { λ 1 ( x ) , λ 2 ( x ) }   u ( x , 0 ) = u 0 ( x ) x ∈ Ω λ 1 ( x ) ∼ d Γ ( x ) α   + b. c. on Γ Theorem (C, Martinez, Vancostenoble – CRAS 2009) 0 ≤ α < 2 null controllability holds α ≥ 2 null-controllability fails the proof uses topological lemma to construct adapted weight Carleman’s estimate to provide observability inequality Hardy’s inequality to control degenerate terms P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 23 / 46

boundary degeneracy inverse problems Outline Examples of degenerate parabolic equations 1 Existence, uniqueness, and regularity 2 Null controllability for boundary degeneracy 3 Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems Controllability for Grushin-type operators 4 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 24 / 46

boundary degeneracy inverse problems inverse problems in 1 d 0 ≤ α < 2  u t − ( x α u x ) x = g Q T = ( 0 , T ) × ( 0 , 1 )    u ( t , 1 ) = 0 t ∈ ( 0 , T )  � u ( t , 0 ) = 0  for 0 ≤ α < 1 t ∈ ( 0 , T )  ( x α u x ) | x = 0 = 0 for 1 ≤ α < 2     u ( 0 , x ) = u 0 ( x ) x ∈ ( 0 , 1 ) want to ‘determine’ g by measurements of u stability estimates uniqueness results T u x ( · , 1 ) u ( T ′ , · ) g ← − 0 1 Figure: ‘boundary measurements’ P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 25 / 46

boundary degeneracy inverse problems references uniformly parabolic case ( α = 0 ) Bukhgeim & Klibanov (1981), Klibanov (1992), Isakov (1998), Klibanov & Timonov (2004) (H¨ older stability by local Carleman estimates) Emanouilov & Yamamoto (1998, 2001) (Lipschitz stability by global Carleman estimates) degenerate case α ∈ [ 0 , 2 ) C– Tort & Yamamoto (2010) (Lipschitz stability n = 1) C– Martinez & Vancostenoble (Lipschitz stability n = 2) P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 26 / 46

boundary degeneracy inverse problems Lipschitz stability: boundary measurements  u t − ( x α u x ) x = g ( t , x ) Q T = ( 0 , T ) × ( 0 , 1 )    u ( t , 1 ) = 0 t ∈ ( 0 , T )   � u ( t , 0 ) = 0 for 0 ≤ α < 1 t ∈ ( 0 , T )  ( x α u x ) | x = 0 = 0 for 1 ≤ α < 2     u ( 0 , x ) = u 0 ( x ) x ∈ ( 0 , 1 ) Theorem 0 ≤ α < 2 there exists t 0 ∈ ( 0 , T ) and k ≥ 0 such that � � ∂ g T ′ = T + t 0 � � � � � g ( T ′ , x ) ∂ t ( t , x ) � ≤ k where � � � 2 � � T � 1 � g � 2 | u tx ( t , 1 ) | 2 dt + C | ( x α u x ( T ′ , x )) x | 2 dx = ⇒ L 2 ( QT ) ≤ C t 0 0 where C = C ( α, k , t 0 , T ) P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 27 / 46

boundary degeneracy inverse problems uniqueness: boundary measurements  u t − ( x α u x ) x = f ( x ) r ( t , x ) Q T = ( 0 , T ) × ( 0 , 1 )    u ( t , 1 ) = 0 t ∈ ( 0 , T )   � u ( t , 0 ) = 0 for 0 ≤ α < 1 ( IP ) t ∈ ( 0 , T )  ( x α u x ) | x = 0 = 0 for 1 ≤ α < 2     u ( 0 , x ) = u 0 ( x ) x ∈ ( 0 , 1 ) r ∈ C 1 ([ 0 , T ] × [ 0 , 1 ]) given such that at T ′ = T + t 0 we have 2 ′ , x ) ≥ d > 0 r ( T ∀ x ∈ [ 0 , 1 ] ( ∗ ) f 1 , f 2 ∈ L 2 ( 0 , 1 ) u 1 , u 2 ∈ L 2 ( 0 , 1 ) solutions of ( IP ) � T � f 1 − f 2 � 2 | ( u 1 − u 2 ) tx ( t , 1 ) | 2 dt = ⇒ L 2 ( 0 , 1 ) ≤ C t 0 � 1 | ( x α ( u 1 ( T ′ , x ) − u 2 ( T ′ , x )) x ) x | 2 dx + C 0 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 28 / 46

