Numerically repeated support splitting and merging phenomena in a porous media equation with strong absorption. Kenji Tomoeda

Transcription

1 Journal of Math-for-Industry, Vol. 3 (C-), pp. Nuerically repeated support splitting and erging phenoena in a porous edia equation with strong absorption To the eory of y friend Professor Nakaki. Kenji Tooeda Revised on October 5, Abstract. Nonlinear diffusion equations exhibit a wide variety of phenoena in the several fields of fluid dynaics, plasa physics and population dynaics. Aong these the interaction between diffusion and absorption suggests a rearkable property in the behavior of the support of solution; that is, after support splitting phenoena appear, the support erges, and thereafter the support splits again. Moreover, nuerically repeated support splitting and erging phenoena are observed. In this paper, aking use of the properties of the particular solutions, we construct an initial function for which such phenoena appear. Keywords. nonlinear diffusion, support splitting, support erging, finite difference schee. Introduction The interaction between diffusion and absorption appears in the flow through an absorbing ediu occupying all of R, and is described in the for of the following initial value proble for the degenerate parabolic equation with an additional lower order ter: v t = (v ) xx cv p, x R, t >, () v(, x) = v (x), x R, () where >, < p <, c is a positive constant, v denotes the density of the fluid, cv p describes voluetric absorption, and v (x) C (R ) is nonnegative and has copact support. This equation () is called the porous edia equation with absorption. In the behavior of solutions of ()-() there appear total extinction in finite tie and finite propagation of the support, which are caused by the interaction between diffusion and absorption. Moreover, nuerical support splitting and erging phenoena are observed (see Fig.). Fro analytical points of view, the existence and uniqueness of a weak solution, the total extinction in finite tie and the finite propagation of the support are proved by Oleinik, Kalashnikov and Chzou[], Kalashnikov[5, ] and Knerr[9]. Rosenau and Kain[] tried the nuerical coputation to ()-() and suggested the possibility of the support to split. Chen, Matano and Miura[] constructed the initial function for which the support of the solution splits into ultiple connected coponents in a finite tie. Fro a nuerical point of view, Nakaki and Tooeda[] constructed the finite difference schee which realizes such a phenoenon, and obtained the sufficient condition iposed on v (x) for which the support splits. Kersner[] proved the support splitting property in the initial-boundary value proble to () by constructing supersolutions. For support splitting and erging phenoena, to the best of our knowledge, we have not been able to find any results. This otivates us to investigate such phenoena. The difficulty is to deterine whether or not the support splitting and erging phenoena occur, because the convergence of nuerical solutions does not always iply that the exact solution also keeps the sae phenoena. In this paper, our approach is based on the properties of the particular solutions of () under the following Assuption A. + p = ( >, < p < ). Unfortunately, in the case where + p ( >, < p < ), we are unable to construct the explicit solution of (). This is the reason why we are concerned with the specific case stated in Assuption A. Figure : An illustration of support splitting and erging phenoena.

2 Journal of Math-for-Industry, Vol. 3 (C-). Behavior of nuerical support Setting u = v, we rewrite () () as follows: u t = uu xx + (u x) ( )cχ u>}, (3) u(, x) = u (x) (v (x)). () We note that the effect of absorption is expressed as the constant ( )c under Assuption A. Our difference schee approxiates the proble (3) () instead of () ()(see []). Now we show soe nuerical solutions of u n h to (3)-() with =.5 and c = 3 in the following figures. The convergence of the nuerical solutions to the exact one of ()-() is established (see Theore 3. in []). Nuerically repeated support splitting and erging phenoena are observed in the neighbourhood of x =. This initial function is constructed so that the assuption in Theore is satisfied, which iplies that the occurrence of such phenoena is justified fro analytical points of view. In the following we show the construction of it. Initial function. t =. Connected t =.57D Merged t =.73D + Split t =.75D + Merged A close up of the previous figure t =.D + Split t =.D + Merged t =.757D Split t =.37D + Split, and Extinction.7 A close up of the previous figure Figure : Nuerical support splitting and erging phenoena for (3), where =.5 and c = 3.

