Global existence and quenching for a model of adiabatic shear band formation

Nonlineur Analysis, Theory, Printed in Great Britain.

GLOBAL

Methods

& Applicudons,

Vol. 21, No. 5, pp. 337-348,

1993. 0

EXISTENCE ADIABATIC

0362-546X/93 $6.CO+ .OO 1993 Pergamon Press Ltd

AND QUENCHING FOR A MODEL SHEAR BAND FORMATION*

OF

J. J. L. VEL~QUEZ Departamento de Matemkica Aplicada, Facultad de Matemlticas, (Received

1992; received for publication 1 March 1993)

9 January

Key words and phrases:

Universidad Complutense, 28040 Madrid, Spain

Adiabatic shear band formation, global existence, quenching.

1. INTRODUCTION

THIS PAPERis concerned with the following problem. Find functions U(X,t) and P(t) such that U, + P(t)u = k Au - u-~ 4%

in Q

0) = udx)

u(x, t)-pdx f

when x E Q, t > 0,

=

b

cl

au0 an=

$x,

(1.1)

(1.2) = 1,

wpdx

3‘

when x E 51 and t 2 0,

(1.3) (1.4)

where Q is a bounded open set in RN (N I 1) with smooth boundary, ]a[ stands for the measure of a,

k > 0, p > 0

and (1.5)

u,,(x) E C(Q) is such that

z&x)-p d/X= 1. f 62 Problem (l.l)-(1.4) has been proposed as a model for adiabatic shear band formation in a material when thermal softening, due to heat generated by plastic deformation, is stronger than hardening mechanisms; cf. [l] and references therein. If we drop the term P(t) in (1.1) and dispense with condition (1.3), then as is well known the solutions of the corresponding problem may become zero somewhere in finite time. This phenomenon is usually termed as quenching (cf. [2-51 and references therein). If quenching occurs, the solutions cannot generally be continued for all times. See [6] for a survey of results in this direction. In this paper we prove the following theorem. THEOREM1.1. Assume that hypotheses (1.5) hold. Then, if p > 1 and N = 1, there exists a unique global solution u(x, t) of (1 . l)-( 1.4). Moreover, u satisfies u(x, t) 2 Ce-”

for any t > 0,

where c, 13,are some positive constants depending only on

p

and uO.

*Partially supported by CICYT Grant PB90-0235 and EEC Contract SCI-0019-C. 337

(1.6)

J. J. L. VEL~QUEZ

338

Since ii(t) = 1 and /I(t) = -1 is a solution of (l.l)-(1.4), it is clear that global solutions may exist for any choice of p, C! and N. However, quenching can also occur if p < 1 or N > 1. In fact, we shall prove the following theorem. T~~0~~~1.2.AssumethatO 0 large enough, there exists a radially symmetric positive function U,,(X)E C(n) such that quenching occurs at some time T < +co for the corresponding solution of (1 .l)-(1.4). The case p = 1, N = 1 seems more difficult to handle and is not considered in this paper. Theorems 1.1 and 1.2 are proved in Sections 2 and 3, respectively. 2. PROOF OF THEOREM

1.1

LEMMA 2.1. Assume that (1.5) holds. Then there exists a 6 > 0 such that there exists a unique, positive classical solution of (1 . l)-( 1.4) in the cylinder Qs = a x (0,6).

Proof.

It is convenient to change the independent

variables as follows (2.la)

Then, exp(p jID(s)ds)

{n v(x, 0-p&

= 1

(2.lb)

in fi x IR+,

(2.2a)

and u solves v, = ~Au - (I

u-~,>-‘~~~J”u-~

au

(2.2b)

in Sz,

u(-%0) = %(X)

(2.2c)

for x E %2 and t 2 0.