boundary degeneracy inverse problems locally distributed measurements  u t − ( x α u x ) x = g Q T = ( 0 , T ) × ( 0 , 1 )    u ( t , 1 ) = 0 t ∈ ( 0 , T )   � u ( t , 0 ) = 0 for 0 ≤ α < 1 t ∈ ( 0 , T )  ( x α u x ) | x = 0 = 0 for 1 ≤ α < 2     u ( 0 , x ) = u 0 ( x ) x ∈ ( 0 , 1 ) r r r r T u t u ( T ′ , · ) g ← − r r r 0 1 ω stability estimates uniqueness results P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 29 / 46

control with interior degeneracy Grushin-type operators Ω := ( − 1 , 1 ) × ( 0 , 1 ) , ω ⊂ ( a , b ) × ( 0 , 1 ) with 0 < a < b < 1 1 y ♥ Ω ω 1 x a b − 1 0  ∂ t u − ∂ 2 x u − | x | 2 γ ∂ 2 y u = χ ω ( x , y ) f ( t , x , y )    γ > 0 u ( t , ± 1 , y ) = 0 , u ( t , x , 0 ) = 0 = u ( t , x , 1 ) ( G )    u ( 0 , x , y ) = u 0 ( x , y ) P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 30 / 46

control with interior degeneracy existence and uniqueness of solutions � � � In H = L 2 (Ω) with scalar product ( f , g ) = f x g x + | x | 2 γ f y g y dxdy Ω V = C ∞ 0 (Ω) a ( f , g ) = − ( f , g ) ∀ f , g ∈ V D ( A ) = { f ∈ V : ∃ c > 0 such that | a ( f , h ) | ≤ c � h � H ∀ h ∈ V } and � Af , h � = a ( f , h ) ∀ h ∈ V A : D ( A ) ⊂ H → H generator of a semigroup e tA of contractions in H Theorem T > 0 , u 0 ∈ L 2 (Ω) , f ∈ L 2 (( 0 , T ) × Ω) ⇒ ∃ ! u ∈ C ([ 0 , T ]; L 2 (Ω)) : ∀ t ∈ ( 0 , T ) , φ ∈ C 2 ([ 0 , T ] × Ω) = � t � � u ( ∂ t φ + ∂ 2 x φ + | x | 2 γ ∂ 2 [ u ( t ) φ ( t ) − u ( 0 ) φ ( 0 )] = y φ ) + χ ω f φ Ω 0 Ω P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 31 / 46

control with interior degeneracy approximate controllability approximate controllability ⇐ ⇒ unique continuation Garofalo (1993): unique continuation for elliptic operator A = ∂ 2 + | x | 2 γ ∂ 2 y for parabolic operators: Proposition (BCG 2012) Let T > 0 , γ > 0 , let ω ⊂ ( 0 , 1 ) × ( 0 , 1 ) , and let g ∈ C ([ 0 , T ]; H ) ∩ L 2 ( 0 , T ; V ) be a weak solution of � ∂ t g − ∂ 2 x g − | x | 2 γ ∂ 2 y g = 0 ( t , x , y ) ∈ ( 0 , ∞ ) × Ω g ( t , x , y ) = 0 ( t , x , y ) ∈ ( 0 , ∞ ) × ∂ Ω If g ≡ 0 on ( 0 , T ) × ω , then g ≡ 0 on ( 0 , T ) × Ω . P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 32 / 46

control with interior degeneracy null controllability  ∂ t u − ∂ 2 x u − | x | 2 γ ∂ 2 y u = χ ω ( x , y ) f ( t , x , y )  u ( t , ± 1 , y ) = 0 , u ( t , x , 0 ) = 0 = u ( t , x , 1 ) ( G )  u ( 0 , x , y ) = u 0 ( x , y ) 1 y ♥ Ω ω 1 x a b − 1 0 Theorem (BCG 2012) 0 < γ < 1 ⇒ ( G ) null controllable ∀ T > 0 γ > 1 ⇒ ( G ) not null controllable ∃ T ∗ � a 2 / 2 γ = 1 & ω = ( a , b ) × ( 0 , 1 ) = ⇒ such that ( G ) is null controllable ∀ T > T ∗ not null controllable ∀ T < T ∗ P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 33 / 46

