APPENDIX A Preliminary Lemmas for Stability

In this appendix, we state and prove five preliminary lemmas used in chap-ter 7 which describe the behavior of the time-dependent solution, u((, t) near the boundaries (

=

⁰⁰ and (

= -ct

and a sixth lemma dealing with the long time behavior of u( (, t) for all (.

Proof of Lemma 7.1: \Ve will begin by showing that if

(1.1 ) then

lim W((, t) = ~Fr

(-00 (1.2)

uniformly in 0 :::;

t :::;

T. Then argue that the above statement and the statement of the lemma are equivalent.

Let €

>

0.1'

>

O. Note IVo( is bounded for ( ;::: -el,O :::; t :::; T by assumption.

Therefore~ there exists a (0

>

0 such that

(1.3) for (

>

(0 by assumption (7.21). Let

V((,t)

=

^H/r

+

~

+

Ne,3tHo-(. ( 1.4) Define

Then

LV = (a((

+

^{ct) -} a(((

+

^{ct) -} b1((

+

^{ct) _ _} (3)1\; /3tHo-(

M(U) c e . ^(1.6)

Since, a. a(, bI , and A1(U) are finite and differentiable, we may choose

f3

large enough that

LV::;

o.

^(1.7)

Then

L(V - W) ::; 0, (1.8)

since LH!

=

0 by (7.17). Evaluation shows

V((,O) - 11'((,0)

=

IVr

+ ~ +

^{Ne(o-( - 11'o(()}

~

0, (1.9) and

V(-ct t) -_, IV(-ct _, t) = IV _T

+

~ 2

+

Ne/3t+(o+ct - 11'(-ct ' t)

>

- 0 , (1.10) for a large enough choice of N by (1.9) and the fact that. 11'(

-ct,

t) is bounded by MI as in (7.24). By the maximum principle

V((, t) - 11/((, t) ~ 0 (1.11)

or

(1.12) This implies that

(1.13) for ( ~ X where X is smallest value of ( so that

(1.14) Similarly, if we define

(1.15) we can show

(1.16)

for ( ;::: (0 where (0 is the smallest value of ( where Nt e{Jlt+(o-(

<

~. Thus we have

and hence, undoing the change of variables,

lim U((, t) - Ur ((

+

ct) =

o.

(-00 e((+ct) ^(1.19)

Since e( (

+

ct)

>

0 and is one-periodic in (

+

ct, we have lim U((, t) - U_{r ((}

+

ct)

=

0,

(-00 ( 1.20)

or going back to our original variables

lim I«u((, t)) - I{(ur ((

+

^ct))

= o.

ulliformly in 0 ::; t ::; T. Then we will argue that t.his statement is equivalent to

Since

a, a"

^bI , and A1( U) are finite, we may select I~ sufficiently large so that

L(V - w)

:5 o.

^{(1.31 )}

Now,

(1.32) Since lim,_oo IVa, =

a

by assumption, there exists a X2 = X2(T) ~

a

such that

(1.33)

for ( ~

x

2 • Now choosing N sufficiently large to take care of ( E [-ct, X2 ] implies (1.34 ) Likewise,

V(-ct,t)-IVd-ct,t)

=

~eJ3(t-T)+Nf.{3tHo+cl -lVd-ct,t). (1.35)

\Ve know 11V,( -ct, t)1

:5

A12

<

00 by the boundary condition (7.19) and can thus choose N sufficiently large depending on T so that

V( -ct, t) -

Wd

-ct, t) ~

a

^(1.36)

for

t

E [0, T]. Thus, we can use the maximum principle to say

V((, t) - Wd(, t) ~

o.

^(1.37)

This implies that there exists a X3 = X3(T) such that Wd(, t)

:5

^f. for ( ~ X 3.

