Large time behavior of temperature in two-phase heat conductors
aa r X i v : . [ m a t h . A P ] F e b Large time behavior of temperature in two-phase heatconductors ∗ Hyeonbae Kang † Shigeru Sakaguchi ‡ Abstract
We consider the Cauchy problem for the heat diffusion equation in the wholeEuclidean space consisting of two media with different constant conductivities. Thelarge time behavior of temperature, the solution of the problem, is studied wheninitially temperature is assigned to be 0 on one medium and 1 on the other. We showthat under a certain geometric condition of the configuration of the media, temperatureis stabilized to a constant as time tends to infinity. We also show by examples thattemperature in general oscillates and is not stabilized.
Key words. heat diffusion equation, two-phase heat conductors, Cauchy problem, self-similar solu-tions, stabilization, oscillation.
AMS subject classifications.
Primary 35K05 ; Secondary 35K10, 35K15, 35B40, 35C06
This paper concerns with the Cauchy problem for the heat diffusion equation in the wholeEuclidean space which is occupied by two heat conducting media with different constantconductivities. It deals with the question of stabilization of temperature (the solution ofthe problem) as time tends to infinity when the initial data is given by the characteristicfunction of a medium. The reason why we introduce such a specific initial data is to clarifygeometry of the composite media. We find a condition on the configuration of the mediaunder which temperature converges to a constant as time tends to infinity. We also showby examples that temperature in general oscillates and does not converge. ∗ This research was partially supported by the Grant-in-Aid for Scientific Research (B) ( ♯ † Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212,S. Korea ([email protected]). ‡ Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, TohokuUniversity, Sendai, 980-8579, Japan ([email protected]). R N with N ≥ R N \ Ω constitute the two media. We suppose that ∂ Ω = ∅ and ∂ Ω isconnected. Denote by σ = σ ( x ) ( x ∈ R N ) the heat conductivity distribution of the wholemedium given by σ = σ + in Ω ,σ − in R N \ Ω , (1.1)where σ − , σ + are positive constants such that σ − = σ + . We consider the Cauchy problemfor the heat diffusion equation: u t = div( σ ∇ u ) in R N × (0 , + ∞ ) ,u = X Ω on R N × { } , (1.2)where X Ω denotes the characteristic function of the set Ω. We look for a geometric condi-tion on Ω such that the unique bounded solution u = u ( x, t ) to (1.2) tends to a constantas t → ∞ . As far as we are aware of, the question of stabilization for the multi-phase heatconductors has not been considered before. Recently, a geometric question related to thediffusion over such multi-phase heat conductors has been dealt with in [CSU, S].To gain a better understanding of the condition for the stabilization given later in(1.7), let us first consider conic regions. Let S N − be the unit sphere in R N . For a subset A of S N − such that ∂A = ∅ , we set Ω A to be the cone over A , namely,Ω A = { x ∈ R N : x = rω, r > , ω ∈ A } . (1.3)Denote by σ A = σ A ( x ) ( x ∈ R N ) the conductivity distribution of the whole medium given2y σ A = σ + in Ω A ,σ − in R N \ Ω A . (1.4)The following proposition can be proved using the self-similarity of the solution (seethe proof in the next section). Proposition 1.1
Let u A = u A ( x, t ) be the unique bounded solution of problem (1.2) with Ω , σ replaced by Ω A , σ A , respectively. It holds that < u A (0 , < and lim t →∞ u A ( x, t ) = u A (0 ,
1) (1.5) uniformly in x belonging to any fixed compact set in R N . We now present a condition on the shape of the domain which is sufficient for stabi-lization of the solution. Let A be a subset of S N − such that there is a point p ∈ A with − p A . For A ⊂ S N − and such a point p , we say A is starshaped with respect to p ∈ A if for every point ω ∈ A the shortest geodesic connecting ω and p in S N − is contained in A , or equivalently, if tω + sp ∈ A for any ω ∈ A and a pair of nonnegative numbers t, s such that tω + sp ∈ S N − . If A ⊂ S N − is starshaped with respect to p ∈ A , thenΩ A ⊂ Ω A − sp (1.6)for all s >
0. Here and throughout this paper Ω A − y denotes the translate of Ω A by y ,that is, Ω A − y = { x − y : x ∈ Ω A } . In fact, if x ∈ Ω A , then x = tω for some t >
0. Since A is starshaped with respect to p , x + sp ∈ Ω A for all s >
0, and hence (1.6) follows.The sufficient condition for the stabilization of the solution to (1.2) to hold on thedomain Ω is as follows: Ω A ⊂ Ω ⊂ Ω A − hp (1.7)for some h >
0. Roughly speaking, this condition means ∂ Ω lies in (Ω A − hp ) \ Ω, but isof arbitrary shape. For such domains we have the following theorem.
