Global Weak Solutions and Uniqueness for a Moving Boundary Problem for a Coupled System of Quasilinear Diffusion-Reaction Equations arising as a Model of Chemical Corrosion of Concrete Surfaces

Global Weak Solutions and Uniqueness for a Moving Boundary Problem for a

Coupled System of Quasilinear Di usion-Reaction Equations arising as a Model of Chemical Corrosion of Concrete Surfaces

(Part 3) by

Michael Bohm⁽¹⁾ and I.G. Rosen⁽²⁾

Abstract

We show existence and uniqueness for global weak solutions of a moving boun- dary problem for a coupled system of three quasi-linear diusion-reaction equa- tions. The model is briey described. The proofs are based on Schauder's and Banach's xed point theorems, the one-dimensional setting and they make use of relatively general and realistic assumptions on the production terms provid- ing bounds on the weak solutions of the problem. The paper extends previously known results with constant coecients to a quasi-linear setting.

AMS classi cation: 35 Q 80, 35 K 45, 35 K 57, 35 R 35, 35 D 05, 35 K 60, 35 B 30, 35 B 50

Key words: Moving boundary problem, quasilinear systems, reaction-diusion equations, well-posedness, maximum estimates, porous media, corrosion, one- dimensional

(1) Institut fur Angewandte Mathematik, Humboldt-Universitat zu Berlin, Un- ter den Linden 6, 10099 Berlin, Germany. e-mail: mbohm@mi.uni-koeln.de.

Part of this paper was written while MB was at the Mathematisches Institut der Universitat zu Koln. In particular, he thanks Prof. Bernhard Kawohl for his hospitality. Partially supported by DFG and NSF.

(2) Dept. of Mathematics, University of Southern California, DRB 155, 1042 W 36^th Place, Los Angeles, CA 90089-1113, USA, email: rosen@mtha.usc.edu.

Partially supported by NSF.

1

1. Introduction 1.1 Preliminaries

In this note we show existence and uniqueness for global weak solutions of a moving boundary problem for a coupled system of quasi-linear diusion-reaction equations.

The model describes the advancement of the corrosion front in the concrete walls of sewer pipes, where, due to the reaction with sulfate generated from hydrogen sul de arising from the sewage, calcium carbonate parts of the concrete wall are transformed into gypsum.

This transformation leads to density changes of the wall, subsequently in- creased stresses in the porous matrix and several other destabilizing eects. Here we concentrate on the non-mechanical chemical phase-change and neglect the density change.

Later on, the present model will be incorporated into a mechanical one.

This is the main motivation for considering weak solutions in this paper.

The general feature of the model is: Hydrogen sul de enters from the inside the part of the pipe, which is already corroded (i.e. the gypsum part), dif- fuses through the water- lled and the air- lled parts of the gypsum, reacts in the water- lled pores to sulfate and moves to the corrosion front, which is the (idealized) interface between corroded part (porous gypsum) and uncorroded part of the wall (also: cf. rem. (1.3.1 and 2)).

We imagine the cross section of the pipe as a circular ring with the outer radius R and initial (i.e. before corrosion) thickness d, draw a horizontal line through the center of the (cross section of the) pipe, position on this line an x- coordinate axis with its origin coinciding with the position of the inner boundary of the pipe wall before the onset of corrosion and pointing to the right. Thus the x-axis is at approximately the same level as the average sewage surface.

Corrosion is considered in the direction of the x-axis, i.e., to the right. The position of the corrosion front is denoted by s(t) witht 0 denoting time. For more on this and on related models cf. BoDJR], De], BoDR], BoR].

The main result is theorem 2 (cf. section 3).

Before formulating the problem we make some general remarks on diusion- reaction equations for porous media in the following section.

