A NONLINEAR STEADY STATE TEMPERATURE PROBLEM FOR A SEMI-INFINITE SLAB

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1 A NONLINEAR STEADY STATE TEMPERATURE PROBLEM FOR A SEMI-INFINITE SLAB DANG DINH ANG We prpse t investigate the fllwing bundary value prblem: (1) wxx+wyy = 0,0<x<ir,0<y<<x>, (2) w{0,y)=fi(y), 0^y<cc, (3) w(ir, y)=ft(y), Ogy<, (4) -wv(x,0)=g[x;w(x,0)],0<x<ir, where/1,/2 are real-valued cntinuus functins n [O, w [vanishing at infinity, and G is a nnlinear real-valued functin f tw real variables satisfying the fllwing cnditins: G[x; u] is strictly decreasing in u, G[x; c] =0 where c is a cnstant such that (5) c^supsupx(fi(x),f2(x)), (6) G[x; u] is cntinuus in x and u jintly. Physically, w is the temperature, and the cnditin (4) with G satisfying (5) and (6) is a generalizatin f Newtn's law f cling; wv(x, 0) =l(c w(x, 0)), land c0 being cnstants. Mann and Blackburn [l ] investigated a particular case f the freging prblem crrespnding t/: = 0=/2, c = l, and G independent f the first argument x. The reasn we cnsider this prblem again is tw-fld: first, the argument given in [l] in the prf f the existence therem (lc. cit. Therem 3) des nt directly carry ver t the present mre general case, and secnd, we wish t present a methd fr cnstructing the slutin in certain particular cases, whereas the authrs in [l] were mre cncerned with existence therems. Our main tl is a fixed pint therem f H. Schaefer [2]. By a slutin f the bundary value prblem (l)-(6), we mean a functin w harmnic in the pen strip, satisfying (2)-(6), cntinuus in O^x^ir, 0^y<c, and vanishing at infinity. We pstulate w t be f the frm (7) w = wi+w2, where wu w2 are harmnic in the pen strip, and (8) wi(0, y)=wi(ir, y)=0, 0^y< 00, (9) -wi,y(x, 0)=G[x; w(x;0)], 0<x<ir, (10) w2(0,y)=fi(y), (11) w2(ir, y)=f2(y), (12) wtjx, 0)=0, 0<x<t. By prceeding as in [l], ne finds Received by the editrs September 21,

2 304 DANG DINK ANG [February l r l -J lg- 1 2 exp( y) cs(x z) + exp( v) Wi(x, y) (13) V ir J exp( y) cs(x + z) + exp( y) G[z;w(z;0)]az, 0<x<tt, 0<;y<, We nw seek an integral representatin frmula fr w2. We use the technique f Furier transfrms. Our peratins with Furier transfrms are purely frmal. We are cncerned with the final representatin frmula which will be justified fr its sake. Let (14) }i(w) = f fi<s) cs wçdç, }2(w) = f f2(t) cs wtdt. 7T t/ T J 0 Nte that the pssibility that these integrals diverge des nt cncern us. Let (A(w) exp(iwy wx) + B(w) exp(iwy w(w x)))dw. Then by (10) and (11), and frmally using the Furier inversin frmula, we get }i(w) cs wydw, /, 00 f2(w) cs wydw. /I 00 Equating»2(0, y) = fi(y), w2(v, y) = f2(y), gives (18) A(w) + B(w) exp ( wir) =}i(w), (19) A(w) exp( wir) + B(w) =J2(w). Hence (20) A(w) = A(w)(]i(w) f2(w) exp( wt)), (21) B(w) = A(w)(J2(w) -}i(w) exp(-wtr)), where (22) A(w) = (1 - exp(-2w))-1.

3 í969] A NONLINEAR STEADY STATE TEMPERATURE PROBLEM 305 In view f (15), (20), (21), we have 2 (23) w2(x, y) = f (fi(t)d(x, y; f) +f2( )E(x, y; t))dt, ir J a where (24) /, 00 A(w)(exp( wx) cs wy cs iffdf, exp( 2wir + wx)) (25) /> 00 A(w)(exp( w(ît x)) exp( w(ir + x))) cs wy cs wf áf. Lemma 1. (i) w2 is harmnic in the pen strip. (ii) Umy-,W2(x, y) exists fr O^xsáir and defines a functin cntinuus n [0, ir]. (iii) limv_ w2,y(x, y)=0, 0<x<7r. (iv) limx-. w2(x, y) =/i(y), linix^ w2(x, y) =/2(y). Prf. The prf f (i), (ii), (iii) is straightfrward. The prf f (iv) prceeds by apprximating the kernels D and E by the Pissn kernel. Details are mitted. In view f Lemma 1 and equatin (13), the bundary value prblem (l)-(6) reduces t the fllwing integral equatin (26) 1 ct u(x) = I G[z; u(z)]k(x, z)dz + w2(x, 0), 0 g x â t, IT J 0 where u(x) =wi(x, 0) and 1 cs(x z) (27) K(x, z) = lg 1 cs(x + z) Define 1 CT (28) F«(x) = I G[z; u(z)]k(x, z)dz + w2(x, 0), 2ir J 0 Ú x g Then «is a slutin f (26) if and nly if «is a fixed pint f T. In the sequel, ur arguments will be cuched in the language f fixed pint therems. Let E be the Banach space f cntinuus functins n [0, jr]. Then it is clear that T takes E int E. It is als clear that T is cmpletely cntinuus, since the functin lg(l cs(x±z)) is integrable in z.