-_=()

an

Let us denote by S(t) the semigroup associated to the heat equation with homogeneous Neumann conditions. By assumption, there exists u > 0 such that uO(x) 2 r in Sz. We now consider the operator (Tu)(+,t)

= S(t)u,

-

[:S(t

- s)(fQ

v”(~,s)dr)ll+p)‘pKp(.,s)d.s

(2.3a)

in the space x, =

v, E C(b x [O, S]): 5 5 9 I 211UJm

(2.3b)

1

t

endowed with the norm (/(~,11) = s~p,,~~~~JIv)(~,s)I),, where 6 > 0 is to be selected presently. is a complete metric space. Then Clearly, (X8, IIIIll)

IIw) - mollcu + 0

as t -+ 0,

339

Adiabatic shear band formation

where I is the identity operator. On the other hand, if u E X,, then vbp 5 (~/2)-~ and (So u-’ dr) -(l+p)‘p I (211uoll_)l+p. It follows that

whence

Ilmoc, 0 - u,C)ll, 5 IIW) - I)u,ll, +

2p$&+pt

5;

provided that t is small enough. This shows that T(X,) c X, if 6 is small enough. Moreover, for some constant L, , JTu(*, t) - Tti(*, t)J - s) / (~nzFp(y,~)dy)-‘l+p)‘p~-p(~,s)

5 j:S(t t I L,

w

-

- (~~6-p(y,s)dy)-o+p~‘pb”(~,s)

1ds

~)llv(*,~) - a(-,s)ll,~

0

Therefore, T is a strict contraction in X, for 6 > 0 sufficiently small. This yields the existence of a unique solution v of (2.2) for small times. By standard results, u E C”(Q&, where Qa = Sz x (0,6) and v > 0 in Q6 = fi x [0, 6). This in turn determines [by (2.1)] a unique solution of (1 . l)-( 1.4) in Qa . l The following result is a key point in the proof of theorem 1.1.

LEMMA 2.2. Let u(x, t) be a positive solution of (2.2) in some cylinder QT = a x (0, t), which is positive in the closure QT. Then, if N = 1 and p > 1, there exist constants C, 0, depending on k, p, u. and a but not on T, such that

ID

for 0 I t I T.

V(X,t)-2p dx 5 0 ect

(2.4)

Set w = v-2p. Then w satisfies

Proof. wt

=

k

AW

_

k’2p2i

‘)

. !?f

+ W

dW

an=0

zp(f

WI/2

&)-‘1+p)‘pWl/2~+3/2

in

QT,

(2.5)

0

for x E 82,

OstsT.

(2.6)

J. J. L. VEL~QUEZ

340

By averaging (2.5) over Q we obtain d WdX dt (f fl > + 2p({c

wl’zdx~o+p”p

{ w1’2p+3’2

for 0 < t < T.

(2.7)

We now claim that

where here and henceforth we shall denote by C a generic contant which possibly depends on k, p, u. and Q, but not on T. We readily see that (2.7) and (2.8) yield

s

w(.x,t)dxl(fnw(x,O)dx)ecf

61

so that the proof is concluded as soon as (2.8) is shown. To prove this inequality, we make use of the well-known Gagliardo-Nirenberg inequality

IlfIIt”(n)5 Cllf IlLglfIlaw’J(*)t

(2.9)

where 1 I CY,p I 00, r > N, and a = (l//3 - l/(r)(l/p + l/N - l/r)-‘, 0 5 a I 1. N = 1, r = 2, Q = l/p + 3 and p = 1 gives that Application of (2.9) with f = CD= We'd,

c

@l/p+3

dx

Jnc(.i.odx>(“5p)‘3p((~.~“~) +(1.

lvwd+)(I+2p~‘3p

I

I

c(

jn

Q, dx)(I+511/p(

So (D2 &T+~P)/~P

+

c(

s.

~ d,)(l+Q’)/*(

JQ Iv~,2

tir+2p)‘3p

(2.10) Since p > 1, (1 + Sp)/3p > (1 + p)/p. (~*~tiyl+“‘/3p

~

Therefore,

by Schwartz’s inequality,

(~~~~~+p)‘p(f.~2~~-l)‘3p

whence (~*@~)o+5p)‘3p(~c~2c+o+2p~‘3p

5 (f~+o+~~“(~*~~d%j.

