Department of Mathematics Go Hasada We consider a generation theorem and an approximation theory for nonlinear semigroups in a general Banach space which provide mild solutions to a semilinear problem of a form

The semigroup approach to the approximation-solvability of semilinear evolution equations

半線形発展方程式の近似可解性に対する半群論的接近

Department of Mathematics Go Hasada We consider a generation theorem and an approximation theory for nonlinear semigroups in a general Banach space which provide mild solutions to a semilinear problem of a form

(SP) u ⁰ (t) = (A + B)u(t), t > 0; u(0) = x ∈ D.

Let (X, | · |) be a real Banach space and D a subset of X . We employ a lower semicontinuous functional ϕ such that D ⊂ D(ϕ) = {x ∈ X; ϕ(x) < ∞} and denote by D α = {x ∈ D; ϕ(x) < α}, α > 0, the level sets of D. X ^∗ denotes the dual space of X. The duality set is defined by F x = {x ^∗ ∈ X ^∗ ; hx, x ^∗ i = |x| ² =

|x ^∗ | ² }. Given a pair x and y in X, we define the lower semi-inner product hy, xi i by the infimum of the set {hy, f i; f ∈ F x}. A nonlinear operator B : D ⊂ X → X is said to be locally quasidissipative on D(B) with respect to ϕ if for each α ≥ 0 there exists ω α ∈ R such that hBx − By, x − yi i ≤ ω α |x− y| ² for x, y ∈ D α . It is said that a function u ∈ C([0, ∞), X ) is a mild solution to (SP) if u(t) ∈ D for t ≥ 0, Bu ∈ C([0, ∞), X) and the integral equation

u(t) = T(t)x + Z _t

0

T (t − s)Bu(s)xds is satisfied for each t ≥ 0.

In what follows, the operators A and B are assumed to satisfy the following conditions:

(A) A : D(A) ⊂ X → X generates a (C 0 )-semigroup T = {T (t); t ≥ 0} such that |T (t)x| ≤ e ^ωt |x| for x ∈ X , t ≥ 0 and some ω ∈ R.

(B) The level set D α = {x ∈ D; ϕ(x) ≤ α} is closed for each α ≥ 0 and B : D ⊂ X → X is nonlinear and continuous on each D α .

Theorem 1 (The generation theorem). Let a, b ≥ 0 and suppose that the operators A and B are assumed to satisfy conditions (A) and (B). Then the following statements (I),(II) are equivalent:

(I) There is a nonlinear semigroup S = {S(t); t ≥ 0} on D satisfying the following properties:

(I.1) S(t)x = T (t)x + Z _t

0

T (t − s)BS (s)xds for t ≥ 0 and x ∈ D.

(I.2) For α > 0 and τ > 0, there is ω 1 = ω 1 (α, τ ) ∈ R such that

|S(t)x − S(t)y| ≤ e ^ω

^(α,τ)t |x − y| for x, y ∈ D α and t ∈ [0, τ ].

(I.3) ϕ(S(t)x) ≤ e ^at (ϕ(x) + bt) for x ∈ D and t ≥ 0.

(II) The semilinear operator A+B satisfies the explicit subtangential condition and the semilinear stability condition below:

(II.1) For x ∈ D and ε > 0, there is (h, x h ) ∈ (0, ε] × D such that 1

h |T(h)x + hBx − x h | ≤ ε and ϕ(x h ) ≤ e ^ah (ϕ(x) + (b + ε)h).

(II.2) For α > 0, there is ω α ∈ R such that for x, y ∈ D α

lim

h↓0

1

h (|T(h)(x − y) + h(Bx − By)| − |x − y|) ≤ ω α |x − y|.

We next consider the approximate evolution problems

(SP; n) u ⁰ _n (t) = (A n + B n )u n (t), t > 0; u n (0) = x n ∈ D n

where D n ⊂ X and ϕ n : X → [0, ∞] are proper lower semicontinuous functionals such that D n ⊂ D(ϕ n ) = {x ∈ X ; ϕ n (x) < ∞}. In what follows we say that {x n } n≥1 is a {ϕ n }-bounded sequence if x n ∈ D n for each n ≥ 1 and sup

n≥1 ϕ n (x n ) < ∞.

We assume that for each n ≥ 1 the operators A n and B n with the domain D n satisfy the hypotheses stated below:

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(A n ) A n : D(A n ) ⊂ X → X generates a (C 0 )-semigroup T n = {T n (t); t ≥ 0} on X and there is ω n ∈ R such that |T n (t)x| ≤ e ^ω

^t |x| for each x ∈ X , t ≥ 0.

(B n ) The level set D n,α = {x ∈ D n ; ϕ n (x) ≤ α} is closed in X for each α ≥ 0 and B n : D n ⊂ X → X is nonlinear and continuous from D n,α into X .

