A nonlinear evolution problem arising from particle transport with nonhomogeneous boundary conditions

Nonlinear Analysis, Theory. Methods & Applicafiom,

l/Z, pp. 33-44, 1998 Ltd All rights reserved. Printed in Great Britain 0362346X/98 $19.00+0.00 Vol. 31. No. 0

Pergamon

1998 Elsevier Science

PII: !30362-546X(%)00049-1

A NONLINEAR EVOLUTION PROBLEM ARISING FROM PARTICLE TRANSPORT WITH NONHOMOGENEOUS BOUNDARY CONDITIONS A. BELLENI-MORANTE Dipartimento di Ingegneria Civile, Via S. Marta 3, I-50139 Firenze, Italy (Received 28 August 1995; received for publication 28 March 19%) Key words and phrases: Particle transport, problems.

nonhomogeneous

boundary conditions, evolution

1. INTRODUCTION

Particle transport problems with nonhomogeneous boundary conditions are of great interest for applications. For instance, the entering fhrx of some contaminant particles may be assigned on a part of the boundary of a cavity containing a gas at low pressure [l]. Then, the transport of the contaminant within the cavity is studied as long as it reaches another part of the boundary. Moreover, in astrophysics, one may be interested in evaluating the radiation flux emerging from an interstellar cloud, which is subject to an entering radiation coming from a group of stars [2]. In [3], nonhomogeneous boundary conditions were considered and the solution of an integral version of the corresponding evolution problem was found in terms of a Neumann series. The proof that such a series is also a solution of the original integrodifferential problem was not given. Case normal mode expansion was used in [4] for problems with plane symmetry. In [5], the theory of linear semigroups was employed to prove in a rigorous way that a suitable function, suggested by physical considerations, satisfies a three-dimensional nonhomogeneous problem. Finally, existence and uniqueness of the solution of a transport problem with nonhomogeneous boundary conditions and with plane symmetry was rigorously proved in [6] by using the theory of linear semigroups and by introducing a suitable fictitious source term. However, the evaluation of such a source may often be difficult. In this paper, we use some results of nonlinear semigroup theory to find the explicit (strict) solution of a particle transport problem in a three-dimensional region in terms of the given nonhomogeneous boundary condition. Such a result is obtained without introducing any fictitious source and without resorting to physical considerations to write the solution “a priori”. Even if some physical interpretation in this paper are similar to those implied by the procedures used in [5] and [6], nonlinear semigroup techniques greatly simplify mathematical manipulations and can be used for more complicated boundary conditions. 2. THE PHYSICAL

PROBLEM

AND THE ABSTRACT

FORMULATION

Let Y c R3 be a convex region bounded by the regular surface W and assume that x E V, y E Wand that n(y) is the outward directed unit vector, normal to W at y. 33

A. BELLENI-MORANTE

34

If w(x, u; t) is the density of the particles, which, at time t 2 0, are at x and have velocity v = vu, u > 0, then w satisfies the transport equation [7] w(x, u; t) du’,

; w(x, u; t) = -uu * VW - uow + u(a,Mn) x E

v, u

E

s, t >

ss

(1)

0,

where S = (u : 1~1 = 1) is the spherical surface of radius one, cos e), du’ = sin 8dBd9. Moreover, Q and cr, are positive interactions between particles and the host medium within section and cr = a, + o, is the total cross section (oC is Equation (1) is supplemented by the initial condition

u’ = (sin 8 cos 9, sin 0 sin 9, constants, which represent I/: o, is the scattering cross the capture cross section).

XEKllES,

(2)

w(x, u; 0 = w&G u), and by the nonhomogeneous

boundary condition y E aK u -n(y) < 0

W(Y,u; t) = %(Y, u; 0,

(3)

where w, and wb are given nonnegative functions. Note that ulu - n(y)\ do, wb(Y, u; t) is the particle flux entering I/ through the surface element da,, centered at y. To proceed further, we write the particle density w as follows w(x, u;v) = Wi(X,u; t) + wz(x, u; t). In (4), w1 is the density of the collision with the host medium particles which, at time t, have It follows from the preceding a ZW’ = --vu %(X,

(4)

particles which, at time t, have not yet undergone any (first-flight particles), whereas w2 is the density of the undergone at least one collision. definition that wi and w2 satisfy the systems * VW,

WO)

w,(ys u; t, =

=

-

UOWI,

xEV,UES,t>o,

x E v, u E s,

WI&U),

wb(y,u;

(54

t),

; w2 = -uu * VW, - uaw, + u(aJ47r)

y E dK

ss

u

- n(y)<

VW

0

(5c)

[w2(x, u’; t) + wI(x, u’; t)] du’, (6a)

xEI/,UES,t>o, w,cG

u; 0) =

w,&,

Wz(Y,u; 0 = 0,

u),

x E v, u E s,

y e c?K u -n(y) < 0.