1.2 Some generalities

The underlying model is the diusion-reaction equation speci ed for a porous medium, which can be written in a variety of ways: To be more speci c, consider

2

the diusive-reactive ow in a porous domain having porosity = (xt), tortuosity = (xt), source- and sink rates fi, diusion coecient E, and de ne a concentration u such that

8 measurable (and physically reasonable) subdomains ⁰ :

R

0 udx= (mass- or molar) content of the diusive species in (1) Formulating FICK's (mass- or molar) ux relation as

j =^;E^r( u) (2)

or, alternatively, as

j =^; E^ru (3)

conservation of mass yields either

@( u)

@t ^;div(E^r( u)) =f¹ (4)

or @( u)

@t ^;div( E^ru) =f² (5) and the reaction rate(-s) f_i are, in either case, of the form

fi= fi( uv) (6)

with the dead variable v standing for other reactants. We are dealing with a simple reaction, for which the corresponding f's will be a product of powers of the porosity and of the participating concentrations.

In this note we will employ (2), (4). Substitution w:= u yields

@w@t ^;div(E^rw) =f: (7) Note that the porosity (formally) disappears and w = u (not u) is the physically relevant density of the dissolute mass.

At this point we would like to point out that the literature dealing with concise derivations of diusion-reaction equations for porous media dealing with space dependent porosities and tortuosities seems to be very rare. The case of constant and is dealt with in an abundance of papers. Homogenization results such as in HoJ] deal mainly with constant eective diusion coecients or a diusion law like (3) is assumed. Averaging techniques as in BeBa], BeC] e.g., do not yield more insight. On the other hand, the situation is similar to the one forow in porous media and the question how Darcy's law (for non-constant coecients) seems, in general, not quite clear (cf. the discussion in Sch], e.g.), although the equivalent of (3) seems to be preferred.

Moreover, the dierence between (2) and (3) is not merely formal { at least, if one has relations like = (u(xt)) in mind.

3

1.3 The model

The resulting model is the following coupled system of diusion-reaction equa- tions with a moving boundary

@v_i

@t ^;(A_iv_ix)_x =f_i 0< x < s(t) t >0 i= 123 (1) vi(x0) =v⁰i(x) 0< x < s⁰ (2) v_i(0t) =_i t >0 (3)

;A_iv_ix =g_i at x=s(t) t >0 (4) s⁰(t) =L²v^3m(s(t)t) t > 0 m= const: 1 (5)

s(0) =s⁰ >0 (6)

where

Ai =Ai(xs(t)v(xt)) v = (v¹v²v³) v=v(xt) (7) B_i =B_i(xsv(xt)) _i = _i(xsv(xt)) (8)

f_i =f_i(xsv(xt)) f_i :=

8

<

:

K(v^2;B¹v¹) i= 1 K²(B²v^1;v²)^;K³v² i= 2

K³v² i= 3 ⁽⁹⁾ L² =L²(s(t)v(xt)) (10) A_ir =A_ir(s(t)v(xt)) (11)

g_i =g_i(s(t)s(t)v(xt)v(t)) g_i(ssvv) :=

(Air(sv)^nv_s^i;;v_d^o i= 12

L²(sv)v^3m+1+ i= 3 (12)

Some remarks.

1. (9) models the exchange between water- and air- lled parts of the corroded part (gypsum, e.g.) and the reaction of hydrogen sul de to sulfate (De], BoDJR],BoDR]). For the sake of expositional simplicity, the position of the inner boundary of the corrosion product remains xed at its initial position (= 0). The corresponding modi cations taking the lower density (and there- fore the larger volume) of the corrosion product into account has been dealt with in BoR]. s(0) > 0 means that at the beginning of the process we are considering, there is already some corrosion product. The case s(0) = 0 is more of mathematical than of practical interest, since the very beginning of the actual corrosion is likely to require a completely dierent model. One- dimensional mbp's for problems which are related to ours and which are dealing with a single equation have been addressed in AnR], e.g. Mathe- matically, the case s(0) = 0 leads to some sort of a degeneracy.

4

2. Although the chemical reaction transforming the sulfur ions (as part of hy- drogen sul de) to sulfate is a simple rst order reaction, for which engineers sometimes use m0:5. For the mathematics of the problem this is essen- tial since it destroys local Lipschitz-continuity of the term on the right hand side of (1.3.5).