Suppose that for each n there exists a locally Lipschitzian semigroup S n = {S n (t); t ≥ 0} on D n such that

(1) S n (t)x n = T n (t)x n +

Z _t

0

T n (t − s)B n S n (s)x n ds, (2) ϕ n (S n (t)x n ) ≤ e ^at (ϕ n (x n ) + bt)

for t ≥ 0 and x n ∈ D n .

In order to handle the limits of solutions to (SP; n), we next define a set D e by D e = {x ∈ X ; x is a limit of some {x n } with x n ∈ D n for n ≥ 1} and a functional Φ : X → [0, ∞) by

(3) Φ(x) =

( inf

n

n→∞ lim ϕ n (x n ); x n ∈ D n for n ≥ 1, x n → x as n → ∞ o

for x ∈ D e

∞ otherwise.

We also define the level sets of Φ by D α = {x ∈ X; Φ(x) ≤ α}, α ≥ 0.

Consistency Condition

(C) The following conditions are satisfied:

(C1) For x ∈ X and τ > 0, T n (t)x → T (t)x as n → ∞, uniformly with respect to t ∈ [0, τ ].

(C2) If x ∈ D, x n ∈ D n , lim

n→∞ ϕ n (x n ) < ∞, and x n → x as n → ∞, then B n x n → Bx in X as n → ∞.

(C3) The following conditions hold:

(a) For each x ∈ D there is a sequence {x n } such that x n ∈ D n , lim

n→∞ ϕ n (x n ) < ∞ and x n → x as n → ∞.

(b) There is β ≥ 0 such that D n,β 6= ∅ for n ≥ 1.

(c) If x n ∈ D n , lim

n→∞ |x n | < ∞ and lim

n→∞ ϕ n (x n ) < ∞, then lim

n→∞ d(x n , D α ) = 0, for α > lim

n→∞ ϕ n (x n ) Stability Condition

(S1) For x ∈ D and for a {ϕ n }-bounded sequence {x n } n≥1 with x n → x as n → ∞, sup

n≥1 |S n (t)x n − x n | → 0 as t ↓ 0.

(S2) There is a separately nondecreasing function ω : [0, ∞) × [0, ∞) → [0, ∞) such that

|S n (t)x n − S n (t)y n | ≤ e ^ω(α,t)t |x n − y n | for t ≥ 0, α ≥ 0, x n , y n ∈ D α and n = 1, 2, · · · .

Theorem 2 (The approximation-solvability theorem). Let ψ be the functional defined by (3).

Suppose that conditions (C) and (S) are satisfied.

Then there exists a locally Lipschitzian semigroup S = {S(t); t ≥ 0} on D, satisfying S(t)x = T (t)x +

Z _t

0

T(t − s)BS(s)xds,

Φ(S(t)x) ≤ e ^at (Φ(x) + bt) for t ≥ 0 and x ∈ D.

Moreover, if x ∈ D and {x n } is a {ϕ n }-bounded sequence with x n → x as n → ∞, then S n (t)x n → S(t)x and the convergence is uniform on bounded subintervals of [0, ∞). If in particular, B is one-sided G-differentiable, S(t)x is C ¹ on [0, ∞) and S(t)x ∈ D(A).

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Lemma 1. For each x ∈ D e there is a sequence {x n } such that x n ∈ D n , x n → x as n → ∞ and ψ(x) = lim

n→∞ ϕ n (x n ).

Theorem 3 (The convergence theorem). Suppose that conditions (C) and (S2) are satisfied. Then condition (S1) is equivalent to the statement below:

If x ∈ D, {x n } n≥1 is a {ϕ n }-bounded sequence and x n → x as n → ∞, then S n (t)x n → S 0 (t)x as n → ∞ for t ≥ 0, and the convergence is uniform on bounded subintervals of [0, ∞).

An application to population dynamics

We consider a semilinear system modeling nonlinear age-dependent population dynamics with pair formation, which takes form

(P D)

 

 

 

 

 

 

 

 

 

 

 



u t + u x = −d 1 u − Z _∞

0

p(t, x, y, 0)dy + Z _∞

0

Z _∞

0

{σ + (1 − d 1 )d 2 }pdydz

v t + v y = −d 2 v − Z _∞

0

p(t, x, y, 0)dx + Z _∞

0

Z _∞

0

{σ + d 1 (1 − d 2 )}pdxdz

p t + p x + p y + p z = −{d 1 (1 − d 2 ) + (1 − d 1 )d 2 + d 1 d 2 + σ}p

(BC)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



u(t, 0) = Z _∞

0

Z _∞

0

Z _∞

0

β 1 pdxdydz

v(t, 0) = Z _∞

0

Z _∞

0

Z _∞

0

β 2 pdxdydz

p(t, x, y, 0) = ξ(t, x, y) p(t, 0, y, z) = p(t, x, 0, z) = 0

(IC) u(0, x) = u 0 (x), v(0, y) = v 0 (y), p(0, x, y, z) = p 0 (x, y, z).

