PROPERTIES OF THE BOUNDARY FLUX OF A SINGULAR DIFFUSION PROCESS

Chin. Ann. Math.25B:2(2004),175- 182.PROPERTIES OF THE BOUNDARY FLUX OFA SINGULAR DIFFUSION PROCESS***YIN JINGXUE*WANG CHUNPENG**AbstractThe authors study the singular difusion equation= div(p°IVu[P -2V2),(x,t)∈Qr=8x (0,T),wherehC R" is a bounded domain with appropriately snmooth boundary on, p(x) =dist(x,052) and prove that ifa≥p- 1, the equation admits a unique solution subjectonly, to a given initial datun without any boundary value condition, while if0 1,a> 0, and p(x) = dist(x, an). If a = 0, then the equation (1.1)becomes the evolutionary p Laplacian equation8研= div(VuP-2Vu). .In particular, if p = 2, then the equation is just the casscal heat conduction equation. For) + 2, it is more naturaltmusetthe equation to describe the heat conduction, since it reflectseven more exactly the physical reality, for example, ifp > 2, the solutions of such equationmay psss the properties of fnite speed of propagation of perturbations (see for example[1, 2). There are a tremendous amount of related works forsuch equations, see for example[1-8).Manuscript received April 29, 2003. Revised October 20, 2003.*Department of Mathematis, Jili University, Changchun 130012, China.Department of Mathematics, Jiin University, Changchun 13002, China.... Pprouaili: mawgeaijauedu.roject supported by the 973 Project of the Ministry of Science and Technology of China, the Outstand-ing Youth Foundation of China (No.10125107) and the Department of Mathematics of Jilin University.中国煤化工MYHCNMHG176YIN,J. x. & WANG, C. P.For the equation (1.1), the difusion cefficient depends on the distance to the boundaryand vanishes on the boundary. Thus the equation degenerates on the boundary. Thememoir by F. Tricomi [9], as well as subsequent investigations of equations of mixed type,elicited interest in the general study of elliptic equations degenerating on the boundary of thedomain. The 1951 paper of M. V. Keldy [10] played a significant role in the developmentof the theory. It was this paper that first brought to light the fact that in the case ofelliptic equations degenerating on the boundary, under definite assumptions a portion ofthe boundary may be free from the prescription of boundary conditions. Later, G. Fichera[11,12] and O. A. Oleinik [13, 14] developed and perfected the general theory of secondorder equations with nonnegative characteristic form, which, in particular, contains thosedegenerating on the boundary.The equation considered by Fichera and Oleinik is linear, and the second order deriva-tives of coefficients of principal part are bounded. They obtained the existence and unique-ness of solution for the Dirichlet problem, and investigated the properties of solutions too.Their results can be applied to the equation (1.1) withp= 2 and a 2 2, revealing that thereis no fux on the boundary no matter how the outer temperature varies.In this paper, we study the singular diffusion equation (1.1) withp> 1 anda> 0. Weare more interested in the behavior of the heat transfer process governed by (1.1) near theboundary. Since the diffusion coefficient vanishes on the boundary, it seems that there is noheat flux across the boundary. However, the fact might not coincide with what we image.The purpose of this paper