control with interior degeneracy observability  ∂ t v − ∂ 2 x v − | x | 2 γ ∂ 2 y v = 0  ( G ∗ ) v ( t , ± 1 , y ) = 0 , v ( t , x , 0 ) = 0 = v ( t , x , 1 )  v ( 0 , x , y ) = v 0 ( x , y ) ∃ C T > 0 such that ∀ v 0 ∈ L 2 (Ω) observable in [ 0 , T ] × ω � T � � | v ( T , x , y ) | 2 dxdy ≤ C T | v ( t , x , y ) | 2 dxdy ( O ) Ω 0 ω Theorem (BCG 2012) ( G ∗ ) 0 < γ < 1 ⇒ ∀ T > 0 observable ( G ∗ ) γ > 1 ⇒ not observable ∃ T ∗ � a 2 / 2 ( G ∗ ) γ = 1 & ω = ( a , b ) × ( 0 , 1 ) = ⇒ such that is observable ∀ T > T ∗ not observable ∀ T < T ∗ P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 34 / 46

control with interior degeneracy method: Fourier decomposition  ∂ t v − ∂ 2 x v − | x | 2 γ ∂ 2 y v = 0  ( G ∗ ) v ( t , ± 1 , y ) = 0 , v ( t , x , 0 ) = 0 = v ( t , x , 1 )  v ( 0 , x , y ) = v 0 ( x , y ) ∞ √ � v ( t , x , y ) = v n ( t , x ) ϕ n ( y ) with ϕ n ( y ) := 2 sin ( n π y ) n = 1 � 1 where v n ( t , x ) := 0 v ( t , x , y ) ϕ n ( y ) dy satisfies  ∂ t v n − ∂ 2 x v n + ( n π ) 2 | x | 2 γ v n = 0 ( t , x ) ∈ ( 0 , T ) × ( − 1 , 1 )  ( G ∗ v n ( t , ± 1 ) = 0 t ∈ ( 0 , T ) n )  v n ( 0 , x ) = v 0 , n ( x ) x ∈ ( − 1 , 1 ) � 1 ∞ � � | v ( T , x , y ) | 2 dxdy = | v n ( T , x ) | 2 dx Ω − 1 n = 1 � b � ∞ � | v ( t , x , y ) | 2 dxdy = | v n ( t , x ) | 2 dx ω =( a , b ) × ( 0 , 1 ) a n = 1 P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 35 / 46

control with interior degeneracy uniform observability  ∂ t v − ∂ 2 x v − | x | 2 γ ∂ 2 y v = 0 ( 0 , T ) × Ω  ( G ∗ ) v ( t , ± 1 , y ) = 0 , v ( t , x , 0 ) = 0 = v ( t , x , 1 ) t ∈ ( 0 , T )  v ( 0 , x , y ) = v 0 ( x , y ) ( x , y ) ∈ Ω 1 y Ω ω 1 x a b − 1 0  ∂ t v n − ∂ 2 x v n + ( n π ) 2 | x | 2 γ v n = 0 ( t , x ) ∈ ( 0 , T ) × ( − 1 , 1 )  ( G ∗ v n ( t , ± 1 ) = 0 t ∈ ( 0 , T ) n )  v n ( 0 , x ) = v 0 , n ( x ) x ∈ ( − 1 , 1 ) observability for ( G ∗ ) in ω uniform observability for ( G ∗ ⇐ ⇒ n ) in ( a , b ) � 1 � T � b | v n ( T , x ) | 2 dx ≤ C | v n ( t , x ) | 2 dxdt ∀ n ≥ 1 − 1 0 a P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 36 / 46

control with interior degeneracy step 1: dissipation speed A n : D ( A n ) ⊂ L 2 ( − 1 , 1 ) → L 2 ( − 1 , 1 ) define by D ( A n ) := H 2 ∩ H 1 A n ϕ := − ϕ ′′ + ( n π ) 2 | x | 2 γ ϕ 0 ( − 1 , 1 ) , λ n := the first eigenvalue of A n so that � ∂ t v n − ∂ 2 x v n + ( n π ) 2 | x | 2 γ v n = 0 ( t , x ) ∈ ( 0 , T ) × ( − 1 , 1 ) v n ( t , ± 1 ) = 0 t ∈ ( 0 , T ) satisfies � 1 � 1 | v n ( T , x ) | 2 dx ≤ e − λ n ( T − t ) | v n ( t , x ) | 2 dx ∀ t ∈ [ 0 , T ] ( D n ) − 1 − 1 Lemma (dissipation speed) ∃ c ∗ > 0 2 λ n ≤ c ∗ n (ub) ∀ γ > 0 such that 1 + γ 2 (lb) ∀ γ ∈ ( 0 , 1 ] ∃ c ∗ > 0 such that λ n ≥ c ∗ n 1 + γ P. Cannarsa (Rome Tor Vergata) degenerate parabolic operators April 4, 2012 37 / 46

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