u := u(t, x) : single male individuals with age x at time t v := v(t, y) : single female individuals with age y at t

p := p(t, x, y, z) : pairs of maturasion z at t consisting of a male with age x and a female of age y d 1 := d 1 (t, x) : mortality rate of males of age x at t

d 2 := d 2 (t, y) : mortality rate of females of age y at t σ := σ(t, x, y, z) : separation rate of a pair p(t, x, y, z)

β 1 := β 1 (t, x, y, z) : rate at which male offspring is born from a pair p(t, x, y, z) β 2 := β 2 (t, x, y, z) : rate at which female offspring is born from a pair p(t, x, y, z) ξ := ξ(t, x, y) : rate at which pairs are formed by u(t,x) and v(t, y)

The semilinear differential equation (PD) with (BC) and (IC) in X = L ¹ (0, ∞)×L ¹ (0, ∞)×L ¹ ((0, ∞) ³ ) is converted to semilinear problem in X = (0, ∞) × X .

(P D) ⁰

½ u ⁰ (t) = Au(t) + F u(t), t > 0 u(0) = u 0

(BC) ⁰ Lu(t) = Bu(t), t > 0

Here u = (u, v, p) and u 0 = (u 0 , v 0 , p 0 ). (PD)’ means (PD) with the initial condition (IC) where the operator A denotes the differential operator on X and Fu(t) represents the right hand sides of (PD).

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The semilinear boundary condition (BC)’ means (BC).

Banach space X = (0, ∞) × X is equipped with the norm k(t, u)k = |t| + |u| X . A subset C of X is defined by C = (0, ∞) × L ¹ ₊ (0, ∞) × L ¹ ₊ (0, ∞) × L ¹ ₊ ((0, ∞) ³ ). Let A be a linear operator on X such that D(A) = {0} × D(A) and A(0, u) = (−Lu, Au) for u ∈ D(A). And also Let B be a nonlinear operator on X such that B(0, u) = (B u, F u) for u ∈ D(B). Put U(t) = (0, u), and a lower semicontinuous functional ϕ : (0, ∞) × X → [0, ∞] defined by ϕ(t, u) = t + kuk X for (t, u) ∈ C. We also put C α = {(t, u) ∈ C; ϕ(t, u) ≤ α}, for α > 0, we have a semilinear problem in X ,

½ U ⁰ (t) = AU(t) + BU (t), t > 0 U(0) = (0, u 0 ).

We next treat the discretization of (PD) and (BC) by means of the finite difference method. We denote A ∼ B that B is a discretization of A.

t ∼ nh, x ∼ ih, y ∼ jh, z ∼ kh d 1 (t, x) ∼ d 1,i , d 2 (t, y) ∼ d 2,j , ξ(t, x, y) ∼ ξ _i,j ⁿ ,

σ(t, x, y, z) ∼ σ ⁿ _i,j,k

(DE)

 

 

 

 

 

 

 

 

 

 

 



u ⁿ⁺¹ _i+1 = (1 − hd 1,i )u ⁿ _i − h ² X ∞

j

ξ ⁿ _i,j + h ³ X ∞

j

X ∞

k

{ρ + (1 − d 1,i )d 2,j }p ⁿ _i,j,k

v ⁿ⁺¹ _j+1 = (1 − hd 2,j )v _j ⁿ − h ² X ∞

i

ξ _i,j ⁿ + h ³ X ∞

i

X ∞

k

{ρ + d 1,i (1 − d 2,j )}p ⁿ _i,j,k

p ⁿ⁺¹ i+1,j+1,k+1 = h

1 − h{(1 − d 1,i )d 2,j + d 1,i (1 − d 2,j ) + d 1,i d 2,j + σ ⁿ _i,j,k } i

p ⁿ _i,j,k

β 1 (t, x, y, z) ∼ β 1,i , β 2 (t, x, y, z) ∼ β 2,i

(DC)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



u ⁿ⁺¹ ₀ = h ³ X ∞

i

X ∞

j

X ∞

k

β 1,i p ⁿ _i,j,k

v ⁿ⁺¹ ₀ = h ³ X ∞

i

X ∞

j

X ∞

k

β 2,i p ⁿ _i,j,k

p ⁿ⁺¹ _i+1,j+1,0 = ξ _i,j ⁿ p ⁿ _0,j,k = p ⁿ _i,0,k = 0

References

[1] P. Georgescu, T. Matsumoto, S. Oharu and T. Takahashi, Approximation and convergence theo- rems for nonlinear semigroups associated with semilinear evolution equations, GAKUTO International Journal Advances in Mathematical Sciences and Applications, Vol. 11 (2001)

[2] P. Georgescu and S. Oharu, Generation and characterization of locally Lipschitzian semigroups asso- ciated with semilinear evolution equations, Hirosima Math. J. 31 (2001), 133-169.

[3] T. Matsumoto, S. Oharu and H. R. Thieme, Nonlinear perturbations of a class of integrated semi- groups, Hiroshima Math. J. 26 (1996), 433-473.

[4] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations”, Springer, 1983.

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