3 Kenji Tooeda 3 3. Particular solutions and support splitting phenoena Under Assuption A let us construct Kersner s solution with support expanding property([7]) and Galaktionov and Vazquez s solution with support splitting property([3, ]) in the for u(t, x) = f(t) + g(t)h(x) to (3). Then it holds that f + g h = (f + gh)gh xx + (gh x) ( )c, (5) where denotes the partial derivative with respect to t. Putting h(x) = x, we have the syste of differential equations f = fg ( ( )c, ) g = + g. Solving (), we have Kersner s solution with u(, x) = [( (x/ρ) )] + for two paraeters (> ) and ρ (> ): (see Fig. 3) u(t, x) = A + ( + a)t} [B A + ( + a)t} + DA + ( + a)t} x ], + (7) where [g] + = axg, }, A A(ρ, ) = ρ, () B B(, c, ρ, ) = ( + DA)A +, D D(, c) = c a, a =. () The extinction tie T (, c, ρ, )(> ) and the support [x (t), x + (t)] ( t T ) are given by and T (, c, ρ, ) = + a ( B D ) + } x ± (t) = [ ± B A + ( + a)t} + DA + ( + a)t} ] + respectively. Lea. T (, c, ρ, ) satisfies T (, c,, ρ) ( )c and li T (, c,, ρ) = ρ ( )c (9) }, () (see F ig. 3). () Putting h(x) = x in (5), we siilarly obtain Galaktionov and Vazquez s solution with u(, x) = εx + ˆ for two paraeters ε > and ˆ > : u(t, x) = E ( + a)t} [ DE ( + a)t} + G E ( + a)t} where E E(ε) = ε DE)E x ] +, () and G G(, c, ˆ, ε) = (ˆ u (x) }} ρ Figure 3: The initial function of Kersner s solution(7). v Lea. G satisfies t x x Figure : Kersner s solution. li G(, c, ˆ, ε) =. (3) ε Moreover, in the case where G(, c, ˆ, ε) < the following inequalities ˆ ( )c < ˆt(, c, ˆ, ε) < ˆT (, ε) and ˆ li ˆt(, c, ˆ, ε) = () ε ( )c hold and u satisfies (see Fig. 5 and ) ( ) u(t, x) > (t, x) [, ˆT (, ε)) R \ S, (5) where u(t, x) = ((t, x) S), () li u(t, ) =, (7) t ˆT (,ε) li u(t, x) = (x ), () t ˆT (,ε) ˆt(, c, ˆ, ε) = E + a ( G D ) + }, (9) E ˆT (, ε) = + a, () ) S = (t, x) t [ˆt(, c, ˆ, ε), ˆT (, ε), + x E ( + a)t} [ G D E ( + a)t} ] } +.()

4 Journal of Math-for-Industry, Vol. 3 (C-) εx + ˆ ˆ x Figure 5: The initial function of Galaktionov and Vazquez s solution (). t ˆT ˆt Blow-up (x ) u = u > u > x Figure : The support of Galaktionov and Vazquez s solution (). Leas and are shown by siple calculations. In Lea the appearance of the region S for a sufficiently sall ε iplies the support splitting phenoena. As a consequence of these leas, we obtain Theore. For arbitrary nubers and ˆ ( > ˆ > ) there exist constants ρ(> ) and ε(> ) such that < ˆt(, c, ˆ, ε) < T (, c,, ρ). () Moreover, for these constants ρ and ε suppose u (x) satisfies [ ((x ± ξ)/ρ) }] u (x) εx + ˆ (3) + for soe ξ(> ). Then there exists t (ˆt(, c, ˆ, ε) < t < T (, c,, ρ) ) such that supp u(t, ) splits into at least two disjoint sets on the interval [ ξ, ξ]. Reark. The last assertion of the theore is proved by the coparison theore([]) which is concerned with the initial functions. Reark. The inequality () is satisfied for a sufficiently sall nuber ε and a sufficiently large nuber ρ.. Support splitting and erging phenoena Using the initial functions of Galaktionov and Vazquez s solution and Kersner s solution, we construct an initial function () for which support splitting and erging phenoena appear. Let two functions εx + ˆ and [( ((x ± ξ)/ρ) )] + be tangent to each other at soe points (±ˆx, γ) (see Fig. 7). ˆx ρ }} εx + ˆ ˆ γ }} α ξ ξ α x Then we have ξ = ( ˆ) [ ˆx ρ Figure 7: The initial function u (x). u (x) ( ) ε + A, () ((x ± ξ)/ρ) }] + εx + ˆ on R, (5) where ξ ξ(, c, ε, ˆ,, ρ). Fro () it follows that there exists t = t at which x ±(t) attains the axiu; that is, t t (, c,, ρ) [ ab = + a D( + a) x ±(t ( + a)b ab ) = + a D( + a) We can iediately state } + ], () }. (7) Theore. Assue that there exist positive constants ˆ, ε, (> ˆ) and ρ such that < ˆt(, c, ˆ, ε) < t (, c,, ρ), () ξ < x ±(t (, c,, ρ)). (9) Let the initial function () satisfy (3). Then there exists t (ˆt(, c, ˆ, ε) < t < t (, c,, ρ) ) such that supp u(t, ) splits into at least two disjoint sets on the interval [ ξ, ξ]. Moreover, [ ξ, ξ] supp u(t (, c,, ρ), ), which iplies that the support is erged on [ ξ, ξ]. In the following let us consider the paraeters ε, and ρ satisfying () for an arbitrarily given nuber ˆ(> ). Let (> ˆ) be an arbitrarily fixed nuber. According to Reark of Theore we can take ε sufficiently sall and ρ sufficiently large so that () holds, which iplies the occurrence of the support splitting property. However, it sees difficult for us to prove that the second inequality of () holds for such a sufficient large nuber ρ. The reason of the difficulty coes fro the following fact: t (, c,, ρ) = T (, c,, ρ) ( a + + a + a ) + } ( B D ) +, (3) which yields li ρ + t (, c,, ρ) =, (3)