On the other hand, using Young’s inequality: ab I euy + CEbY’, where

3p >I

y=1+2p

+Y, ’

Y-l

(2.11)

341

Adiabatic shear band formation

and E > 0 is to be selected later, we see that

Substituting (2.11) into (2.10) and taking E = k(2p + 1)/4p2, (2.8) follows.

n

LEMMA 2.3. Let V(X,t) be a solution of (2.2) in some cylinder Qr = Sz x (0, T), 0 < T I 00. Then, if v > 0 in Qr, the following inequalities are satisfied

for 0 < t < T,

(2.12a)

(2.12b)

for 0 < t < T.

(2.12b) follows at once from Jensen’s inequality, since p(t) = teP is convex whenever and t > 0. On the other hand, averaging (2.2a) over Sz yields

Proof. p > 0

-$(~*u(x,t,dx) =-(~nu(x,t)-pdX)(l+p~‘p(~*u(X,t)-~dX) whence (2.12a). LEMMA 2.4.

10

(2.13)

n

Assume that N = 1. Then the following estimates hold Ilf(*, 011L2cn)

5

for 0 I t 5 T,

Cr eelr

(2.14)

where -o f(x, t) =

(S n

(a,

O)-P dx

>

+PvP

(4% O)-P9

for any q E (1,m) and any 6 E (0, T) 5

C2 ee2’

for 6 c t < T,

(2.15)

.wQ)

where Cr and 13~(respectively, C, and 0,) depend on k, p, u. and Sz (respectively, k, p, u,,, Sz, and 6) but not on T.

q

J. J. L. VELAZQUEZ

342

Proof. Inequality (2.14) is a direct consequence of (2.4) and (2.12). To derive (2.15) we may take fi = (-R, R) for some R > 0. We extend u to the whole line as a periodic, even function, symmetric with respect to x = -R, x = +R so that the extended function 6(x, t) satisfies: fit = kc,, + K(x, t)

O
for --oo < x < co,

fi(x, t) = i&(x)

for --Q) < x < +co.

Notice that, since h”(x, t) I 0, w, Now

t) 5 IIQ.llm

for any t E (0, T).

ts

fix t E (6, T), and set t,, = maxlO, t - 1). By standard results, we have that E(x, t - s) *

6(x, t) = E(x, t - t(J * 6(x, to) +

h(+, s) ds,

to

where * denotes Therefore,

convolution

in the space variable,

gcx,%(x,t t) =

to) *

qx, to)

and E(x, s) = (47~s)-“~ exp(-x2/4s).

fax I aE -(x,t

+

-s)*~(*,s)ds.

to

Let X,(x) = 1 when x E &2and zero otherwise. Then

(2.16) We now observe that, for any g E L2(&2)and any q E [l, 00)

x, gc5 I/ ( >li 5 t1*go

Ct-3'411gllL2(n)

(2.17)

Lwa

for some C = C(q, R), whenever t E (0, l), where g is a periodic extension of g symmetric with respect to x = +R, x = -R. Let us assume (2.17) for the moment. Then

5

C i

i

Sincet-t,,r6>OandG
1(t -

C(t -

s)-~‘~eeTsds

t,)-3’4

eeFt.

substituting (2.18) into (2.16), (2.15) follows.

(2.18)

Adiabatic shear band formation

343

It remains yet to show (2.17). This can be done by means of classical (and rather cumbersome) caloric estimates; we shall sketch the corresponding argument for completeness. Clearly,

I/(

I

Q

g(-,t)*go

xfl

Lq(w

R+2nR

X

l/2

q

d-x

_R+2nR k(O/2dt >> -q/2

1

=

C-1 4J7t

1: (1x1 ([:I:% ~-3q’211glle~,n,

lx -

r12e-‘““z’2~dy)l’2~dr

= S.

Notice that, for IuI I 2, . . . Ri2nR _R+2nR

Ix

-


e-(x-t)z’2’

d<

whereas,

ym

<

s

Ix

-


e-(X-t)2’2t

dr

=

ct3’2,

s

for InI > 2, if r E (-R + 2nR, R + 2nR) we have that I<[ 2 5R. Therefore, if we deduce that Ix -
s R+2nR

-R+2nR

d<5 &$ Ix- rl 2e-(X-t)2/2t

r2 e-W/WE2/W

Furthermore,

= t(&t)

e-(8/25)(t2/‘)

Notice also that, if 0 < t I 1, and y1 = -(R there holds Ri2nR

(2e-W/WR2/2’) &Z.

