A qualitative description of microstructure formation and coarsening phenomena for an evolution equation

A qualitative description of microstructure formation and coarsening phenomena for an evolution equation Tom´as Caraballo and Renato Colucci Abstract. We provide a qualitative description of microstructure formation and coarsening phenomena for the solutions of a singularly perturbed fourth order evolution equation arising in the study of phase transitions. In particular we study stationary and traveling wave solutions and we construct a class of approximate solution which mimics the principle features of the dynamics. Finally we present several simulations in order to illustrate the results. Mathematics Subject Classification (2010). Primary 35B36, 35B10, 35B25, 35B40; Secondary 82C26. Keywords. Forward-Backward Diffusion, Phase Transitions, Traveling waves. 1. Introduction Forward-backward equations ([12], [18] and references cited therein) arise in many contexts such as nonlinear elasticity ([15]), image processing and phase transitions, the interest of researchers in this problem is motivated by the rich observed phenomenology. The evolution equation ut=W00(ux)uxx,(1.1) is ill-posed due to the change of sign of the diffusion coefficient (see [6]), or analogously, the corresponding minimum problem for the energy F(u) = ZI W(ux)dx, admits infinite many minimizers. Here, the function W(p) = (p2−1)2is the so called double well potential and I= (0,1). This work has been partially supported by FEDER and the Spanish Ministerio de Econom´ıa y Competitividad project MTM2015-63723-P and the Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa) under Proyecto de Excelencia P12-FQM-1492. 2 Tom´as Caraballo and Renato Colucci In general it is convenient to consider a regularized version of the problem, such as ut+ε2uxxxx =W00(ux)uxx,(1.2) where 0 < ε << 1. If the parameter εis sufficiently small, it is natural to expect that we have a good approximation of the original problem. Equation (1.2) has been studied in [1] and [2], where the authors pointed out the existence of three well separated time scales with different dynamical behavior. In a first time scale of order Tε=O(ε2) they observed the formation of microstructure (see [14]) of spatial wave length of order O(ε). Microstructure appears in regions where the first derivative of the initial datum u0falls in the non convex region of the potential W(·) (or equivalently in the region in which the diffusion coefficient of (1.1) is negative). In the second time scale of order O(1) ,regions without microstructure exhibit heat equation-like behavior. The separation between regions with and without microstructures may give rise to a free boundary problem. In the third time scale of order T−1 ε=O(ε−2), the equation exhibits a finite dimensional behavior, the solutions are approximately the union of segments with slopes ±1 and the number of degrees of freedom for the system is approximately represented by the number of jumps of ux. This time scale is characterized by very slow motion and the number of jumps decreases very slowly giving rise to the so called coarsening phenomenon. In this paper we consider the third time scale, that is the long term behavior. This is well described by the global attractor A([19]), in the sense that any fixed trajectory, for a large time, is well approximated by portions of trajectories which lie in the attractor (see [16], corollary 10.15 pag. 277, termed as tracking property). For this reason, it is very interesting to study the structure of the global attractor and the dynamics on it. For gradient systems, such as (1.2), whose Lyapunov function is given by Fε(u) = 1 2ε2ZI u2 xxdx +1 2ZI W(ux)dx, it is well known (see [16]) that the attractor is given by the unstable manifold of the set E=E(ε)of stationary solutions: A=Wu(E), if Eis discrete we have an even simpler expression: A=[ z∈E Wu(z). Unfortunately, in our case, we cannot conclude that the set Eis discrete. However, we still consider interesting enough to study the set Ein order to obtain some information