4 306 DANG DINH ANG February Lemma 2. T has at mst ne fixed pint. The prf f this lemma prceeds exactly as in the prf f Therem 4 f [l]. Lemma 3. Let T\=\T, 0<Xgl. Prf. Let If u is a fixed pint f T\, then inf inf(fi(x),f2(x)) ^ug c. s X r * 1 2 exp( y) cs(x z) + exp(-y) v (x, y) = I lgit J 1 2 exp( y) cs(a; + z) + exp( y) Then, by Lemma 1, G[z;w(z, 0)] dz + Xw2(x, y). limv(x,y) = Xfi(y), x >0 limt>(s,y) = X/2(y), - lim (*,y) = G[x,u(x)\. v-> ay T prve the duble inequality f the lemma, we use the maximum mdulus principle fr harmnic functins. (i) u is ^ c. Suppse n the cntrary that u(x)>c fr sme x. Let xm be such that u(xm) =max u(x). Then G[xm; u(xm)] <0 by the definitin f G, since u(xm)>c. On the ther hand, v is nt cnstant. Hence, by the maximum principle, -y-l(v(xm, y) - v(xu, 0)) > 0, y > 0. Since vy(xm, Q)=G[xm, u(xm)], we have a cntradictin. This cntradictin prves (i). (ii) m is ^inf* inf (fi(x),f2(x)) = a. Suppse n the cntrary that fr sme x, we have u(x) <a. Let xm be such that u(xm) = inf u(x). Then u(xm) =inf v(x, y) by the maximum principle. The prf prceeds n the same lines as in part (i). We nw state ur existence therem. Therem 1. T has a unique fixed pint u. Furthermre, a^u^c. Prf. Suppse that T has n fixed pint. Then by a therem f Schaefer [2], there exist a sequence (w ) f elements f E and a se-

5 1969] A NONLINEAR STEADY STATE TEMPERATURE PROBLEM 307 \ quence f real numbers 0<X <1, such that un = \ntu, w But by Lemma 3, a^un^c. This cntradictin prves that T has a fixed pint u, which is unique by Lemma 2. Furthermre, a^u^c, by Lemma 3. Thus the therem is prved. Therem 2. The bundary value prblem (l)-(6) has a unique slutin. Prf. Each fixed pint f the peratr T gives rise t a slutin f the bundary value prblem (l)-(6) in an bvius way. Hence by Therem 1, the given bundary value prblem has a slutin. Let v(x, y), v'(x, y) be tw slutins f the bundary value prblem. Let w(x, y) v(x, y) v'(x, y). Then w vanishes n the edges x = 0, 0^y<«>, and x = 7r, 0^y<. Using the maximum mdulus principle, and the prperties f the functin G[x, u(x)], we can prve that w vanishes n the base O^x^x. Then, again by the maximum mdulus principle, w must be the null functin. This prves the therem. The prblem f actually cnstructing the slutin f the prblem is ne f great interest. If T is a cntractin, i.e., if Fm Tu'\\ a\\u u'\\, 0<a<l, then the fixed pint f T can be btained by successive apprximatin. We shall shw that even when T is nly nnexpansive, i.e., Fm Tu'\\ ^ \\u u'\\, the fixed pint f T can still be btained by successive apprximatin. Therem 3. Let T be nnexpansive. Then the fixed pint u f T can be btained by successive apprximatin. Mre precisely, let Tn=a T, 0<«<1. Ifan >1 and a»0, then lim T%w = ufr any w in E. Prf. By T we mean f curse the Mth iterate f Tn. Fr each n, Tn is a cntractin, and hence has a fixed pint u, un = Tnun. By Lemma 3, the sequence (un) is bunded, and, by the abve relatin, is relatively cmpact. Hence (un) has a cluster value. Nw, each cluster value f (m ) is a fixed pint f T. Since by Therem 1, T has a unique fixed pint, it fllws that (u ) has a unique cluster value u. Since (a,) is relatively cmpact, (un) cnverges t u. Nw, if w is any element f E, Tnw un\\ = tiw TnUn\\ ^ an\\w m. Since (u ) is bunded, the therem fllws.

6 308 DANG DINH ANG Acknwledgements. The authr had several helpful discussins with Prfessr Len Knpff n the subject matter f this paper. References 1. W. R. Mann and J. F. Blackburn, A nnlinear steady state temperature prblem, Prc. Amer. Math. Sc. 5 (1954), H. Schaefer, Über die Methde der a priri-schranken, Math. Ann. 129 (1955), University f Saign and University f Califrnia, Ls Angeles