will exhibit the fact, which is different from the usual imagination,that a, the exponent characterizing the vanishing ratio of the diffusion cofficient near theboundary, does determine the behavior of the heat transfer near the boundary. We will showthat if the ratio is relatively small, the outer temperature may affect the difusion process ofthe inner temperature of the object, while if the ratio is relatively large, there is no fux onthe boundary no matter how the outer temperature varies. Exactly, we will prove that p- 1is the critical value for the exponenta. If0 0. Similar to the theory for evolutionary pLaplacian equation, for anyus,0 stisying Uue,0∈L∞(M) and p'|Vus,0IP∈L'(M2), the above problem admits a uniqueweak solution e∈C(O,T; [2())0LP(0,T:;W1,(2)) with哉∈L2(Qr), in the sense that,for any test function ψ∈C(Qr), ue satisfies the fllowin integral equalityand (2.2), (2.3) hold in the trace sense.Lemma 2.1. Assumeuo∈L∞(2), p°|VuolP ∈L'(2),loL and IU2v+_01o"( are uniformly bounded, and Ue.o conerges to uo inWeat(). Then the wveak. soltion of the first initial boundary problem (2.1)-(2.3)4 isconuergent in [2(Qr) and the limit function is the weak solution of the equation (1.1) withinitial value condition (1.3).Proof. Using the maximum principle and a rather standard technique, we may easilyshow that there exists a contant C depending on lu010(, lIpPIVue ol(q), 9 andindependent of ε such thatIulL(Qr)≤C, .dxdt≤C.So there exist a function u and an n-dimensional vetor function ζ= (1... ,5n) such thatue∈(0,[;[2(2))∩L°(Qr)，H∈L(Qr)， 1ζ∈ Lp/(/ -)(Qr)Ft中国煤化工MYHCNMHG178YIN, J. X. & WANG, C. P.andue→uin L2(Qr),Vue→Vu in Le(Qr),oueBuor-oinL2(Qr),p°|Vue!P- 2Vue→ζ in Dp/(-lI)(Qr;R").In addition, u satisfes (1.3) in the trace sense. To prove that u satisfies the equation (1.1),we note that for any test function 4∈Co (Qr), the integral equality(2.4)holds, which implies, by lettingε→0, that(2.5)JJ (凯叶ζv咧=0It remains to show that for any 4∈C(Qr),[/&. pIuP-3Vu Vydrdt=/f. 7vyded.(2.6)Let0≤ψ∈Co(Qr) and ψ= 1 on suppp. Choosing4 = ψue in (2.4), we see thatp"u[VueP-ZVue. vbpdrdt.Let v∈C(0,T; L2(A))∩L∞(Qr) and p°|VP∈L(Qr). It is obvious thatJ[&,ψp:(VvueIP-2Vue - |V0|P -2V0) . (Vue - Vy)drdt≥0.Therefore-f{。ψP②l|Vue|P- -2Vue . Vudxdt-feψp°|Vv|P- -2V0. (Vue - Vr)dxdt. j[o (∞°-p)Vo1P° 2Vv:(Vu4e - Vo)derdt≥0.Lettingε→0 and noticing that$(ρ° - p%)\VvIP -2Vv. (Vue - Vo)dxdt|≤sup1。二2[[ 01vuP-IVvue - Vudrdt,(x,t)∈QrpAwe obtainuζ .Vpdxdt岵Jodad- &,;ψp°|VolP-2Vv.(Vu- Vv)drdt≥0.中国煤化工MYHCNMHGPROPFERTES OF THE BOUNDARY FLUX OF A SINGULAR DIFFUSION PROCESS79By the choice that中= bu in (2.5), we see thatTherefore时-O°IVp -2V0).(Vu - V)drdt20.(2.7)Choosingu=u- λ with λ> 0 in (2.7), we getwhich implies, by lttingλ→0, thatJS. ( -1u1-0叫) Vyulrd≥0.If we choose入< 0, we get the inequality with opposite sign. ThusJ(e, 姑-1叫Vyuda =0.Noticing that ψ= 1 on suppp, we see that (2.6) holds. The proof is complete.Proof of Theorem 1.1. We first prove the existence. For all e > 0, choose ue ,0 suchthat |ue ol() and lpl|Veu,1I'L+(q) are uniformly bounded, and Ue,0 converges to uo inWoet(Q). Let ue be the weak solution of the first initial-boundary problem (2.1)- (2.3). FromLemma 2.1, we see that Ue is convergent in L*(Qr) and the limit function u satisfies theequation (1.1) with the initial condition (1.3). Now we prove u also satisfes the boundarycondition (1.2), thus u is the weak solution of the first initial-boundary problem (1.1)-(1.3).Since门< 1 andp-a> 1, there exists a constantβ∈(*,1) such thatp-号>1. .Sinceβ< 1 andp-号> 1, there exists a constant γ∈(1,p- 9) such that βr< 1. Therefore(vu!dadt = Ji.。