Spatial bounds for resolvent families and applications to PDES with - PowerPoint PPT Presentation

Spatial bounds for resolvent families and applications to PDE’S with critical IWOTA nonlinearities Chemnitz, 14 August 2017 Luciano Abadias Departamento de Matem´ atica Aplicada y Estadstica Centro Universitario de la Defensa Zaragoza

1 Historical Motivation 2 The fractional Cauchy problem with memory effects 3 Uniform Stability

1 Historical Motivation 2 The fractional Cauchy problem with memory effects 3 Uniform Stability

Historical Motivation First order case ◮ We consider  x ′ ( t ) = Ax ( t ) + f ( t, x ( t )) , t ∈ (0 , τ ]  (1) x (0) = x 0 ∈ D ( A ) ,  where X is a Banach space, − A : D ( A ) → X is a sectorial linear operator of angle 0 ≤ θ < π/ 2 .

Historical Motivation First order case ◮ We consider  x ′ ( t ) = Ax ( t ) + f ( t, x ( t )) , t ∈ (0 , τ ]  (1) x (0) = x 0 ∈ D ( A ) ,  where X is a Banach space, − A : D ( A ) → X is a sectorial linear operator of angle 0 ≤ θ < π/ 2 . ◮ In fact, if f is time independent, it is well known that if f : X 1 → X α (0 < α ≤ 1) such that � f ( x ) − f ( y ) � X α ≤ C ( R ) � x − y � X 1 , α > 0 , � x � X 1 , � y � X 1 ≤ R, then (1) is locally well posed.

Historical Motivation First order case ◮ We consider  x ′ ( t ) = Ax ( t ) + f ( t, x ( t )) , t ∈ (0 , τ ]  (1) x (0) = x 0 ∈ D ( A ) ,  where X is a Banach space, − A : D ( A ) → X is a sectorial linear operator of angle 0 ≤ θ < π/ 2 . ◮ In fact, if f is time independent, it is well known that if f : X 1 → X α (0 < α ≤ 1) such that � f ( x ) − f ( y ) � X α ≤ C ( R ) � x − y � X 1 , α > 0 , � x � X 1 , � y � X 1 ≤ R, then (1) is locally well posed. ◮ X α := D (( − A ) α ) and � x � X α := � ( − A ) α x � .

◮ Let T : K → K with K ( τ, µ ) = { x ∈ C ([0 , τ ] , X 1 ); x (0) = x 0 , � x � ∞ ≤ � x 0 � X 1 + µ ) } , where � t ( Tx )( t ) = e tA x 0 + e ( t − s ) A f ( x ( s )) ds. 0

◮ Let T : K → K with K ( τ, µ ) = { x ∈ C ([0 , τ ] , X 1 ); x (0) = x 0 , � x � ∞ ≤ � x 0 � X 1 + µ ) } , where � t ( Tx )( t ) = e tA x 0 + e ( t − s ) A f ( x ( s )) ds. 0 ◮ � t ( t − s ) α − 1 ds ( � f (0) � X α � ( Tx )( t ) � X 1 ≤ � e tA x 0 � X 1 + M 0 + C sup {� x ( s )) � X 1 } ) , 0 ≤ s ≤ t � t ( t − s ) α − 1 ds sup � ( Tx )( t ) − ( Ty )( t ) � X 1 ≤ CM {� x ( s ) − y ( s ) � X 1 } , 0 0 ≤ s ≤ t where it is used that � e tA x 0 � X 1 − α ≤ Mt α − 1 � x 0 � , t > 0 .

Example  u t = ∆ u + u | u | ρ − 1 , in Ω ⊂ R 3 ,  u = 0 in ∂ Ω , u (0) = u 0 . 