3. Local solutions for very general one-dimensional quasilinearsingleparabolic equations are obtained in FaP]. FrRZ] consider a mbp for three weakly coupled semilinear parabolic equations in a context which is vaguely related to ours. Am] is likely to yield a wealth of (time-wise) local solutions for our problem, although one would have to do similar work as in this note and in BoR] to obtain estimates andglobal weak solutions. At the practical level, Ri] is conceptually related to this paper. He considers a single semi-linear parabolic mbp for diusion through an ash layer. The methods in Ri] and in FrRZ] yield classical solutions and they are not applicable to quasilinear problems. The problem of polymer swelling, which is formally related to the expansion of the corrosion product in this paper, is considered in several articles (FaMP],AnR],CoRT], e.g.).

2. Notations. Technicalities

L^p(G),W^{s p}(G),H^s(G),C⁰(G) andC^0;(G) := ^\

0<<C⁰(G) are the usual Lebesgue-, Sobolev- and Holder spaces with the norms^jj_p,^kk_{s p},^kk_sand^jj0, resp. V := ^fv = (v¹v²v³) : vi ² H¹(01) and vi(0) = 0^g and H := L²(01)³ are normed by ^kv^k:=

3

P

i^=1jv_iy^j2_L²⁽⁰¹⁾

!

1=²

and ^jv^j:=

3

P

i^=1jv_i^j2_L²⁽⁰¹⁾

!

1=²

, resp.

Usually we won't distinguish between the norm in H and in L²(01). For ap- propriate functions w = w(y:::) and v(t) = v(t:::) and we write wy := ^@w_@y and v⁰ := ^@v_@t, resp. Let S = (0T) be a (time-)interval, X a normed space.

M ^! N] stands for the set of all maps from the set M into the set N. L^p(SX), C^k (SX) and W^{s p}(SX) are the usual spaces of B-measurable functions ² S ^! X], of k times F-dierentiable functions ² S ^! X] having -Holder continuous derivatives and of Sobolev-functions ²S ^!X] (with cor- responding integrability assumptions on all (up to the s-th) derivative (ifs is an integer) and de ned by interpolation, if s is not an integer), resp., cf. KuJF], Ze], Zi], GGZ]).

For functions w =w(xt) we set w(t) :=w(t),w⁰(t) := ^@w_@t(t), w_x := ^@w_@x. References within the same section are made without pre x, references to other sections contain the number of that section. (3.4) means reference (4) in section 3, e.g. Usually, c stands for a nonnegative constant.

5

The following lemma collects some repeatedly used arguments. In some way, the rst part with = ¹² is one of several essential gaps which the proofs go through.

Lemma 1.

Let " >0, ²¹²1.

(i) There are constants ^c= ^c(), c">0 such that

jv^j1 c^^jv^{j1; k}v^k ^c("^kv^k+c"^jv^j) ⁸v ²V : (ii) Let s²W¹¹(S)v'²Vs⁰(t) 0 a.e. Then there is a constant c:

s⁰(t)(yvy') =s⁰(t)^fv(1)'(1)^;(yv'y)^;(v')^g

s⁰(t)(yvyv_{) = 12}s⁰(t)^fv(1)^2;jv^{j2g 1}₂^js⁰(t)^jfc^1jv^{j2(1; )k}v^{k2 ;j}v^j2g : (iii) Let ^c"c" be as above, v²V, s > 0. Then

1s^jv(1)^j2 1

s^jv^j21 (^c)^2fs^{2 ;1j}v^{j2(1; )g}(s^;1kv^k)²

"

s^2kv^k2+c_"(^c)^{1;2 j}v^j2 :

s⁰

s^jv(1)^j2 s⁰

s^jv^j21 (^c)²s⁰s^{2 ;1j}v^{j2(1; )}(s^;1kv^k)²

(iv) "

s^2kv^k2+c"(^c)^1;2 s^2;11; (s⁰)^{1;1 j}v^j2 :

Proof.

For the rst part of (i) cf. Ag]. All the other parts are based on this and on obvious applications of Young's inequality and on integration by parts.