K 1vu.["dxdt{(z.t)∈Qr:?1vu[>1}|Vue|"dxdts{/{e;p587dxdt+ 10 pl|vu.1/B+/dxdt≤{]。osnhradt+fp:(1l + |Vu.{P)dxdt≤C,where C is a positive constant independent of e. Thus Vue is uniformly bounded in L"(Qr).So u satisfies the boundary condition (1.2).Now, we prove the uniqueness. Let u and。be two weak solutions of the first initial-boundary problem (1.1)-(1.3). From the definition of solutions, we see thatpolu-)drdt=- 1f。 o(VuP-Vu- IvP-2V0). VvpdxdtSt中国煤化工MHCNMHG180YIN,JX. & WANG, C. P.holds for any∈Cg(Qr). For any fixed s∈[0, T], after an approximate procedure, we maychoose X0.s(u -川) as a test function in the above equality, where X[0,a] is the characteristicfunction on [0, s]. ThusJ[。(u - 0)0(4-0)dadt8t=-|p°(|VyIP -2Vu- |VP -2V0) . V(u- v)dxdt≤0,and hence(u(x,s) - o(,))2dx= I (u(x,0) - (,0)2dx +u一),=2/)[.(u-n)2(uθtdxdt≤0,which implies thatu(x,8) = o(x,8),a.e. (x,8)∈ Qr.The proof is complete.Proof of Theorem 1.2. The existence has already been done in Lemma 2.1. Weneed only to show the uniqueness. DenoteSe= {x∈S | dist(x,0n)>e}.Letξε∈Co(Ne) such thatξe=1on Sl2ze,0≤ξε≤1 and1v&eI≤σ,where C is a constant independent ofe. Let u and v be two weak solutions of the equation(1.1) with initial value (1 3). From the definition of solutions, we get2dxdt=- I1。. °(VuI 2Vu_ IVvP -2V0). Voldritfor any 4∈Co(Qr). For any fixed s∈[0,T], after an approximate procedure, we maychoose X[0,j(u- v)ξe as a test function in the above equality, where X[0.a! is the characteristicf(u-oxs.u 2dardt8=-1[。p*(|Vu1P-2Vu - |V01P 2V0). V((u - o)e。)dxdt.中国煤化工MYHCNMHGPROPERTIES OF THE BOUNDARY PLUX OF A SINGULAR DIFFUSION PROCESS181Therefore(u(x,8) - (x,))2Eedx= I (u(x,0) - 0(x, 0))°ξe dxdt-21/.Sep°(IVulp-2Vu- |VP 2V0) . V(u - v)dxdt-2/1 (u- 0)p(IVu[P- 2Vu - |VvP 2V0). VEedxdt≤2// |u - oIp°(VuI -1 + |Vo1p-)|V&,ldxdtp°(VuP + |Vo1P)dxdt)') (p-1)/pp1y6e]Pdad)/D(p-1)/p≤Ce(a+1-D)/0( [" f。p"(VuP + Ivup)dard)(中- 1)/psc(。" lhim p(w +10))“where C isa constant independent ofe. Since p°|VulP,p°|VIP∈L'(Qr), lttinge→0,we see that(u(x,8)- (x,))2dx ≤0,which implies thatu(x,8)=v(x,s)，a.e. (x,s)∈ Qr.The proof is complete.References1] Kalashnikov, A. S., Some problems of the qualitative theorynonlinear degenerate second orderparabolic equations, Russian Math. Surveyse, 42:2(1987), 169 222.22] DiBenedetto, E., Degenerale Parabolie Equations, Springer-Verlag, New York, 1993.3] Bouguima, s. M. & L.akmeche, A., Mutiple slutins of a nonlinear problem involving the plapacian,Comnm. Appl. Nonlinear Anal, 7:3(2000), 83 96.4 ] Nabana, E., Uniqueness for positivesolutions of p-Laplacian problem in an annulus, Ann. Fac. Sci.Toulouse Math., 81(999), 143-154.5] Reichel, w. & Walter, W., Radial solutions of equations and inequalities involving the p-Laplacian, J.Inequal. Appl, 1:1(1997), 47-71.{6} Dambrosio, w. Muliple solutions of weakly-coupled systens with p-Laplacian operators, ResulsMath, 36:1-2(1999), 34- 54.中国煤化工MYHCNMHG182YIN,J. x. & WANG, C. 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