Theorem 1.2
If the region Ω in R N satisfies (1.7) for some h > , A ⊂ S N − and p ∈ A with − p A , where A is starshaped with respect to p , then the unique bounded solution u of problem (1.2) satisfies that lim t →∞ u ( x, t ) = u A (0 ,
1) (1.8) uniformly in x belonging to each compact set in R N . σ = σ ( x ), not necessarilytwo-phase conductivity distribution. Theorem 1.3
Let m ≤ M be positive constants. There exists a domain Ω in R N suchthat for any conductivity σ = σ ( x ) ( x ∈ R N ) satisfying < m ≤ σ ( x ) ≤ M for every x ∈ R N , (1.9) the unique bounded solution u of problem (1.2) satisfies that < lim inf t →∞ u (0 , t ) < lim sup t →∞ u (0 , t ) < . (1.10)We remark that since Theorem 1.3 is proved only using the Gaussian bounds for thefundamental solutions of diffusion equations (see (3.1)), it holds also for the diffusionequations of the form u t = N X i,j =1 ∂∂x i (cid:18) a ij ( x, t ) ∂u∂x j (cid:19) where the coefficients satisfy the following for some positive constants m, Mm | ξ | ≤ N X i,j =1 a ij ( x, t ) ξ i ξ j ≤ M | ξ | for x, ξ ∈ R N and t > . Such diffusion equations have been dealt with in [K].This paper is organized as follows. In section 2, we prove Proposition 1.1 and Theorem1.2 by introducing the one-parameter families of solutions { u kA } , { u k } as in [K]. Section 3is to prove Theorem 1.3. Proof of
Proposition 1.1. Note that the function u kA defined by u kA ( x, t ) = u A ( kx, k t ) for k > , σ replaced by Ω A , σ A , respectively. Thus it followsfrom the uniqueness of the solution of (1.2) that the solution u A is self-similar, namely, u A ( kx, k t ) = u A ( x, t ) (2.1)for every ( x, t ) ∈ R N × (0 , + ∞ ) and every k >
0, from which we infer that u A (0 , t ) = u A (0 ,
1) for every t > . (2.2)Note that 0 < u A < R N × (0 , + ∞ ) by the maximum principle.4y the H¨older estimate for u A as in [Z, p. 526], there exist constants C > < θ < | u A ( x, − u A (0 , | ≤ C | x | θ for every x ∈ R N . (2.3)It then follows from (2.1) that | u A ( x, t ) − u A (0 , t ) | ≤ C | x | θ t − θ for every ( x, t ) ∈ R N × (0 , + ∞ ) . (2.4)Combining (2.2) with this yields (1.5). Proof of
Theorem 1.2 . Let u = u ( x, t ) be the unique bounded solution of problem (1.2).As in [K, EKT], we introduce the one-parameter family of functions { u k } by u k ( x, t ) = u ( kx, k t ) for ( x, t ) ∈ R N × (0 , + ∞ ) and k > . Let us show that u k converges to the self-similar solution u A as k → ∞ uniformly in eachcompact set in R N × (0 , + ∞ ). By the maximum principle we have0 < u k ( x, t ) < x, t ) ∈ R N × (0 , + ∞ ) and every k > . (2.5)Moreover, each u k solves the problem (1.2) with Ω , σ replaced by Ω k , σ k , respectively,where Ω k = { x ∈ R N : kx ∈ Ω } and σ k = σ + in Ω k ,σ − in R N \ Ω k . As in [K, Lemmas 1 and 2], we have the following two lemmas which come from (2.5)together with the H¨older and energy estimates for solutions of parabolic equations ofsecond order with discontinuous coefficients (see, for example, [LSU, Chapter III]).