An analogous argument, taking

V((,t) = _':e/3(t-T) - Ne J3tHo -(

2

will yield H',((, t) ~ ^-f. for ( ~ X4

=

X4(€, T) for some X4 E R. Thus

(1.38)

(1.39)

uniformly in 0 ~ t ~ T. Then we have

lim

a

^{(U((, t) - Ur ((}

+

ct)) = 0, (1.40)

(-00 ( e((+ct)

lim e((

+

ct)(U - Ur)( - e((

+

ct)dU - Ur) = O. (1.41)

(-00 e2 ( (

+

^ct)

Since e((

+

ct)

>

0 and is one-periodic in (, we know

Looking at only the second term in the limit and letting El = min( e( (

+

^{ct)( and}

E'2 = max( e((

+

ct)" we have

El lim (U - [Tr) ~ lim inf e( (

+

ct)d ^{[T -} Ur )

(-00 (-OC

(1.43) By Lemma i.1~ both the bounding limits go to zero. Therefore the middle part of (1.4:3) also goes to zero. Thus we are left with

lim e((

+

ct)(U - Urk

=

0,

(-00 (1.44 )

and since e((

+

el)

>

0 and is one-periodic in (, we again know lim (U(s, t) - Ur{s

+

^ct))(

= o.

(-00 (1.45)

Changing back into our original variables, we see

(1.46)

or

(1.47) where

e

is an intermediate value of u and ^lir. Taking the derivative we have

(1.48)

By Lemma i.1, tIle second term in (1.48) vanishes and J('

>

0 implies

for any t E [0, T]. The proof is complete.

Before stating our next lemma, we will observe the following:

Remark A.1 Lei 111 and 112 be any solutions of (7.:1) {i.e. full time-dependent, traveling wave, or.steady state solutions}. Let U₁ and U₂ be any solutions of {7.11};

1-V_l and 11'2 ^t any solutions of (7.17). Then the following are equivalent:

l

^{Xl X2}

^{lUI -}

^1l21ds

^<

^f,

l

^{Xl X2}

^{lUI -}

^U21ds

^<

^C'f.,

l

^{Xl X2}

^{IW I}

^-lV2

^{lds <}

^{C"f., (1.55)}

fol' the same ^f. where ^X}'X2 E Rl and C',C"

>

0 are constants depending on K(u) ande((+ct).

Proof:

(1.56) where

e

is an intermediate value between ^III and 112. Since e((

+

^{ct) and K(u) are}

bounded, positive functions, we have

(1..57) where C' ~ mOJ.:uK(u). Similarly,

(1..58) where C" ~ mClx(u,(+ct) e~~uc't)' The proof is complete.

Proof of Lemma 7.4: Let

-This operator is derived by writing down (7.2) once with a traveling wave solution and again with a full, time-dependent solution:

Int.egrate (1.65) once from ( to 00 and we obtain LV;. It is easy to verify that L \.'2

=

O. Now since

V2((,0) -

J,

[00 [u'(s - so,s) - uo(s)]ds

+ ~ _

- fcoo[u'(s - so,s) - ur(s)]ds

+

fcoo[ur(s) - uo{s)]ds

+ i,

^(1.66)

where the second integral is finite by assumption (7.8) for 0 :::; (

<

^00. Vole can choose ^So

2::

0 large enough to make the first integral as large as we need so that the sum of the two integrals is posith'e. Thus, 1,:!((,0)

2::

0 for all (

>

O. vVhen ( = -ct, we can show ~(-ct, t)

2::

0 by calculating

1I2t (-ct,t) -

1

⁰⁰ (u' - U)tds

+

c[u' - U]k=-d -ct

- (D{u', (+ ct)u( - /((u', (+ ct)

+

^CUi

D(u,

(+

^ct)u,

+

^1\(u, (+ ct) - cu((

+

ct, t))I~-ct

+

c[u' - u]k=-ct.