-R+2nR

5 Ct for some universal constant C > 0. y2 = (R + 2nR)/dt, and InI > 2,

+ 2nR)/Jt,

+Yzct1 e-@/25)A2/‘)

i

Ri2nR

dr

-R+2nR

=

Jt

e-W25)t2d( I Jt s Yl(f)

Summing up our previous results, we conclude that

and the proof is concluded.

n

As a last step, we prove the following lemma.

e-@/W2 d<. sk

2 nR

344

J. J. L. VEL.&QUEZ

LEMMA 2.5. Let u(x, t) be as lemma 2.2. Then there exist positive on k, p, i2 and u0 but not on T, such that v(x, t) 5 MemL*

for x E G

and

M and L, depending

constants

for any T* < T.

t E ]O, T*],

(2.19)

Proof. By lemma 2.1, u(x, t) 1 q > 0 for some x E fi and t 5 6 where 6 > 0 is small enough. Assume

now that v(x,, , t,) < MeWLtn

for some sequence

(x,,

t,)

c

fi

where M > 0 and L > 0 will be selected later. By (2.15) and Morrey’s have that

x

[a,

T*],

imbedding

(2.20) theorem,

1u(x, t) - z&F,t)J 5 C, e e3*l~ - Xl* for any x, X E fi, cx E (0, 1) and t E (6, T), on k, p, Q, uO, CYand 6.

where C, and 6$ depend

It then follows from (2.20) and (2.21) that there exist [x, - D,, x,] with D, L (M/(C,) e-(L+e3)*n)1’asuch that v(x, t,) c 2MeeLtn qx, Q-P

k

2

(2.21)

Z,, = [x,, x,, + D,]

or

if x E Z,

“-zLtn (g

?R On the other hand,

intervals

we

e-CL+‘ya.

(2.22)

3

by (2.4) l/2

v(x,

i n Comparing contradiction

t,p

dx I

v(x, tp 11

&

5 p

e(C’2)*” 5 p

ec6/2e

>

(2.22) and (2.23), we readily see that for any fixed M > 0, we arrive by selecting CY> l/p and L > 0 large enough, whence the result. n

at a

End of the proof of theorem 1.1. Let us denote by T the supremum of the times for which v(x, t) exists. Assume that T < +a. By lemma 2.5, u(x, t) 2 MemL* in Sz x [0, T), and we may extend u by continuity up to t = T, so that the previous bound continues to hold. By lemma 2.5, we may then continue u(x, t) to some time interval (T, T + a), but this contradicts the definition of T, thus showing that T = co. H 3. THE PROOF OF THEOREM

1.2

In this section we shall establish the quenching result by adapting a well-known comparison technique which was introduced in [7] in the context of blow-up problems, and has been applied successfully since for extinction and quenching problems (cf. for instance [S, 91). For simplicity, we shall assume that k = 1 in (1.1). (This may be done without loss of generality by resealing the spatial variables.) Set -o+P)/P v(x, t)ydx , y(t) = (3.1) n >

345

Adiabatic shear band formation

where Sz = BR(0) in rdv. When v is a radially symmetric solution, equation (2.2a) reads v, = v, +

N-

1 r

v, - j$t)v-P.

(3.2)

First, we show that as long as V(X,t) > 0, for suitable initial values U,,(X)and R, y(t) 2 2. Consider now the auxiliary function J(r, t) = ?+i(z& - A(r)F(v)),

(3.3)

where functions A and F will be selected in the sequel. As long as v is positive 1

NLJ = Jt - J, + r

J, - (p~-~-‘y(t)

+ 2X(r)F’(v))J

N- 1 N1 A’/(r) + - r A’(r) - -L(r) r2

= rN-l

+ A(r)(puP-‘y(t)F(v)

+ y(t)v-PF’(v)

F(v) + 2A(r)/‘(r)F(v)F’(v)

+ F”(v)v;)

We take

1 .