(6b)

(6~)

Note that the nonhomogeneous boundary condition affects directly only the density of first-flight particles, see (k), whereas the scattering phenomenon transforms w,-particles into w2-particles, see (6a). Moreover, if we add (5a) and (aa), (5b) and (6b), (5~) and (6c), then we obtain system (1) + (2) + (3) provided that wrO + w20 = w, . Remark I. (6) is a standard particle transport system with non re-entry boundary conditions and with a “source” term u(a,Ma) js w, du’.

Nonhomogeneous

boundary conditions

35

In order to write the abstract version of the coupled systems (5) and (6), we introduce the Banach space X = L’(Vx S) with the usual norm [8] lk-ll=

ss

du

s

v If@, u)l dx.

We also define the following linear operators: Kf = (l/4@

f(x, u’) du’,

D(K)

= X,

R(K)

= X,

(7)

Tf = -vu.Vf, D(T)

= (f : f E X, u . Vf E X, f satisfies the boundary

R(T)

c X.

condition (SC)),

(8)

condition (5c)),

(9)

Finally, we consider the nonlinear operator Af = -vu. D(A)

Vf,

= (f: f E X, u - Vf E X, f satisfies the boundary

R(A) c X.

Note that A is a nonlinear operator because its domain D(A) is not a linear subset of X. In Sections 3 and 4, we shall first consider a time independent wb; for the time dependent case, see Section 5. By using definitions (7), (8) and (9), systems (5) and (6) can be put into the following abstract form: d dt wi(t) = Aw, - vowI,

t > 0;

w,(O) = w10,

d dt wz(f) = Tw, - vcrw, + va,Kw2 + va,Kw,,

t > 0;

w*(O) =

(10)

w20.

(11)

In (10) and (II), w,(t) = wl(., .; t) and w2(t) = w2(., .; t) are now to be considered as functions from [0, +oo) into the space X, whereas dw,/dt and dw,/dt are derivatives in the strong sense.

3. THE OPERATORS

K,

T

AND

A

The following lemmas will be used to show that (T - vcd) is the generator of the linear semigroup [Z(t), t 2r 0) and that (A - vd) is the generator of the nonlinear semigroup (Y(t), t 2 01.

LEMMA 1. (a) The linear operator K is bounded, and l[Kll = 1; (b) given any z > 0, the linear operator T, = I - z(T - vcd) has a bounded inverse, and 11T,-’ )) 5 (1 + vaz)-‘.

A. BELLENI-MORANTE

36

Proof. (a) It follows directly from definition (7) that llKf]I s ]lfl] vf EX. On the other hand, \l~fll = llfll vf EX+ = (f:fe X,f(x, u) L 0 at a.e. (x, u) E VX S]. (b) Given z > - l/va and g E X, we consider the equation

z(T- ud)]f

[I -

= g

where the unknown f must be obviously sought in D(T). Due to definition (8), we have (1 + mz)f(x

- su, u) - z$f(x

- SW u) = g(x - SW u)

because u. Vf(x- su,u)= -(d/d.s)f(x,su,u). Hence, we obtain - su, u) exp(-bs)]

-$[f(x

= $ g(x - su, u) exp(-bs)

where b = (1 + uaz)/uz and b > 0 provided that z > 0. It follows that we have f (x, u) = -

1

W.0) exp(-bs)g(x

- su, u) d.s

u.2 s 0

(12)

where so(x, u) is such that y = x - so(x, u)u E aK and where we took into account that f must belong to D(T) and so f (x - so(x, u)u, u) = f (y, u) = 0 because u - n(y) c 0. Relation (12) shows that

s

so(x.N

(T,%(x, u) = ;

exp(-bs)g(x

- su, u) ds,

z > 0, g E x.