3. Transformation onto a xed domain. Results

(1.3.1-12) can be reformulated on a xed domain by the transformation (xt)²0s(t)]S ^7!(yt)²01]0T]

y :=x=s(t) u(yt) :=v(xt)^;(t) u= (u¹u²u³) = (¹²³): ⁽¹⁾ 6

Using the transformed coecients

A_i =A_i(ys(t)u(yt)+(t)) :=A_i(xs(t)v(xt)) (2:a) fi =fi(ys(t)u(yt)+(t)) :=f_i(xv(xt)) (2:b) gi =gi(ys(t)u(1t)+(t)v(t)) :=g_i(xs(t)v(xt)v(t))

(i = 12 y= 1) (2:c)

g³ =g³(ys(t)u(1t)+(t)v(t)) :=g³(u³+³) (y= 1) (2:d) L² =L²(s(t)u(1t)+(t)) :=L²(s(t)v(s(t)t)) (2:e) Air =Air(s(t)u(1t) +(t)) :=Air(s(t)v(s(t)t)) (2:f)

hi =hi(ss⁰y) := s⁰

s y ⁽²:g)

(1.3.1-6) transforms to

@ui

@t ^; s²¹(t)(A_iu_iy)_y =f_i^;0_i+h_i(ss⁰yu_iy) (3) ui(y0) =:u⁰i(y) :=v⁰i(ys⁰)^;i(0) y²(01) (4)

ui(0t) = 0 t > 0 (5)

;

s(1t)Ai @ui

@u _y^{=1 =}gi(1s(t)u(1t)+(t)v(t)) (6) s⁰(t) =L²(u³(1t) +³(t))^m t >0 (7)

s(0) =s⁰: (8)

We will use the following assumptions on the transformed coecients Ai : 01]s⁰¹)(IR⁺)^{3 !}IR y ^7!Ai(ysu) is measurable for all (su)²s⁰¹)(IR⁺)³ (su)^7!Ai(ysu) is continuous for ally ²01]

there are bounds Aij >0 :Ai⁰ Ai(ysu) Ai¹

for a.a. y and for all sui= 123,

9

>

>

>

>

=

>

>

>

>

(9)

BiKi: 01]s⁰¹)(IR⁺)^{3 !}IR are non-negative, bounded, measurable and

(su)^7!Ki(ysu) Bi(ysu) are continuous, i = 123:

9

=

(10) L²Air : s⁰¹)(IR⁺)^{3 !}IR⁺ are continuous, bounded

and non-negative, i= 12: ⁽¹¹⁾

For an \auxiliary problem" we will need the following coecients A^i := ^Ai(yt) A^{^}i ²L¹(GS) such that there are constants

A_ij >0 with A_i⁰ A^{^}_i(yt) A_i¹ a.e. (A_ij - cf. (9)), (12) 7

K^_i := ^K_i(yt) 0 a.e., ^K_i²L¹(GS) i= 12 (13) B^i := ^Bi(yt) 0 a.e., ^Bi ²L¹(GS) i= 12 (14) A^_ir := ^A_ir(t) 0 a.e., ^A_ir²L¹(S) i= 12 (15) f^i := ^fi de ned with ^Ki B^{^}i in analogy to fi, (16) L^² := ^L²(yt) L^{^2 2}L¹(GS): (17) We call the quadruple (su),u = (u¹u²u³), a weak solution of (2)-(8) if

s²W¹¹(S) (18)

u²L²(SV)^\H¹(SV)^\L¹(S((01])³)^\S ^!L¹(01)³] (19) and ⁸'= ('¹'²'³)²V, a.a. t²S:

(u⁰(t)') + _s²¹⁽_t⁾_i^P3

=1

(Aiuiy'iy) + _s⁽¹_t⁾_i^P3

=1

gi(1s(t)u(1t) +(t)u(1t) +(t))'i(1)

=_i^P3

=1

(fi(s(t)u(t) +(t)u(t) +(t))'i) +^P3

i⁼¹(⁰_i(t)'_i) + ^P3

i⁼¹s⁰⁽t⁾

s⁽t⁾(yu_iy'_i)

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

(20)

u(0) =u⁰ (21)

s⁰(t) =L²(1s(t)u(1t)+(t))(u³(1t) +³(t))^m for a.a. t²S (22)

s(0) =s⁰: (23)

In complete analogy to (18)-(23) we de ne a weak solution for the system (3)-(8) withA_iB_iK_iL²A_ir

(in the de nition of gi) replaced by coecients A^_i = ^A_i(ty):::A^{^}_ir = ^A_ir(ty) L^{^2} = ^L²(yt) (cf. (12)-(17)). The corresponding substitutes for (18)-(23) will be marked as (^c18)^;(^c23).