Evaluating by the boundary conditions (7.4) and (7.5), we see

- - c

^r

+ c

^{r -} (D( u' , (

+

^ct)

lI( - /( (

u' . (

+

ct) ) k=-ct - Cl

+

^{c(u' -} u)l~ct

+

c[u' - u]k=-ct

(1.67)

- -(D(u',(

+

ct)u( -1\(u',(

+

cf))k=-d - ^Cl

+

^{c(u' -} u)k=o:,. (1.68) By Lemma 3.2, we know that u'( ( - so, (

+

ct) - u r( (

+

ct)

< e->',

for (

2::

s}, t

2::

0 and by Lemma 7.1, we know lim,_oo(u - u_{r )} = O. Therefore, we have

c(u' - u)k=oo

=

c(ur - ur)

=

0, (1.69) and thus,

V;{ -ct, t) - ~(O, 0)

+

^{fot[-(D(ul, (+} cr)u( - /((u', (+ cr))k=-CT - ct]dr. (1.70)

\Ve can see that ~he integral in (1.70) is finite for all t by the following argument.

for any ((, t). Now we will show an upper bound. Let

1

^{< ,}

h((' t) _ct[u(s, t) - u (s - so, s

+

d)]ds

J

^{-ct f:}

+ -eo

[u/(s

+

ct) - u'(s - so, s

+

ct)]ds

+ 2'

^(1.76)

where u(s, t) is a time-dependent solution of Richards' equation and u'(s-so, s+ct) is a traveling wave solution. Now, define the parabolic operator

L· = D('· )82 D(u,(+ct)-D(u',(+ct) a

since the integrand is positive by Lemma 3.1. Therefore, the maximum principle for parabolic operators tells us that V:z

2::

O. Thus,

or

j '

-ct [UI - u]ds ~

j'

^-00 [UI - U ds

'] + 2'

^{E (1.82)}

By Lemma 3.2, we may choose X

=

X(E) such that

o ~

l'o)uI - u']ds

~ ~

^(1.83)

for ( ~ X. Then combining (1.86) and (1.87), we have

l')uI(s

+

^{ct) -} u(s, t)]ds

~

^{Eo (1.84 )}

The proof is complete.

Proof of Lemma 7.6: First we employ a change of variables

u(;r, t) = v(x, t)q(x)

=

u(x, t)(1 - (:JeO'(r-xl»), (1.8.5 ) where a: and f3 are positive constants to be determined so that ~

:s;

q(x)

:s;

1 for

,'l: E [Xl, ^,'l·r]. Then

L'( v)

=

^{a(x, t)v}xx

+

^{b'(x, t} )vx - c(x, t)v - Vt

=

^{0, (1.86)}

where

b'( ^{x, t)}

c(x, t) (1.87)

Since X comes from a bounded domain, we may choose an a: larger than supC:1bl ) and then a f3 = f3(a:,xr - xI) small enough that ~ ~ q(x) ~ 1, for all x E [XI,Xr ].

Then

inf c(x, t)

=

⁸

^2::

^2a:j3(aa

+

b)

>

O.

(rtt) (1.88)

Next, for every E

>

0 there exists a T = T(c:)

>

0 such that 1~1(~ll

<

^E and I~Wll

<

c:

for t

>

T( E). Next, consider the auxilliary functions

w±(x, t)

=

^Ale-co(t-T)

+

^2t

±

v(x, t), (1.89)

where /'vI

>

0 and Co

>

0 are constants. By choosing JVf large enough and using our bounds on the boundary values, we have

For t ~ T, we have

- M

+

2E

±

v(x,T) ~ 0, x E Q

_ M e-co(t-T)

+

2E

±

91( (t)) 2: 0, t

~

^T,

q XI

_ M e-co(t-T)

+

^2<:

±

9(2(t))

~

^{0, t}

~

^T.

q Xr

(1.90)

L'(w±) = -c(x, t)(Ale-^{co (t-T)}

+

^2E)

+

cQJl1e-co (t-T) ::;; 0, (1.91) if we select Co = 8. Thus, we can apply the maximum principle to show that w±(.'t, t) ~ 0 for all x E ^[XI, xr ] and t ~ Tor

Iv( x, t)

I ::;;

^E

+

^{1\1 E-co(t-T)} ( 1.92)

for t ~ T(E), which implies when sending t --+ 00 that

lim sup lu(x. t)1 ::;; E. (1.93)

t-oo

The proof is complete.

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