Lemma 2.1
For every τ > , there exist two constants C > and < θ < such that | u k ( x, t ) − u k ( y, s ) | ≤ C ( | x − y | θ + | t − s | θ ) for every ( x, t ) , ( y, s ) ∈ R N × [ τ, + ∞ ) and for every k > . Lemma 2.2
For every
T > and ρ > there exists a constant C > such that Z T Z B ρ (0) |∇ u k | dxdt ≤ C for every k > . Here, B ρ (0) = { x ∈ R N : | x | < ρ } . X Ω k ( x ) , σ k ( x ) converge to X Ω A ( x ) , σ A ( x ), respec-tively, as k → ∞ , for almost every x ∈ R N . Therefore, by using the diagonal process withthe aid of the estimates (2.5), Lemma 2.1 and Lemma 2.2, we infer from the uniquenessof the solution of problem (1.2) that u k converges to u A as k → ∞ uniformly in eachcompact set in R N × (0 , + ∞ ). Hence in particular u (0 , t ) converges to u A (0 ,
1) as t → ∞ .Moreover it follow from Lemma 2.1 that | u ( x, t ) − u (0 , t ) | ≤ C | x | θ t − θ for every ( x, t ) ∈ R N × (0 , + ∞ ) , which yields the desired conclusion (1.8). We utilize the Gaussian bounds for the fundamental solutions of diffusion equations dueto Aronson [A, Theorem 1, p. 891] (see also [FS, p. 328]). Let g = g ( x, ξ, t ) be thefundamental solution of u t = div( σ ∇ u ). Then there exist two positive constants λ < Λsuch that λt − N e − | x − ξ | λt ≤ g ( x, ξ, t ) ≤ Λ t − N e − | x − ξ | t (3.1)for all ( x, t ) , ( ξ, t ) ∈ R N × (0 , + ∞ ), where the constants λ, Λ depend only on N and thebounds m, M of σ .To construct the domain Ω with the desired property, we first choose two domains A, B in S N − such that A ⊂ B ( S N − and H N − ( B ) λ N +22 > H N − ( A )Λ N +22 , (3.2)where H N − is the standard ( N − α := H N − ( A ) and β := H N − ( B ) (3.3)for simplicity of expression. According to (3.2), we may choose a small number ε with0 < ε < − ε ) β + εα ] λ N +22 > [(1 − ε ) α + εβ ] Λ N +22 . (3.4)Let δ, R be two numbers such that 0 < δ < < R and Z δRδ e − s s N − ds = (1 − ε ) Z ∞ e − s s N − ds. (3.5)We then define a sequence of numbers { r n } by r = 0 and r n = δR n − , n ∈ N , (3.6)6nd a sequence of sets { E n } in R N by E k − = { x ∈ R N : x = rω, r k − ≤ r ≤ r k − , ω ∈ A } ,E k − = { x ∈ R N : x = rω, r k − ≤ r ≤ r k , ω ∈ B } for k ∈ N . At last, we define the domain Ω to be the interior of the set ∞ S n =0 E n .Since the initial condition of the problem (1.2) is the characteristic function of Ω, thesolution u ( x, t ) is given by u ( x, t ) = Z Ω g ( x, ξ, t ) dξ. It follows from (3.1) that for every t > λt − N Z Ω e − | ξ | λt dξ ≤ u (0 , t ) ≤ Λ t − N Z Ω e − | ξ | t dξ. (3.7)Let us calculate the both sides of (3.7). Using the notation (3.3), we have λt − N Z Ω e − | ξ | λt dξ = λt − N ∞ X k =1 "Z E k − e − | ξ | λt dξ + Z E k − e − | ξ | λt dξ = λt − N ∞ X k =1 " α Z r k − r k − e − r λt r N − dr + β Z r k r k − e − r λt r N − dr = λ N +22 ∞ X k =1 " α Z r k − √ λtr k − √ λt e − s s N − ds + β Z r k √ λtr k − √ λt e − s s N − ds . (3.8)Replacing λ by Λ yields thatΛ t − N Z Ω e − | ξ | t dξ = Λ N +22 ∞ X k =1 " α Z r k − √ Λ tr k − √ Λ t e − s s N − ds + β Z r k √ Λ tr k − √ Λ t e − s s N − ds . (3.9)Let us consider the sequence of times { t n } defined by t n = 1 λ R n − , n ∈ N . According to (3.7) and (3.8), we have u (0 , t n ) ≥ λ N +22 " α Z ∞ e − s s N − ds + ( β − α ) ∞ X k =1 Z r k √ λtnr k − √ λtn e − s s N − ds (3.10)Since β > α , we have u (0 , t n ) ≥ λ N +22 " α Z ∞ e − s s N − ds + ( β − α ) Z r n √ λtnr n − √ λtn e − s s N − ds By the definition (3.6) of r l , r n − √ λt n = δ and r n √ λt n = δR , and hence we have Z r n √ λtnr n − √ λtn e − s s N − ds = Z δRδ e − s s N − ds = (1 − ε ) Z ∞ e − s s N − ds, u (0 , t n ) ≥ λ N +22 ((1 − ε ) β + εα ) Z ∞ e − s s N − ds for every n ∈ N , and hencelim sup t →∞ u (0 , t ) ≥ λ N +22 ((1 − ε ) β + εα ) Z ∞ e − s s N − ds. (3.11)We now consider the sequence of times { T n } defined by T n = 1Λ R n − , n = 2 , , . . . . In the same way as above, one can show using (3.9) that u (0 , T n ) ≤ Λ N +22 ((1 − ε ) α + εβ ) Z ∞ e − s s N − ds, and hence lim inf t →∞ u (0 , t ) ≤ Λ N +22 ((1 − ε ) α + εβ ) Z ∞ e − s s N − ds. (3.12)Therefore, it follows from (3.4), (3.11) and (3.12) thatlim inf t →∞ u (0 , t ) < lim sup t →∞ u (0 , t ) . (3.13)Since α < β , we have from (3.7), (3.8) and (3.10) that for every t > u (0 , t ) ≥ λ N +22 α Z ∞ e − s s N − ds, (3.14)which shows that lim inf t →∞ u (0 , t ) > . (3.15)Observe that 1 − u solves problem (1.2) where Ω is replaced by Ω c = R N \ Ω, and the factthat Ω B c ⊂ Ω c with B c = S N − \ B . Here we used the notation (1.7) for Ω B c . Hence, bythe same argument as that for (3.14), we infer that for every t > − u (0 , t ) ≥ λ N +22 H N − ( B c ) Z ∞ e − s s N − ds, which yields that lim sup t →∞ u (0 , t ) < . (3.16)This completes the proof. 8 eferences [A] D. G. Aronson, Bounds for the fundamental solutions of a parabolic equation, Bull.Amer. Math. Soc., 73 (1967), 890–896.[CSU] L. Cavallina, S. Sakaguchi and S. Udagawa, A characterization of a hyperplanein two-phase heat conductors, arXiv: 1910.06757v1, Commun. Anal. Geom., toappear.[CT] M.-S. Chang and D.-H. Tsai, On the oscillation behavior of solutions to the heatequation on R nn