(13)

0

Moreover (13) gives

ew(-WI& - su,4 h, In order to evaluate the integral over V, we define (for each assigned u E S) the subset W_(u) = ly : y E t3Ku * n(y) < 0) c W Then, we put for each given y E W(u): x = y +

Correspondingly,

0 I r I s,(y, - u);

dx = lu*n(y)ldcr__dr.

Sb(“-@dr [:exp[-b(r

- s’)]]g(y + s’u, u)l d.s’,

ru,

we obtain

x

Nonhomogeneous boundary conditions

31

because x - su = y + (r - s)u = y + s’u with s’ = r - s. It follows that we have

s SO(Y.

X

--o)

exp[-b(r

- s’)] dr

s’

a v-

=&

da,lu - n(y)1

s&, --II) MY + s’u, @I ds’

s0

lkll

??

and (b) is proved.

LEMMA2. Given any z > 0, let the nonlinear operator A, be defined by the relation A, = I - z(A - ~01). Then, we have: (a) D(A,) = D(zA) = D(A); (b) D(A,) is dense in X; (c) the inverse operator A;’ exists with domain D(A;‘) = X and it satisfies the relation A;‘g

= T,-‘g + wb(y, u) exp[-bs,(x,

u)]

VgEX,

(14)

where y = x - so(x, u)u, b = (1 + vaz)/vz; (d) given any g, g, E X, the following inequality holds

IIA,‘g-

A-‘g,lI

5 &

llg -

(15)

gill;

(e) A;’ satisfies the relation n-l A;“g geX,n=

=

T,-“g

+

C j=O

T,-jw,(y,

u)

exp[-bso(x,

u)],

(16)

1,2 ,....

Proof. (a) If g E D(A), then f = z(Ag) is defined, i.e., g E D(zA); hence D(A) C D(z4). Conversely, if g E D&A), then (zA)g = z(Ag) is defined and so g must belong to D(A); hence, &A) C D(A). We conclude that D(A) = D(d). Finally, @A,) = D(I) f-l &?A) = x n 0(2/l) = D(A). (b) Given g E X, let g(x, u) = g(x, u) - +,(x - so(x, u)u, u). Since D(T) is dense in X because it contains Ct(Vx S), a sequence (g,,,, m = 1,2, . . .) C D(T) can be found, such that g,,, -+ g E X as m + +oo. Then, (g,,, = S, + w,, m = 1,2, . . .) C D(A) and gnl + g + w, = g as m --) +oo. In an analogous way, it can be proved that the graph of A is closed in X x X because the graph of T has such a property.

A.BELLENI-MORANTE

38

(c)Consider the equation

[Z - z&4 - uaZ)[f = g where g is given and the unknown f must be sought in D(A). As in the proof of Lemma 1, we have = t

- su, u) exp(-bs)]

-$]f(x

g(x - su, u) exp(-bs)

and so we obtain

so@, ?? exp(-bs)g(x o .I‘

WI + $

f(x, n) = wb(y, n) exp]-&(x,

- su, u) d.s

because f must belong to D(A) and so f(x - se(x, u)u, u) = f(y, u) = wb(y, u). Relation (14) follows from (13). (d) (14) gives /l;‘g

- A;‘g,

= T,-‘(g - gr)

and (15) follows from (b) of Lemma 1. (e) We have from (14) A,Zg = A,‘[T,-’ g +

wb

exP(-hd

=

T,- ’[T,- ‘g + wbeXp(

=

q-‘g

+

and (16) follows by induction.

(cm1

+

- ho)]

wb eXp( - bS,)

+ z)[xbeXp(-bs,)]

??

By using Lemmas 1 and 2 and some results of [9], see also [lo], following theorem.

we can state the

THEOREM 1. (a) (T - vaZ) generates the linear semigroup (Z(t), t 1 01, such that

Z - i (T - ml)

Z(t)g = lim

tI++CO [

-ng, I

lIz(t)

(b) (A - oal) generates the nonlinear [Z - (Un)(A - uaz)]-“g, lim,,,,

ev(-Wllgll

i

semigroup

IIYWg - wg, II 5 exp(-Wllg

- glII

vgEX,tzo;

{Y(t), t 2 01, such that

vg, g, E x,

Y(t)g =

t 2 0;

(c) Y(t) satisfies the relation

Y(t)g = z@)[g - %exp(-ose)] with so = s&x,

U)

and wb = wb(y, U),

y = X -

+ Wbexp(-o&,),

So(X, U)U.