(3.4)

forOIr
A(r)=f(l+cos(~))=Rh($

(3.5)

It is easy to check that NA”(r) + -I

1

- N+

r

A(r)

whenever 0 I r < R for some constant o > 0 depending only on N. Suppose now that F’(v) I 0. Then, since I’(r)=t(l+cos())-&sin(F)sI we have that A’(r)(-F’(v)) would obtain that

I -F’(v).

LJ L rN-‘l(r)

F(v) (

-G

forrE[O,R],

If in addition F(v) 2 0 and F”(v) 2 0 for r E [0, r], we

+ y(t)(pv-P-lF(v)

+ 2F’(v)

.

+ vmpF’(v))

>

(

>

We want to show that the right-hand side of this equation is nonnegative for a suitable choice of F, u,, . To this end we consider the equation F(v)

-$ (

+ p~-~-‘F(v)

+ 2F’(v)

+ v-~F’(v)

= 0.

>

Invariance under suitable resealing will play an important role in the analysis of (3.7).

(3.7)

J. J. L. VEL~QUEZ

346

LEMMA3.1. There exists a solution F(u) of (3.7) that can be written in the form F(u) = 6G(6%)

= 6G(r)

where 6 = (oR-2)P’@+1) with o as in (3.6).

(3.8a)

Moreover, F(v) 2 0, F’(u) 5 0 and F”(u) 2 0 provided that u I (pR2a-‘)1’(P+1)

(3.8b)

where o is as in (3.6). Proof. In view of (3.7), (3.8), we need G to satisfy 2G(r)G’(r)

+ p<-‘-PC(<)

+ T-pG’(<) - G(r) = 0.

(3.9

Notice that, if G(r) solves (3.9), then G’(P”@+~)) = 0, G’(r) < 0 We now take G(r) = G,(r)

for < < I?“(~+‘).

as the solution of (3.9) which satisfies

G(PwP+l)) =

m,

(3.10)

where m > 0 will be selected later. Clearly, a unique solution of (3.9), (3.10) exists for all r > 0. With such a choice of G = G, we, thus, have that, if F(s) is given by (3.8) F(u) L 0, F’(u) I 0

for u 5 (pRzo-‘)l’(p+l)

(3.11a)

and using (3.9) we readily check that F”(U) 2 0 which concludes the proof.

for u 5 (pR20-1)“(p+1)

(3.11b)

n

We shall presently use the following estimate G(r) 5 mpP’cpi “< -p

for any r E (O,pl’(p+lJ).

(3.12)

To derive (3.12), consider the barrier W(r) = m,lop’(p+l)<-p with ml > m in the interval (0, P l’(p+l)). By (3.10) we have G,,, < W in (cr,p “(p+l)) for some (YL 0. Assume that G,(&) = II%) for some To E (0, P l’(p+l)). By (3.9) we have GA(&,) > W’(&). This readily implies that G,(r) > W(t) for all [ E (p, pl’(p+l) ) where p is the first point where G, and W agree, but this contradicts the fact that G, < Win (a,p”@+‘)); thus, G,(r) 5 ml@‘(p+‘)&j-p for all < E (0,~ “(p+l)). Then letting ml --t m, (3.12) follows. Let now Q(r) be defined as the solution of B’(r) = A(r)F(Q(r))

for r > 0,

Q(O) = s, > 0,

(3.13a) (3.13b)

where F(s) is as in lemma 3.1, and S, is a small positive constant to be fixed later on. If we set Q(r) = R2’(l+P)v f 0

s j72/(1+PJ~(~),

(3.14)

Adiabatic shear band formation

347

we then see that I&) satisfies w’(y) = aP”l+P’h(y)G,(al’o+P’~(y))

forO
1,

(3.15a)

~(0) = R2’(1+P)&,

(3.15b)

where h and A are related through (3.5). Recalling (3.12), we now integrate (3.15) to obtain that v/(y)l+P 5 (R-2(1+J'+&,)1+p + (1 + ~)pP'(~+p)m

forOIy5

1.

