Applications Example 5.1: The Lasota equation

Corollary 5.1: If A is the generator of a global (locally) equicontinuous semigroup

5.3. Applications Example 5.1: The Lasota equation

describes the process of reproduction and differentiation of a population of red blood cells. Lasota equation can be solved by ergodic method (please see R. Rudnikicki [23]). However, we like to apply our Theorem 5.2.2 to solve this problem. For this purpose, let

x x

A ∂

− ∂

= and ^f

( )

^t^,^u ⁼^λ^u, then the Lasota equation is a special case of (5.1.2). Let be the set of all rapidly decreasing test functions whose topology is determined by the seminorms

( )

satisfies uniformly Lipschitz condition corresponding to with Lipschitz constant

pmn

λ (independent of all nonnegative integers m, n), the Lasota equation has a unique solution

Moreover, we may consider more general initial value problem

( ) ( ) ( ) ( ) ( ( ) )

where ϕ is a given function on R possessing bounded derivatives of all orders, and Babalola [1] showed that

u∈X

This shows that f is a locally Lipschitz continuous function with Lipschitz constant

( )

^, ²

Lmn t c = c. Notice that the Lipschitz constant is independent of t, m and n.

According to Theorem 5.2, (5.3.1) has a unique local solution.

Example 5.2: Consider the initial value problem

( ) ( ) ( ) ( ( ) )

2 2

x xx

A u=xu +x u for every ^u^{∈ =}^X ^{S R}

( )

. To find the solution of the initial value problem (5.3.2), we need only to show that is a generator of some

( )

semigroup. Babalola [1] showed that the resolvent operator

A2 C₀,1

(

^: ^mn

)

R λ A satisfies

( )

(

)

^Re ¹

mn mn

p R λ A ≤ λ ⁻ for every complex number λ , where A_mn= 1

A nI− +mI+2I . Hence the resolvent operators ^R

(

^λ^{: A}^mn

)

^and ^R

(

⁻^λ^:^A^mn

)

exist for all positive real number λ and they satisfies that

( )

(

)

mn mn

p R λ A ≤λ⁻ , ^p^mn

(

⁻^λ^:^A^mn

) )

^≤^λ⁻¹ for all positive real number λ. Since ^R(^λ^:^A^mn² )⁼^R

(

^λ^:^A^mn

) (

^R ⁻ ^λ^:^A^mn

)

for all positive real number λ, this implies that

( )

(

^: ²

) ( )

¹ ²

mn mn

p R λ A ≤ λ⁻ ≤λ⁻¹ for all positive real number λ.

This shows the condition (3) of Lemma 5.2 is satisfied. Conditions (1) and (2) of Lemma 5.2 are easy to check, and hence A² =A² be a generator of a

( )

semigroup. By Theorem 5.2, (5.3.2) has a unique local solution.

0, C

Example 5.3: Let H be the space of all real-valued C^∞-functions on whose partial derivatives of all orders belongs to ^L²( )^R^m . The space H is a pre-Hilbert space with inner product

(

v w

)

D v

( )

x D^αw

( )

x dx

n ⁼

∑ ∫

^α^≤n Rm ^α

, for all v,w∈H. (5.3.3)

Hence, for each n=0,1,2, , a norm ⋅ _n is defined by

(

)

_n²¹

n v v

v = , v∈H

(

α α α_m

)

, a seminorm p_α is defined on H by α = ₁, ₂, ,

For each multi-index

( )

⁰

(

R^m(

( )

)²

)

¹²

p_α v = D v^α =

∫

D v x^α ^, ^v^∈^H ^(5.3.4) The totality Γ of these seminorms p_α corresponding to all multi-indices α induces a Fréchet topology for H.

We consider the semilinear initial value problem (for the case m=1):

( , ) ( , ) ( , ), 0, (0, ) ( ),

u t x uxx t x t x x t t

u x u x R

⎧ ∂ = + >

⎪ ∂⎨

⎪⎩

, . R x

∈

= ∈

(5.3.5) where is a -function. One can convert it to the abstract semilinear initial value problem

u0 C^∞

( ) ( ) ( , u( )) (0)

d u t Au t f t t dt

u u

⎧ = +

⎪⎨

⎪ =

⎩

(5.3.6)

where ₂

A= d , u₀∈H and ^{f t u}

( )

^, ⁼^u². In [2, example 6.1] Choe shows that

A generates an equi-continuous -semigroup. As we showed in Example 1, one can easily check that

(

)

f t u is Lipschitz continuous. From Corollary 5.2, (5.3.5) has a unique solution.

In (5.3.6) A can be spread to an elliptic partial differential operator of the order in

I. Miyadera [20] showed that L generates a quasi-equicontinuous -semigroup on H.

According to Corollary 5.2, the initial value problem

( ( ) )

Example 5.4: We consider the initial value problem

( ) ( )

be the set of functions that are continuously differentiable in . To solve the boundary-initial value problem (5.3.8) we are looking for a pair of functions ,

in which satisfy the boundary and initial conditions. Denote

(

Then X is a complete topological locally convex space. Let a vector value function

It is well known that the Cauchy problem .

ies th ondition. For this purpose, we apply the identity

( )

( , ) ( , ) ( , )

( , ) ( , )

max sup ( , ) , sup , sup

t x Q t x Q t x Q

v t x v t x

v v t x

t x

α α α

α ∈ ∈ ∈

⎧ ⎫

= ⎛ ∂ ⎞ ⎛ ∂ ⎞

⎨ ⎜⎝ ∂ ⎟⎠ ⎜⎝ ∂ ⎟⎠⎬

⎩ ⎭

for every v in ^{C Q}¹

( )

α , then

( ) ( )

⁽ ⁾

( )

1 2

1 1

1 2

0 !

n n k k

v v k

v v

e e v v

α α α α

α α

− − −

∞ =

− ≤ −

∑ ∑

( )

¹0

( )

1 ⁽ ¹⁾

( )

1 2

1 !

n n k k

v v

α α

α α α α

− − −

∞ =

≤ −

∑

(

1 2

) (

1 2

)

1 !

n c n

v v c e v v

α α α n α α α

∞

≤ −

∑

≤ − ^.