on the structure of the attractor and, as a consequence, of the long term behavior of the solutions of (1.2). In [3]-[10] the problem of microstructure formation and finite dimension dynamics have been studied. In particular, the following upper bound of the Microstructure formation and coarsening phenomena 3 fractal dimension of the global attractor Aεof (1.2) has been obtained (together with the existence and dimension estimate of exponential attractor, [13] and inertial manifold, [11] ): dF(Aε)≤O(ε−1), where dF(X) is the fractal dimension of Xwhich is based on counting the number of ball of a fixed radius that are necessary to cover X(see [16], [17] for a detailed discussion). This is in accordance with what we expected by looking at the numerical experiments and, in particular, the estimate of the spatial wave length of microstructure. In fact, as the degrees of freedom of the system are well represented by the number of jumps Jxof ux, we obtain an upper bound of their number by Jx≈|I| ε=O(ε−1) (see [7] for a more detailed discussion). In Section 2 below we will prove that it is possible to recover many of these ideas by studying some properties of stationary solutions. Among the general solutions of (1.2) we decided to dedicate some effort to the study of traveling waves on the whole R, since this will allow us to describe some of the previously mentioned phenomenology. In Section 3 we deal with traveling wave analysis, we will see that, in full accordance with the above cited corollary in [16], these kind of solutions will be very close to stationary periodic solutions for a time of order smaller than O(ε2), while for time of order O(ε−2) we have that uxgoes to 1 (−1 respectively) as (x−ct)→ ∞, and uwill be close to a class of unbounded solutions associated to heteroclinic connections in the plane (ux, uxx) (see figure 3 below). This means that uxstarts to loose its oscillatory behavior and converges to ±1 as expected by the observed coarsening phenomenon. In Section 4 we construct an approximate solution which will describe what has been obtained in Section 3. Moreover, we present several numerical simulations to illustrate the results. 2. Stationary Solutions In this section we study some properties of the stationary solutions of equation (1.2) in order to have some qualitative description of the dynamics. A more complete study of the set of stationary solutions is postponed for future investigation. In details we look for the solutions of ε2(uxxxx) = 1 2[W0(ux)]x,(2.1) which can be rewritten, after integration, in the following form ε2uxxx =1 2W0(ux) + K, 4 Tom´as Caraballo and Renato Colucci where Kis a real constant. We rewrite the above third order ordinary differential equation as a first order three dimensional system by using the usual identification u=x1, ux=x2, uxx =x3,(2.2) obtaining    ˙x1=x2, ˙x2=x3, ˙x3=1 ε22x2(x2 2−1) + K. (2.3) If K6= 0 the system has no fixed points. We note that in this case, by integrating the quotient of the last two equations, we obtain 1 2x4 2−x2 2+Kx2+S=1 2(x2 2−1)2+Kx2+S−1 2=1 2ε2x2 3, where Sis another real constant. For simplicity we rewrite the previous expression in the following way: x2 2−3 22 + (x2+K)2+2S−K2−9 4=ε2x2 3.(2.4) We note that, if S > K2 2+9 8, the previous expression is defined for all x2∈R then there are no periodic orbits in the plane x2x3. In figure 2 below we represent the level curves of the difference between the r.h.s. and the l.h.s. of (2.4) for a fixed value of Kand several values of S. We observe the existence of a region with periodic orbits (see figure 2) on the plane x2x3. For example if K=1 2and S= 0 we have x2(x2−1)(x2 2+x2−1) = ε2x2 3, which is defined if x2<−1−√5 2, if x2∈[0,−1+√5 2] or if x2>1. In particular the periodic orbit correspond to the interval x2∈[0,−1+√5 2]. We have x3= 0 for x2∈ {0,−1+√5 2}while the maximum of x3is obtained at (see figure 1 below) 4x2(x2 2−1) + 1 = 0,that is x2≈0.26. Remark 2.1. A periodic orbit in the (x2, x3)plane corresponds to an oscillatory stationary solution, that is, a solution with both oscillatory first and second