(17)

Nonhomogeneous

boundary conditions

39

Proof.

(a) follows from (b) of Lemma 1 [8]. (b) follows from Lemma 2 [9, lo]. (c) Since s,,(x - su, u) = sO(x, u) - s we have from (13) T,-‘~WY,

u) exp[-~s,(x9

=- I

so@. 9 exp(-bs)wb(x

vz s 0 =-

=-

Nil

so@,? ? exp(4.r 02 [i 0 1

1

- su - sO(x - su, u) exp[-as,(x

1 1

+ as) & wb(y, u) exp[-os,(x,

sok 0)

=

vz [S 0

exp(-s/v@

(1 - expl-s,(x,

ds w&, u) exp[-as,(x,

u)/vzl]w,(y,

u)exp[-as,(x,

- su, u)] d.s

u)]

u)]

u)].

Thus, we obtain

VAs a consequence,

T,-‘)lwb(Y, u) expl-os,(x,

u)]] = wb(y, u) exp[-bs,(x,

u)].

(16) becomes n-1

Ai% = T,-“g + C T,-‘(I - T,-‘)[w,exp(-as,)], j=O

4%

=

T,-"[g

-

+,eXp(--OSo)]

w, eXp(-as,).

??

The preceding relation leads to (17) as n + +oo. 4. THE ABSTRACT

+

PROBLEM

(lO)+(ll)

Consider first system (lo), which is independent of the solution w2 of (11). If w1 E D(A) and ti, E D(A) satisfy (lo), then q(t) = wl(t) - a,(t) belongs to D(T) and it is such that (A - vaZ)w, - (A - vaZ)iit, = (T - vaZ)q.

It follows from (10) g q(t) = (T - vaZ)q(t),

1 > 0;

q(0) = qo = 0.

However, (t - vcrl) is the generator of the linear semigroup Z(t) and so the unique strict solution of the preceding system has the form q(t) = Z(t)q, = 0. We conclude that (10) has at most one strict solution. Moreover, it follows form Theorem 1 that such a solution has the form Wl(0 =

W[w,o

-

wb eXp(-oso)]

+

wb eXp(-oso),

t 2 0.

Note that wb exp(-as,) is a stationary solution of (10) and that WI(t) + wb eXp(-us,) t -+ +co because of (a) of Theorem 1.

(18)

as

A. BELLENI-MORANTE

40

Consider then system (1 l), where vcr,Kw, is now a known source term because of (18). The linear operator [(T - vaZ) + ~$1 generates the linear semigroup (Z,(t), t 1 01, which satisfies the equation [8] f W)s

= Wg

+

Z(t - t’)vo,KZ#‘)g

dt’,

VgEX

(19)

s0

and it is such that IIZz(t)II s exp[-v(a

t 2 0.

- a,)t] = exp(-ua,t),

t

It follows that the unique strict solution of (11) has the form

w2w = z2vw20

+

Z2(t - t’)[uo,Kw,(t’)]

(20)

dt’.

0

Substitution of (18) into (20) gives wz(t) = Z#)w,,

+ ua,

lz,(t s0

- w, exp(-as,)]

- t’)KZ(t’)[w,,

dt’

f +

Z,(t - t’)Kw, exp(-as,)

ur7,

dt’.

(21)

s 0

Since (T - uol) = [(T - uaZ + NJ&) - ~$1, generated by (T - ual):

we have for the linear semigroup Z(t),

t

Z(t)g = Z,(t)g - ucr,

Z2(t - t’)KZ(t’)g dt’,

(22)

0

and (18) becomes

W)

= Z2(Nw,o - wb exp(-as,)]

+ w, exp(-oso)

t - vu,

z,(t

-

t’)KZ(t’)[W,,

-

W,

dt’.

eXp(-as,)]

s0 The total particle density w(t) = WI(t) + w2(t) is obtained as follows

+st

w(t) = z2(t)[wo

- wb exp(-use)]

ucr,

z,(t

- t’)KWb

(23)

by adding (21) and (23)

+ wb exp(-aso)

eXp(-Us,)

dt',

0

t 2

0,

(24)

where w. = wlo + w2, and where w. E D(A) provided that wio E D(A) and w20 E D(T). Relation (24) gives w = w(t) explicitly as a function of the boundary datum wb = w,(y, u)* Remark 2. The preceding results hold under the assumption that wb(x - $,(x, u)u, u) exp(-o&,(x, u)) E D(A).