Example  u t = ∆ u + u | u | ρ − 1 , in Ω ⊂ R 3 ,  u = 0 in ∂ Ω , u (0) = u 0 .  ∆ is an unbounded operator on X = H − 1 (Ω) := ( E 1 / 2 ) ′ , where E 1 / 2 is the fractional space associated to ∆ in L 2 (Ω) with Dirichlet boundary conditions, with domain X 1 := H 1 0 (Ω) , and X α ֒ → H 2 α − 1 , α > 1 / 2 , X 1 / 2 = L 2 (Ω) , X α ← ֓ H 2 α − 1 , α < 1 / 2 .

Example  u t = ∆ u + u | u | ρ − 1 , in Ω ⊂ R 3 ,  u = 0 in ∂ Ω , u (0) = u 0 .  ∆ is an unbounded operator on X = H − 1 (Ω) := ( E 1 / 2 ) ′ , where E 1 / 2 is the fractional space associated to ∆ in L 2 (Ω) with Dirichlet boundary conditions, with domain X 1 := H 1 0 (Ω) , and X α ֒ → H 2 α − 1 , α > 1 / 2 , X 1 / 2 = L 2 (Ω) , X α ← ֓ H 2 α − 1 , α < 1 / 2 . For 1 < ρ < 5 , f : X 1 → X α for some 0 < α < 1 . For ρ = 5 , f : X 1 → X, and we are in the critical case.

Example  u t = ∆ u + u | u | ρ − 1 , in Ω ⊂ R 3 ,  u = 0 in ∂ Ω , u (0) = u 0 .  ∆ is an unbounded operator on X = H − 1 (Ω) := ( E 1 / 2 ) ′ , where E 1 / 2 is the fractional space associated to ∆ in L 2 (Ω) with Dirichlet boundary conditions, with domain X 1 := H 1 0 (Ω) , and X α ֒ → H 2 α − 1 , α > 1 / 2 , X 1 / 2 = L 2 (Ω) , X α ← ֓ H 2 α − 1 , α < 1 / 2 . For 1 < ρ < 5 , f : X 1 → X α for some 0 < α < 1 . For ρ = 5 , f : X 1 → X, and we are in the critical case. But for ρ = 5 , by using the Sobolev embeddings, if ǫ is small then f : X 1+ ǫ → X 5 ǫ , while A : X 1+ ǫ → X ǫ .

ε -regular map For ε > 0 we say that a map g is ε -regular relative to ( X 1 , X ) if there exist ρ > 1 , γ ( ε ) with ρε ≤ γ ( ε ) < 1 , and c > 0 such that g : X 1+ ε → X γ ( ε ) satisfying � g ( x ) − g ( y ) � X γ ( ε ) ≤ c (1+ � x � ρ − 1 X 1+ ε + � y � ρ − 1 x, y ∈ X 1+ ε . X 1+ ε ) � x − y � X 1+ ε ,

ε -regular map For ε > 0 we say that a map g is ε -regular relative to ( X 1 , X ) if there exist ρ > 1 , γ ( ε ) with ρε ≤ γ ( ε ) < 1 , and c > 0 such that g : X 1+ ε → X γ ( ε ) satisfying � g ( x ) − g ( y ) � X γ ( ε ) ≤ c (1+ � x � ρ − 1 X 1+ ε + � y � ρ − 1 x, y ∈ X 1+ ε . X 1+ ε ) � x − y � X 1+ ε , The class F ( ν ) Let ε, γ ( ε ) , ξ, ζ, c, δ ′ > 0 , and a real function ν such that 0 ≤ ν ( t ) < δ ′ and l´ ım t → 0 + ν ( t ) = 0 . The class F ( ε, γ ( ε ) , c, ν, ξ, ζ ) denotes the family of functions f such that, for t ≥ 0 f ( t, · ) is an ε -regular map relative to ( X 1 , X ) , satisfying for all x, y ∈ X 1+ ε c ( � x � ρ − 1 X 1+ ε + � y � ρ − 1 X 1+ ε + ν ( t ) t − ζ ) � x − y � X 1+ ε , � f ( t, x ) − f ( t, y ) � X γ ( ε ) ≤ c ( � x � ρ X 1+ ε + ν ( t ) t − ξ ) . � f ( t, x ) � X γ ( ε ) ≤