9

>

>

>

>

>

>

=

>

>

>

>

>

>

(^c18)^;(^c23)

Theorem 1.

Let the coecients satisfy (12)-(17) and let

i ²W¹²(0T) i(t) 0 ⁸t ²0T] i= 123 (24) u⁰i ²L¹(01) u⁰i(y) +i(0) 0 for a.a. y ²01] i= 123 (25)

0< s⁰ < d (26)

v_i ²L²(0T) i= 12 (27) ki max^fu⁰i(y) +i(t)i(t)v_i(t)y²01]t²0T]^g i= 12 (28) k³ max^fu⁰³(y) +³(t)³(t)y²01]t²0T]^g (29)

8

and assume

K^²(yt) ^B²(yt)

K^²(yt) + ^K³(yt) B^¹(yt) for a.a. y²(01) t²S (30) inf_{y t} B^¹(yt) k²

k^{1 and} K^{^2}(yt) ^B²(yt)

K^²(yt) + ^K³(yt) k² k¹

for a.a. y²(01) t²S (31) and T^jL^{^2j1}k^3m< d^;s⁰: (32) Then(i) Problem (^c18)^;(^c23) admits a weak solution (su).

(ii) One has

0 ui(yt) +i(t) ki for a.a. t ²0T] and

for all y²01] s(t)^;s⁰²0b¹] b¹ :=T^jL^~2j1k^{m3 (33)} and there is a constant c depending at most on the constants ki in (28), (29), on the L¹-norms of u⁰ and and on the L¹-bounds for A^i:::L^{^2} in (12)-(17) such that

ku^kL²⁽SV^)\H¹⁽SV⁾ c

ks^k_W¹¹⁽_S⁾ c: ⁽³⁴⁾

For the main theorem we need a variant of (30), (31):

K²⁽y s u⁾⁽B²⁽y s u⁾

K²⁽y s u⁾⁺K³⁽y s u⁾ B²(ysu) for a.a. y²01]

a.a. (su)²s⁰b¹]IR³⁺

)

(35) B¹(ysu) ^k_k²1and ^k_k²1

K²⁽y s u⁾⁽B²⁽y s u⁾ K²⁽y s u⁾⁺K³⁽y s u⁾

for a.a. y²01] (su)²s⁰b]IR³⁺

)

(36) and a variant of (32):

T^jL^2j1k^m3 < d^;s⁰: (37)

Remarks.

1. (37) will be needed to guarantee thats(t)< d. This condition is not needed, if the uxes for u¹ and u² are assumed to vanish at y = 1. The latter assumption can be made for the practical models we have in mind (cf. BoR], BoDJR], BoDR]), which (partly) justi es the title global solutions.

2. (35) and (36) seem to be essential to guarantee the existence of an appro- priate invariant region for the concentrations (also: cf. rem. 2.1 in BoR]).

In practice one has B¹ =B², implying (35).

9

Theorem 2.

Let the coecients satisfy (9)-(11), (35)-(37) and (24)-29). Then there is a solution (su) of problem (18)-(23) satisfying (33).

Remark 3.

The boundedness assumptions on most coecients can be consid- erably relaxed.

Theorem 3.

Let the assumptions of theorem 2 be fullled and assume all the coecients to be locally Lipschitz-continuous. Moreover, assume that there is at least one solution (su) such of (3.18)-(3.23) such that

u²L⁴(0TV): (38) Finally, (38) can be obtained if j ²W¹⁴(S).

Remark 4.

There is a whole variety of other regularity assumptions also leading to uniqueness. Furthermore, even under the assumptions of theorem 1, the solutions of theorem 1 are more regular than stated. Theorem 3 can be extended to a well-posedness statement, such, that the solutions depend locally Lipschitz- continuous on the data and on most coecient-functions. We do not go into detail.

4. Proofs of Theorems 1 and 2 Proof of Theorem 1.

A similar problem with constant coecients has been dealt with in BoR]. The arguments there carry over to the situation of theorem 1.