41

Nonhomogeneous boundary conditions 5. TIME DEPENDENT

BOUNDARY

CONDITION

If W, = +,(Y, U; t), Y E dV, u E S, t 1 -6/u with 6 = sup(s,(x, u), x E V, u E S], then A = A(t) because the domain D(A, t) is time dependent even if the formal expression of A does not depend on t, see (9) and (5~). However, the nonlinear operator A retains most

of the properties which hold in the time independent case, because T is in any case time independent. In particular, system (10) has still at most one strict solution and some results of [l 11can be used. Assume now that t,, c t, < . . - < t,, with to = 0 and t, = t is a partition of [0, t] and that we approximate the boundary conditions during the time interval [tj , tj+,) by its value at During the time interval [t,, , t,) = [0, tl) we have from (18) t

=

tj.

W)

MY,

= z(O]wiO -

u; -~&,

WV)

exp(-WI

+ wb(y, u; tl - so(x, u/u) exp(-as,)

(25)

where we took into account that, at x E V, we find at time t the particles which were in x - .q,(x, u)u at the instant t - so(x, u)/u. During the interval [ti, f2) we have ~~(0 = z(t - ti)]Wi)

WAY,

-

u; t,

-

~~/u)exp(-a~~)l

+ wb(y, u; tz - so/u) exp(-as,). By using (25) at t = ti, we then obtain

+ w,(y,u; - z(t +

wb(Y,

W(t) = W]w,, +

wb(Y,

-

tl

so/u)

tl)[Wb(Y, u;

u;

u;

-

t2

(Y,

-

u;

u; --so/v)

exp(-as,)1

exp(-as,)] tl

-

so/u)

exp(-@_dl

exp(-as,),

so/u)

wb t2

MY,

-

w,(t) = z(t - WW,)[%,

-%/u)

exp(-c%)l

exp(-as,),

so/u)

t, I t I t2.

(26)

Similar expressions for wl(t) can be obtained during the time intervals [t2, t3), [t3, f4), . . . . The preceding approximate procedure suggests that w,(t) might have the form W)

= zW]%, +

wb(y$

u;

wb(Y, t

-

u; so/u)

-%/v) exp(-usoh

whose physical meaning is clear. Note that w,(O) = wiOand that W,(t) = z(t)[wi, the system -$ iit, = (T - uaZ)iir,,

t > 0;

w(-%)I t 1

(27)

0,

- wb(y, u; -&/u) exp(-a&,)] satisfies

ii+(O) = WI0 - wb(y, u; -q/u)

exp(-aso)

(28)

where W,(O) E D(T) provided that wu, E D(A, t = 0). Moreover, if $(t) = a,(*, *; t) with iG’,(x,u; t) = wb(x - q,(x, u)u, u; t - so(x, u)/u) exp[-c&x, u)], then rV, + G1 belongs to D(A, t) because WI(t) E D(T) and G1 E WA, t).

A. BELLENI-MORANTE

42

On the other hand, we have su, u) =

u - Vs,(x -

--$so(x - su, u)

= _

lim

so@ - (s + w,

u) - sob - SK u) h

h-0 = _

lim

so(x - su, u) - h h

h-0

so@ - su, u) =

1,

and so we obtain u - V(w,(y, u; t - so/u) exp[-os,(x, i a = --u wb exp(-cso)at

u)])

exp(-oso),

owb

where we took into account that u - V leaves y unchanged. We conclude that i?,(t) satisfies equation (11) provided that the strong derivative dGr/dt coincides with the partial derivative a[wb exp(-os,)]/at. Thus, we have Q, = (A - UOZ)i?l,

;

t > 0.

As a consequence, WI(t) = n,(t) + i&(t) is the unique strict solution of system (10). In fact, w,(O) = a,(O) = wIo, q(t) E D(A, t), and (28) and (29) give ;t ii, = (T - uaZ)ii), = T(w, - a,) - ua(w, - a,) = Aw,

- Flit1 +

ua8,-

UdW,

and so we have d jj

Wl

=

Awl

-

UOWI

-

d,

-g

Wl,

tt

Wl

=

(A

-

uaZ)w,.