5 Kenji Tooeda 5 where B B(, c, ρ, ) is given by (). The second inequality of () ay fail. To avoid such a difficulty we have to take ρ and sufficiently large so that not only the second inequality of () but also (9) hold. To find the relation between ρ and we tried nuerical coputation in the cases where the ratio ρ = constant and where the ratio ρ in the latter case. Then we have = constant. We obtain the good result Theore 3. For an arbitrary nuber ˆ(> ) there exist ε ε(, c, ˆ)(> ) and (, c, ˆ, ε)(> ) satisfying () and (9). Proof. The first inequality of () can be shown by putting ε ε(, c, ˆ) < Ḓ. To prove the theore it suffices to show the second inequality of () and (9) for a positive constant satisfying the following inequality: ( + > A + )} D( + )A DA, (3) ε where A ρ = constant. Fro (), (3) and Assuption A we have t t (, c,, ρ) [ } + ] ( + DA) = A + a D( + ) [ ( + > A + )} + ] A + a ε [ ( ) + + = + a A ( A + θ ) } ] A ε ε ( ) > + a ε > ε( + a) = ˆT (, ε) > ˆt(, c, ˆ, ε), (33) where θ is a positive constant. Thus the second inequality of () holds. Fro (), (7) and (3) it follows that x ±(t ) = ( + a)( + DA)A + + a a( + DA) D( + a) } A (+) > + DA + D( + ) ( > A + ) ε } A A A = which iplies (9), and the proof is coplete. ( A + ) > ξ, (3) ε The following theore iediately follows fro Theores and 3. Theore. Let > and N( ) be arbitrary fixed nuber and integer, respectively. Then we can choose the sequence ε k, k, ρ k }(k =,,, N) satisfying < ˆt(, c, k, ε k ) < t (, c, k, ρ k ), (35) ξ (, c, ε k, k, k, ρ k ) < x ±(t (, c, k, ρ k )), (3) where ρ k = A k. Moreover, for the initial function u (x) satisfying [ k ((x ± ξk )/ρ k ) }] + u (x) ε k x + k on R (k =,,, N), (37) the support splitting and erging phenoena appear at least N ties on the interval [ ξ, ξ ], where ξ k = ξ(, c, ε k, k, k, ρ k ) (k =,,, N). We show soe nuerical paraeters such that (35)- (37) hold in the case where N = 3. Here ˆt k = ˆt(, c, k, ε k ), ξ k = ξ(, c, ε k, k, k, ρ k ), and t k = t (, c, k, ρ k ) (k =,, 3). Let us explain the paraeters in Table. For a given constant ˆ = =.3, we can choose ε = ε =.9 so that the first inequality of (35) is satisfied. Then Galaktionov and Vazquez s solution takes zero at x = and t = ˆt =.9, and supp u(ˆt, ) begins to split into at least two disjoint sets. On the other hand, we can choose = =.9 satisfying (3) with ε = ε. The support of Kersner s solution with = onotonously expands as t t =. > ˆt, and is given by [ξ x + (t ), ξ + x + (t )] = [.35.3, ] = [.,.3]. Thus supp u(t, ) [ ξ x + (t ), ξ + x + (t )] holds and the support of u is erged at t = t on the interval [ ξ, ξ ]. Siilarly, we can choose ε i, ˆt i, i, ξ i, and t i (i =, 3), which iplies that the support splitting and erging property appears at least 3 ties on [ ξ, ξ ]. We show the initial function u (x) in Figures and 9. Table : Nuerical exaples related to Theore. =.5, c = 3., A = =.3 ε =.9 ˆt =.9 =.9 t =. x+(t ) =.3 ξ =.35 ρ =.7 ε =.7 ˆt =.9 =.9 t =.3 x +(t ) = 5.73 ξ = 5.99 ρ =.9 ε =. ˆt =.5 3 =.3 t 3 =.333 x +(t 3 ) = 9.9 ξ 3 = 7. ρ 3 = 3.59