This shows that ^{F t U}^α

( )

^, satisfies local Lipschitz co^ndition on ^{C Q}¹

^{( )}

^α ^×^{C Q}¹

^{( )}

^α for every α∈N . It is easy to check that F t U satisfies

( )

condition on d hence, by Theorem 5.2, (5.3.8) ique local solution in Q.

local Lipschitz

X, an has a un

References

[1] V. A. Babalola, Semigroups of operators on locally convex spaces, Trans.

Amer.Math. Soc.199(1974), 163-179.

[2] Y. H. Choe, -Semigroups on a locally convex space J. Math. Anal. and Appal., 106(1985), 293-320.

[3] R. deLaubenfel, "Existence Families, Functional Calculi and Evolution Equations", Lecture Notes in Mathematics (1570) Spring-Verlag, Berlin, 1994.

[4] R. deLaubenfel, Existence and Uniqueness families for abstract Cauchy problem, J.

London Math. Soc. (2) 44 (1991) 310-338.

[5] K. J. Engel. and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts Mathematics, vol. 194, Spring-Verlag, New York, 2000.

[6] L.C. Eran, (1998). Partial Differential Equation, Providence: Amer.Math. soc.

[7] Gerald B. Folland, “Real analysis: modern techniques and their application” John Wiley and Sons, New York, 1999 Second Edition.

[8] G. R.Goldstein, J. A. Goldstein, and E. Obrecht, (1996). Structuture of solutions to linear evolution equation: extensions of d'Alembert formula, J. Math. Anal. App.201, no.2, p461-477.

[9] J. A. Goldstein and J. T. Sandefur, (1987). An abstract d'Alembert formula, SIAM J. Math.Anal.18, p842-856.

[10] J. A. Goldstein (1986). Asymptotics for boundedsemigroups on Hilbert space, in '' Aspectcts of Postivity in functional Analysis,'' p.49-62 (Nagel, R., schlotterbeck, U and Wolff, M.P.H., Eds), North-Holland :Elsevier, Dordrecht.

[11] J. A. Goldstein and G. Shi, (1995) in ''evolution Equations and applications '' (A.McBride and G. Roach,Eds), p.3-17, New York :Longman, Harlow.

[12] J. A. Goldstein, R. deLaubenfels, and J. T. Sandefur, (1993). Regularizer semigroups, iterated Cauchy problems, and equipartition of energy, Monats.Math 115, p47-66.

[13] E. Hille, and R. H. Phillips, (1957). Functional Analysis and semigroups, Providence :Amer.Math.soc.

[14] G. Köthe, “Topological vector spaces I”, Spring-Verlag, New York/heidelberg/Berlin, 1979.

[15] G. Köthe, “Topological vector spaces II”, Springer-Verlag, New York / Heidelberg / Berlin, 1979.

[16] S. Kantorovitz, The Hille-Yosida space of an arbitrary operator, J. Math. Anal.

and Appl., 136(1988), 107-111.

[17] T. Komura, Semigroups of operators in locally convex spaces, J. Fun. An. 2 (1968), 258-296.

[18] Meri Lisi and Silvia Totaro, Photon transport with a localized source in locally convex spaces, Math. Meth. Appl. Sci. 2006; 29:1019-1033.

[19] Meri Lisi and Silvia Totaro, ''Inverse Problem Related to Photon Transport in an Interstellar Cloud'', Transport Theory and Statistical Physics. Vol. 32, Nos. 3& 4, pp.

327-345, 2003

[20] I. Miyadera, Semigroup of operator in Fréchet space and application to partial differential equations, Tôhoku Math. J (2) 11 (1959), 162-183.

[21] A. Pazy, "Semigroups of Linear operators and Applications to Partial Differential Equations", Spring-Verlag, 1983.

[22] M. M. H. Pang, Resolvent estimates for Schrodinger in and the theory of exponentially bounded -semigroups, Semigroup Forum 41 (1990), 97-114.

[23] R. Rudnicki, Chaos for some infinite-dimensional dynamical systems.

Mathematical Methods in the Applied Sciences 2004; 27:723-738., 1968 Second Edition.

[24] S.Y. Shaw, C.C. Kuo , and Y.C. Li, Perturbation of local C-semigroups, Nonlinear Analysis 63 e2569-e2574 (2005)

[25] F. Treves, '' Topological Vector Spaces, Distributions and Kernals''\

AcademicPress: London,1967.

[26] Eduardo V. Teixeira, Strong solutions for differential equations in abstract spaces, J. Differential Equations 214. (2005) p65-91.

[27] K. Yosida, ''Functional analysis,'' Academic Press, New York, 1968 Second Edition.

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