Proof of Theorem 2.

We employ Schauder's xed point theorem. In order to de ne an appropriate xed point-operator, x p²1¹), set

b¹ :=T ^jL^2j1k^3m (k³ - cf. (3.29)) (1) and note that the set

M :=^f(su)²C(0T])L^p(SC(01])³) :s(0) =s⁰ s(t)^;s^{0 2}0b¹] for all t ²0T] ^jui(t) +i(t)^j1 ki a.a. t ²S^g (ki - cf. (3.28),(3.29)) is convex and closed as well as bounded in Y := C(0T])L^p(SC(01])³).

We de ne the ( xed point) operator Q with

dom(Q) :=M and Q: (su)^7!(su) (2) 10

where (su) is the (weak) solution of (3:^c22^;c28) with (su) ²M and A^_i :=A_i(ys(t)u(yt)) B^{^}_i :=B_i(ys(t)u(yt)) K^_i :=K_i(ys(t)u(yt)) A^{^}_ir :=A_ir(1s(t)u(1t)) E^³j :=E³j(ys(t)u(1t)) L^{^2} :=L²(s(t)u(1t)): By theorem 1, Q is well-de ned,

Q(M)L²(SV)^\H¹(SV)^\L¹(SC¹²(01]))^\S ^!L¹(01)³]^\M (3) and there are constants c, independent of (su), such that

js^j_W¹¹⁽_S⁾ c

ku^kL²⁽SV^)\H¹⁽SV^)\L¹⁽SC^(01])3) c for all (su)²Q(M):

9

=

(4) We have

Lemma 1.

(i) Q(M) is relatively compact inY.

(ii) Let(un)L²(SV)^\H¹(SV)^\L¹(SC(01]³) =:X be bounded, u²X, and un ^!u in L²(SH). Then un ^!u in L^r(SC(01])) for all r²1¹).

Proof of Lemma 1 (i).

By Aubin's lemma (cf. Ze]) there is a subsequence (we drop the subsequence index) and a limit u such that un ^! u in L²(SH).

This and L²(SV)^\H¹(SV) ,^! C(SH) imply un ^! u in L^r(SH) for all r²1¹). Let ²24),b:= 4=_a¹ +¹_b = 1. By lemma 2.1(i) and by Holder's inequality:

ku_n^;u_m^k_L⁽_SC⁽_G⁾³⁾ c^ju_n^;u_m^j1_L^{=a=22 (}_SH^)ku_n^;u_m^k1_L⁼²²⁽_SV⁾ for allnm:

Therefore: un ^!u inL(SC(G)³) for all ²24). This and the boundedness of (u_n) in L¹(SC(G)³) imply (i).

(ii) follows similarly.

It remains to show

11

Lemma 2.

The (xed point) operator Q is continuous.

Proof.

Let (s_nu_n), (su)²M, s_n ^!s in C(G), u_n ^!u in L^p(SC(G)), (5i) set (su) :=Q(su), (snun) :=Q(snun), wn :=un^;u, wn:=un^;u. By the estimates in (4) there is a constant d (independent of n) such that

jsn^jW¹¹⁽S⁾+^js^j_W¹¹⁽_S⁾ d

kun^kL²⁽SV^)\H¹⁽SV^)\L¹⁽SC^(01])3) d

ku^k_L²⁽_SV^)\_H¹⁽_SV^)\_L¹⁽_SC^(01])3) d:

9

=

(5ii) We claim

sn ^!s in C(S) and un ^!u in L²(SH): (6) To this end consider the equations in (3:^c18)^;(3:^c23) for (snun), subtract the ones for (su), split some of the expressions, use ' := wn := un ^;u as test function and integrate with respect of time from 0 to t ²(0T]. This results in

jwn(t)^j2+A¹n(t) +G¹n(t) =A²n(t) +A³n(t) +F¹n(t) +G²n(t)

+H¹n(t) +Rn(t) =:Bn(t) +H²n(t) +H³n(t) (7i) sn(t)^;s(t) =L¹n(t) +L²n(t) (7ii) where

A¹n(t) :=

Z t

0

s²_n1()