6 Journal of Math-for-Industry, Vol. 3 (C-) 3 ε x + ε x + ε x + u u u Figure : The initial functions of Galaktionov and Vazquez s solution and Kersner s solution, and u (x), where =.5 and c = ε x + ε x + u. u Figure 9: A close up of Figure. 5. Nuerical support in the initial-boundary value proble We show the nuerical behavior of the support of the solutions of (3) with the conditions: u(, x) = u (x), x (.5,.5), (3) u(t, ±.5) = φ(t), t >. (39) Put =.5 and c =. We consider the following three cases. Case (I). u (x) =.5 and φ(t) =.5; Case (II). u (x) =. and φ(t) = cos(πt); Case (III). u (x) =. and φ(t) = cos(πt). In Case (I) the nuerical solution converges to the solution ū(x) = x ( x.5) as t, which is derived fro a(ū x ) ( )c = on the right side of (3). If u (x) < ū(x) and φ(t) <.5, the coparison theore yields the appearance of the support splitting phenoena. If u (x) >.5 and φ(t) >.5, the support never splits. In Case (II) nuerically repeated support splitting and erging phenoena are observed (see Figure ). The boundary value φ(t) with the period takes the axiu. and the iniu.. On the other hand, the period is in Case (III) and is less than that in Case (II). In this case the support splitting phenoena are not observed (see Figure.). The nuerical coputation suggests that the appearance of the support splitting and erging phenoena depends on the period of φ(t). So, the atheatical analysis for such phenoena is needed.

7 Kenji Tooeda 7.5 t =..5 t =.5.5 u =.5 u = t =.5.5 t =.5.5 u =.5 u = t =.75.5 t =.55.5 u =.5 u = t =..5 t =.57.5 u =.5 u = t =.7.5 t =..5 u =.5 u = t =.95.5 t =.5.5 u =.5 u = Figure : Nuerically repeated support splitting and erging phenoena in Case (II), where =.5 and c =. Figure : Nuerical support no-splitting phenoena in Case (III), where =.5 and c =.

8 Journal of Math-for-Industry, Vol. 3 (C-) Acknowledgeents This work was supported by Japan Society for the Prootion of Science(JSPS) through Grant-in-Aid(No.53) for challenging Exploratory Research. Kenji Tooeda Osaka Institute of Technology, 5--, Oiya, Asahi-ward, Osaka, , Japan E-ail: tooeda(at)ge.oit.ac.jp References [] Bertsch, M.: A class of degenerate diffusion equations with a singular nonlinear ter, Nonlinear Anal.,7 (93) 7 7. [] Chen, X.-Y., Matano, H., and Miura, M.: Finitepoint extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption, J. reine angew. Math., 59(995) 3. [3] Galaktionov, V. A., and Vazquez, J. L.: Extinction for a quasilinear heat equation with absorption I. Technique of intersection coparison, Coun. in Partial Differential Equation,, 9(99) 75. [] Galaktionov, V. A., and Vazquez, J. L.: Extinction for a quasilinear heat equation with absorption II. A dynaical systes approach, Coun. in Partial Differential Equation,, 9(99) [5] Kalashnikov, A. S.: The propagation of disturbances in probles of non-linear heat conduction with absorption, Zh. Vychisl. Mat. i Mat. Fiz., (97) [] Kalashnikov, A. S.: Soe probles of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys, (97) 9. [7] Kersner, R.: The behavior of teperature fronts in edia with nonlinear theral conductivity under absorption, Vestnik. Mosk. Univ. Mat., 33 (97) 5. [] Kersner, R.: Degenerate parabolic equations with general nonlinearities, Nonlinear Anal., (9) 3. [9] Knerr, B. F.: The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one diension, Trans. of the Aer. Math. Soc., 9 (979) 9. [] Nakaki, T., and Tooeda, K.: A finite difference schee for soe nonlinear diffusion equations in an absorbing ediu: support splitting phenoena, SIAM J. Nuer. Anal., () [] Oleinik, O.A., Kalashnikov A. S., and Chzou, Y.-L.: The Cauchy proble and boundary value probles for equations of the type of nonstationary filtration, Izv. Acad. Nauk SSSR Ser. Mat., (95) 7 7. [] Rosenau P., and Kain, S.: Theral waves in an absorbing and convecting ediu, Phys. D (93) 73 3.