(3.16)

We are now in a position to show the following lemma. LEMMA

3.2. There exists uO(x) E C(B, (0)) such that uO(x) > 0 in B, (0) satisfies (3.13a) in BR (0)

and z&x) I (pR*a- ‘)l’(’ +p).

(3.17)

Proof. In view of (3.13)-(3.16), it suffices to take u,,(x) = Q(r) to be a solution of (3.13), and then select S, and m small enough so that w(y) I (pa-‘)l’(l+P’ for y E [0, 11. n End of the proof of theorem 1.2. Let u(x, t) be a solution of (l.l)-(1.3), lemma 3.2, and let u(x, t) be given by (2.1). Since

v, I Au

where uO(x) is as in

as far as u > 0

it follows from (3.17) and the maximum principle that u(x, t) I (pR2a-‘)1’(1+P)

as far as v > 0.

(3.18)

Let y(t) be the function given in (3.1). By (3.14), we have that y(0) = (fQ @D(r)-Pdr)l”“’

= r-I(iIYI

I,,,

r~(??y

dx)(l’pi’p

= KR*

for some K = K(p, m, N) > 0. We now select R > 0 large enough so that y(O) = 2, in which case y(t) L 1 for t E [0, r] and some r > 0. Recalling (3.4), (3.8) and (3.18), we then have that LJ 2 0 in QR,, = BR (0) x (0, t). Notice that the coefficient of J in LJ is smooth in QR,7 since u > 0 there. It then turns out that J 2 0 in QR,r, whence u(r, t) L R2’(1+p)QY(y)

in QR,T,

where I@) satisfies (3.13a) with initial value W(O)= 0. We now remark that f@(y) 2 qy*'(l+P)+&

for any E > 0, and some constant K, = K,(p, N, E).

(3.19)

Inequality (3.19) is easily arrived at by standard ODE methods. Indeed, if G(r) solves (3.9), it is easy to see that H(T) = TpG(r) satisfies ((2HH’ + H’) = 2pH2 + rp”H, a dominated balance argument shows then that H(r) - -GP

log 0-l

as c + 0.

348

Therefore,

J. J. L. VELASQUEZ

G(r) - -cp(2p log <)-’ as < + 0. Recalling (3.15a), we then have that W’(Y) - -MY)(lS/(Y))-P(2P log P(Y))_’

as y + 0,

integrating this relation, we finally obtain W(y)

-

(f+)““+p’(_2C~“1+p’

as y +

0

whence the desired result. If E > 0 is small enough, it then follows from (3.19) that (V(Y)>-’

dy < +m

for N L 2 and any p > 0

BR(o)

(respectively, for 0 < p < 1 if N = 1) and, therefore, y(t) L 2 in QR,7. Iterating this argument, we obtain that y(t) L 2 for any t > 0 such that u (and u) remain positive up to such time. If Qr = BR(0) x (0, T) is such a cylinder, we should have there v, I Au - 2~-~. The fact that T cannot be arbitrarily (3.20) by comparison. n

(3.20)

large under our current hypotheses follows now from

Acknowledgements-The

author is very thankful to Drs A. A. Lacey, H. Ockendon and J. R. Ockendon for several interesting discussions during the preparation of the manuscript, and to A. Friedman for helpful suggestions in the presentation of the results. REFERENCES 1. WRIGHT T. W. & WALTER J. W., On stress collapse in adiabatic shear bands, J. Mech. phys. Solids 35(6), 701-720 (1987). 2. KAWARADA H., On solutions of the initial boundary value problem for r+ = U, + l/(1 - u), RIMS Kyoto U. 10, 729-736 (1975). 3. ACKER A. & WALTER W., The quenching problem for nonlinear parabolic equations, Lecture Notes in Mathematics, Vol. 564, pp. l-12. Springer, Berlin (1976). 4. Guo J. S., On the quenching behaviour of the solution of a semilinear parabolic equation, J. math. Analysis Applic.

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