3

X

i⁼¹(Ai(sn()un())wniy()wniy())d d^{1Z t}

0

kw_n()^k2d (d¹:= (s⁰+b¹)^;2X3

i⁼¹A_i⁰) G¹n(t) :=^X3

i⁼¹

Z t

0

sn1()gi(1sn()un(1)wn(1)0)wn(1)d 0 A²n(t) :=^{Z t}

0

sn(1)²

3

X

i⁼¹(Ai(snun)^;Ai(snu)]uiywniy)d s^{;20 X3}

i⁼¹

jAi(snun)^;Ai(su)]uiy^jL²⁽SH^)kwn^kL²⁽SV⁾

2ds^{;20 X3}

i⁼¹

jAi(snun)^;Ai(su)]uiy^jL²⁽SH⁾=: ^A²n(t) (d - cf. (5)),

A³n(t) :=

Z t

0

1

sn()^{2 ;} 1 s()²

3

X

i⁼¹(Ai(su)uiywny)d 12

jsn^;s^jC⁽⁰T^])d² =: ^A³n(t) (d² := 4s^;40 (s⁰+b¹)^X3

i⁼¹Ai¹d (cf. (5)),

F¹n(t) :=^{Z t}

0

af(un+wnwn)d d^{3Z t}

0

jwn()^j2d G²_n(t) :=

Z t

0 2

X

j⁼¹E³_j(s_n()u_n(1))^;E³_j(s()u(1))]

(u³(1) +³())^m;1+j w³n(1)d =: ^G²n(t): H¹n(t) :=^X3

i⁼¹

Z t

0

(s⁰_n()^;s⁰())s^;1_n ()(yuniywni)d H²n(t) :=^X3

i⁼¹

Z t

0

s⁰_n()

1 s_n ^{; 1}s

(yuniywni)d H³_n(t) :=^X3

i⁼¹

Z t

0

s⁰()

s() (yw_niyw_ni)d L¹_n(t) :=

Z t

0

L²(s_nu_n(1))(u_n³(1) +³())^m]

;(u³(1) +³())^m]w³n(1)d

jL¹_n(t)^j d^{4Z t}

0

jw_n³(1)^j2d d⁴c^{Z t}

0

jw_n³()^jkw_n³()^kd

"^{Z t}

0

kwn(t)^k2d + (d⁴c)²c"

Z t

0

jwn()^j2d

where d⁴ := sup_n vraimaxL²(sn()un()) m (2(d)^m;1) and c stems from lemma 2.1(i). " and c_" are related via Young's inequality,

L²_n(t) :=

Z t

0

L²(s_n()u_n(1))

;L²(s()u(1))](u³(1) +³())^mwn³(1)d

jL²_n(t)^j d^{5Z t}

0

L²(s_n()u_n(1))^;L²(s()u(1))]d =: ^L²_n(t) where d⁵ := sup_n ^ju³(1) +³()^jmjwn³(1)^j<¹ (because of (5ii)).

The remaining expressions in the equation for wn are collectively denoted by Rn (see (7i)).

13

Plugging the relations (3:^c22) for s⁰_n and s⁰, resp., into the expressions H¹_n(t) ::: H³n(t) and using similar estimates as for L²n(t), one obtains

jH¹n(t)^j d^{6Z t}

0

jL²(sn()un())^;L²(s()u())^j2d +"^{Z t}

0

kwn()^k2d +c"d^{27Z t}

0

kun()^k2jwn()^j2d where d⁶ := sup_{n i} ^jwni()^j1d(d+^jj1)^ms^;10 <¹ (cf. (5ii)) and

d⁷ := sup^jL²(s()u(1))^jmsup_n ^fju_n³(1)+³()^jm+^ju³(1)+³()^jmg<

1 (cf. (5ii)). Furthermore:

jH²n(t)^j d^8jsn^;s^j1

whered⁸ := sup_n ^jL²(sn()un(1))^jsup_n ^fjsn^j1+^js^{j1gR0T k}un()^kjwn()^jd <

1 (cf. (5ii)), and

jH³n(t)^j d^{9Z t}

0

kwn()^kjwn()^jd

"^{Z t}

0

kwn()^k2d +c"d^{29Z t}

0

jwn()^j2d where d⁹ := sup^jL²(s()u(1))^js^;10 <¹.