Remark 3. The preceding results hold under the assumptions: (a) i&(t) E D(A, t), t r 0; (b) the strong derivative dGr/dt coincides with the partial derivative with respect to t of [wb(y, u; t - so/u) exp(-us,)] = 6l(x7 u; t). Note that (b) is satisfied if, for each given t, a suitable g = g(t) E X exists, such that wb(x

0 = !‘T

g(t) -

-

so@, U)U,U; t

+ h - so(x, u)/u) - w& - s,,u, u; t - so/u) exp( - as,) h

.

Nonhomogeneous

boundary conditions

43

Finally, by a procedure similar to that of Section 4, it follows that the total particle density w = wi + w2 is given by the relation w(t) = Z,(t)[w,

- wb(y, u; --so/u) exp(-as,)]

r’ + va,

&(t - t’)K[wb(y,

+ wb(y, u; t - so/u) exp( - as0)

u; t’ - so/v) exp(-as,)]

dt’.

(30)

0

6. CONCLUDING

REMARKS

We may summarize the results of the preceding sections as follows. Assume that so = so@, u) (i.e., the polar equation of I~Vwith pole at x E V) and the boundary condition term wb are such that conditions (a) and (b) of Remark 3 are satisfied. Then, the unique strict solution (wi , w2) of the abstract evolution problem (10) + (11) is such that the total particle density w = wi + w2 is given by (30). Note that the first term on the right-hand side of (30) takes into account the effect of the initial particle distribution w,. If w, coincides with the first-flight population in Y at t = 0, then such a term vanishes. The second term represents the contribution to the total density at x E Vand at time t of first-flight particles, which were at y = x - so@, u)u E 8V at time t - so/u and which did not undergo any scattering collision during their trip from y to x. Finally, the third term is due to the particles which undergo one or more collisions during their trip from 8V to x. In fact, a first approximation for the semigroup Z,(t) (whose generator T - vaI + uo,K contains the scattering operator ua,K) can be obtained from (19) as follows t Z,(t)g

= Z(t)g + ua,

Z(t - t’)KZ(t’)gdt’. s0

(31)

Substitution of (31) into the integral term of (30) gives the contributions of second-flight and third-flight particles. We also remark that (30) can be used to find the explicit expression of the emerging flux of particles at a given y’ E al! This is of particular interest in astrophysics, as noted in Section 1. Acknowledgement-This work was partially supported by the Italian “Minister0 dell’Universit8 e della Ricerca Scientifica e Tecnologica” 40% Project “Problemi nonlineari sull’analisi e sulle applicazioni fisiche, chimiche e biologiche”, and 60% Research Funds, as well as by GNFM of Italian CNR. REFERENCES 1. Belleni-Morante, A. and Busoni, G., Outgassing and contamination: strict solution of a Bolzmann like model, Math. Meth. in the Appl. Sci., 1993, 16, 799-817. 2. Belleni-Morante, A. and Moro, A., Time dependent photon transport in an interstellar cloud with stochastic clumps. Transport Theory and Statist. Phys., 1996, 25, 85-101. 3. Case, K. M. and Zweifel, P. F., Existence and uniqueness theorems for the neutron transport equation. J. Math. Phys., 1963, 4, 1376-1385. 4. Newman, P. A. and Bowden, R. L., Solution of the initial value neutron transport problem for a slab with infinite reflectors, J. Math. Phys., 1970, 11,2445-2458. 5. Mika, J. and Stankiewics, R., Nonhomogeneous boundary conditions in neutron transport. Transport Theory and Statist. Phys., 1972, 2, 55-67. 6. Busoni, G., Nonhomogeneous boundary conditions in evolution problems. Riv. Mat. Univ. Parma, 1979, 5(4), 567-576.

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7. Duderstadt, J. J. and Martin, W. R., Transport Theory. Wiley, New York, 1979. 8. Belleni-Morante, A., A Concise Guide to Semigroups and Evolution Equations. World Scientific, Singapore, 1994. 9. Crandall, M. G. and Liggett, T. M., Generation of semigroups of nonlinear transformations on general Banach spaces. Amer. J. Math., 1971, 93, 265-298. 10. Miyadera, I., Nonlinear Semigroups. American Mathematical Society, Providence, RI, 1992. 11. Crandall, M. G. and Pazy, A., Nonlinear evolution equations in Banach spaces, Israel J. Math., 1972, 11, 57-74.