ε -regular map For ε > 0 we say that a map g is ε -regular relative to ( X 1 , X ) if there exist ρ > 1 , γ ( ε ) with ρε ≤ γ ( ε ) < 1 , and c > 0 such that g : X 1+ ε → X γ ( ε ) satisfying � g ( x ) − g ( y ) � X γ ( ε ) ≤ c (1+ � x � ρ − 1 X 1+ ε + � y � ρ − 1 x, y ∈ X 1+ ε . X 1+ ε ) � x − y � X 1+ ε , The class F ( ν ) Let ε, γ ( ε ) , ξ, ζ, c, δ ′ > 0 , and a real function ν such that 0 ≤ ν ( t ) < δ ′ and l´ ım t → 0 + ν ( t ) = 0 . The class F ( ε, γ ( ε ) , c, ν, ξ, ζ ) denotes the family of functions f such that, for t ≥ 0 f ( t, · ) is an ε -regular map relative to ( X 1 , X ) , satisfying for all x, y ∈ X 1+ ε c ( � x � ρ − 1 X 1+ ε + � y � ρ − 1 X 1+ ε + ν ( t ) t − ζ ) � x − y � X 1+ ε , � f ( t, x ) − f ( t, y ) � X γ ( ε ) ≤ c ( � x � ρ X 1+ ε + ν ( t ) t − ξ ) . � f ( t, x ) � X γ ( ε ) ≤ ε -regular mild solution We say that x : [0 , τ ] → X 1 is an ε -regular mild solution to (1) if x ∈ C ([0 , τ ] , X 1 ) ∩ C ((0 , τ ] , X 1+ ε ) and � t x ( t ) = e tA x 0 + e A ( t − s ) f ( s, x ( s )) ds. 0

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably:

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably: ◮ Biology

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably: ◮ Biology ◮ Chemistry

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably: ◮ Biology ◮ Chemistry ◮ Economics

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably: ◮ Biology ◮ Chemistry ◮ Economics ◮ Engineering

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably: ◮ Biology ◮ Chemistry ◮ Economics ◮ Engineering ◮ Medicine

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably: ◮ Biology ◮ Chemistry ◮ Economics ◮ Engineering ◮ Medicine ◮ ...

Historical Motivation Fractional case In recent years, the study of fractional partial differential equations has growth considerably: ◮ Biology ◮ Chemistry ◮ Economics ◮ Engineering ◮ Medicine ◮ ... Specifically, fractional models allow to describe phenomena on viscous fluids or in special types of porous medium.

◮ Let  D α t x ( t ) = Ax ( t ) + f ( t, x ( t )) , t ∈ (0 , τ ] ,  (2) x (0) = x 0  where 0 < α ≤ 1, D α t is the Caputo fractional derivative, − A : D ( A ) → X is a sectorial operator and f belongs to the class F ( ν ) .

◮ Let  D α t x ( t ) = Ax ( t ) + f ( t, x ( t )) , t ∈ (0 , τ ] ,  (2) x (0) = x 0  where 0 < α ≤ 1, D α t is the Caputo fractional derivative, − A : D ( A ) → X is a sectorial operator and f belongs to the class F ( ν ) . ◮ Let ( R α ( t )) t> 0 and ( S α ( t )) t ≥ 0 defined by 1 � e λt ( λ α − A ) − 1 dλ, R α ( t ) := t > 0 , 2 πi γ and 1 � e λt λ α − 1 ( λ α − A ) − 1 dλ, S α ( t ) := t > 0 , 2 πi γ where γ ⊂ ρ ( A ) is a suitable Hankel’s path.

ε -regular mild solution We say that x : [0 , τ ] → X 1 is an ε -regular mild solution to (2) if x ∈ C ([0 , τ ] , X 1 ) ∩ C ((0 , τ ] , X 1+ ε ) and � t x ( t ) = S α ( t ) x 0 + R α ( t − s ) f ( s, x ( s )) ds. 0