Choosing " suciently small (compared with d¹, from the estimate of A¹n(t)) and estimating the terms appearing at the right hand side of (7i) from above as on the preceding lines, one arrives at

jwn(t)^j2+

Z t

0

kwn()^k2d Pn(t) +

Z t

0

Qn()^jwn()^j2d (7iii) where

P_n(t) := ^A²_n(t) + ^A³_n(t) + ^G²_n(t) + ^L²_n(t) +d^8js_n^;s^j1 Qn(t) :=d³+ (d⁴c)²c"+d²⁸c"^kun(t)^k2_"+d²⁹c":

Gronwall's inequality and a straightforward estimate yield

jwn(t)^j2 Pn(t)exp(

Z t

0

Qn()d) Pn(T)d^{10 8}t (7iv) where d¹⁰ := sup_n exp(^R0T Qn()d)<¹ (cf. (5ii)).

14

It remains to show

w_n ^!0 in C(SH): (8i) (5i) implies for a sub-sequence

un^j(t)^!u(t) inC(01])³ f.a.a. t²0T] and

sn^j(t)^!s(t) in IR for all t²0T]: ⁽⁹ⁱ⁾ This, continuity of the coecients involved and Lebesgue's theorem imply

Pn^j(T)^!0 as j ^!1: (9ii) Therefore, by (7iii),

wn^{j !}0 in C(SH) asj ^!1:

The convergence of thewhole sequence (w_n) inC(SH) follows by contradiction.

This yields (8i). Moreover, (8i), (7ii) and (7iii) imply

wn ^!0 in L²(SV): (8ii)

Set #Lⁿ²() :=L²(s()u(1))^;L²(sn()un(1)).

(3:^c22) implies

jsn(t)^;s(t)^{j Z t}

0

j#Lⁿ²()^jj(un³(1) +³())^jmd +

Z t

0

jL²(s()u(1))^jj(un³(1) +³())^m

;(u³(1) +³())^mjd d^{11Z T}

0

j#Lⁿ²()^jd +d^{12Z t}

0

jwn³(1)^jd d^{11Z T}

0

j#Lⁿ²()^jd +d¹²c^{Z t}

0

kwn()^kd where

d¹¹ := sup_n ^j(un³(1) +³())^mj<¹ and

d¹² := sup_n ^jL²(s()u(1))^jm2(d+^jj1)^m;1 <¹: By (8ii) and (9i):

sn^{j !}s in C(S) 15

and, again by contradiction, the whole sequence (s_n) converges. Now, (8.i) and lemma 1(ii) imply un ^!u inL^p(SC(01])). This and the convergence of (sn) implies: Q is continuous.

Remark.

We have not been able to employ Nemyzki-type arguments. More- over, using the boundedness properties (5ii), some of the boundedness assump- tions imposed on the coecients could be relaxed.

5. Proof of Theorem 5

At rst we show uniqueness. The main tools are: The L¹-estimates for the solutions, combined with the other estimates and a repeated use of Gronwall's inequality.

Assume there are two solutions (s_ju_j),j = 12, satisfying (3.18)-(3.23), (3.33) and let u¹ satisfy (3.38). Set

s :=s^2;s and w:=u^2;u¹ :

Let " > 0. (3.18), (3.19), (3.22), (3.23) and local Lipschitz-continuity of L² imply

js(t)^j c^{Z t}

0

jw(1)^j+^js()^jd (1) c^{Z t}

0

jw(1)^jd +t¹⁼²

"

Z t

0

js()^j2d

#

1=²

and

js⁰(t)^j c^jw(1t)^j+^js(t)^j]: (2) Gronwall's inequality and (1) yield

js(t)^j c^{Z t}

0

jw(1)^jd: (3)

This and (2) imply

js⁰(t)^j c^jw(1t)^j+

Z t

0

jw(1)^jd]: (4) Subtract the equation (3.20) for u¹ from the one for u², use ' := w as test function, split the nonlinear terms to arrive at

jw(t)^j2+A(t) +B(t) =C(t) +D(t) +E(t) for a.a. t ²S (5) 16