derivatives. If K= 0 we have a line of fixed points for the three dimensional system: l={(x, 0,0) : x∈R}, and the Jacobian of the system at any point of the line lis given by   010 001 0−2 ε20 , with trace and determinant Tr(J)=0, Det(J)=0. Microstructure formation and coarsening phenomena 5 0.2 0.4 0.6 0.8 1.0 -0.3 -0.2 -0.1 0.1 0.2 0.3 x 2 x 3 Figure 1. The periodic orbit for K=1 2and S= 0. !2 !1 1 2 y !4 !2 2 4 z 3 x 2 x Figure 2. The orbits (x2, x3) for K=1 4, several values of Sand ε= 1. We consider the following change of variables in order to deduce the canonical form for the Jacobian of the system: x1=X, x2=Y, x3=√2 εZ. (2.5) 6 Tom´as Caraballo and Renato Colucci Then the system takes the form:   ˙ X ˙ Y ˙ Z =   0 1 0 0 0 √2 ε 0−√2 ε0    X Y Z +  0 0 √2 εY3 ,(2.6) and the Jacobian at any point of lis    0 1 0 0 0 √2 ε 0−√2 ε0  .(2.7) Of course we have again zero trace and determinant, while the eigenvalues are λ1= 0, λ2,3=i√2 ε:= iω. We note that the second and third equations of system (2.6), do not depend on X. Moreover the system restricted on the plane (Y, Z) can be written in Hamiltonian form: ˙ Y=∂H ∂Z , ˙ Z=−∂H ∂Y .(2.8) with the first integral (see figure 3 below) H(Y, Z) = 1 √2ε−1 2(Y2−1)2+Z2.(2.9) The projection of solutions on the Y Z plane are the level curves of the function H(see figure 3 below). We note that if we are looking for bounded solutions, we need to restrict the analysis to the case in which the system on the plane Y Z possesses bounded solutions. In general we have the following result: Theorem 2.2. System (2.6) admits the following classes of solutions depending on the values of (2.9). 1. If H > 0, there exist unbounded solutions in the Y Z plane. As a consequence, the Xcomponent is unbounded too. 2. If H= 0 it follows that: If (Y0, Z0) = (±,1,0), which correspond to the fixed points of the Y Z system, we obtain the following unbounded solutions: (X, Y, Z)=(X0±t, ±1,0). The heteroclinic connections which connect the fixed points (±1,0) of the Y Z system correspond to unbounded solutions of the three dimensional system since Yremains near ±1asymptotically and as a consequence X→ ±∞. There exist unbounded solutions for |Y0|>1. Microstructure formation and coarsening phenomena 7 !3 !2 !1 1 2 3 y !6 !4 !2 2 4 6 z Y Z Figure 3. The orbits of the Hamiltonian systems (2.8) with =1 √2and with several values for H. We observe that for K= 0 the set of level curves is symmetric with respect to the vertical axis. 3. If the energy is negative and satisfies −1 2√2ε< H < 0,(2.10) there exists a family of periodic orbits inside the region |Y|<1(Observe that the fixed point (0,0) for the restricted system is a center). Proof. The proof is a straightforward consequence of what we have observed above and can be rigorously proved using standard Hamiltonian methods.  In the sequel, we will provide a brief description of these kinds of solutions. 2.1. Periodic Solutions We rewrite for simplicity the condition of existence of periodic solutions in the Y Z plane in the following way: (Y2−1)2−2Z2=p2, where p2=−2√2εH ∈(0,1). Then we look for a periodic orbit (Y, Z) that satisfies Y2−1 p2 −2Z p2 = 1,(2.11) with p∈(−1,1) and p6= 0. As a consequence we have: 8 Tom´as Caraballo and Renato Colucci Remark 2.3. The maximum and minimum values of (Y, Z)for the periodic orbits satisfy: YMAX,MIN =±r1−q2√2|H|ε=±p1−p, and ZMAX,MIN =±r1 2−√2ε|H|=±1 √2p1−p2. Remark 2.4. Recalling that |H|=O(ε−1)and taking into account the change of variable (2.5), we have x2=O(1),x3=O(ε−1), which means max ux=O(1),max uxx =O(ε−1). This is in complete accordance with similar results obtained in [5] for the functional Fε,µ(u, ¯u;I) = 1 2ε2ZI u2 xxdx +1 2ZI W(ux)dx +µ 2ε2ZI (u−¯u)2dx, whose minimizers share many properties with solutions of (1.2) (see [5] for a detailed discussion). Then, to obtain a parametrization of the periodic orbits, for any p2∈(0,1) we set: Z p2 =1−p2 2p2sin2θ, Z =±s1−p2 2sin θ. This expression is suggested by that of ZMAX obtained for θ=π 2. We substitute it in (2.11), yielding (Y2−1)2=p2+ (1 −p2) sin2θ, Y =±r1−qp2+ (1 −p2) sin2θ, which, in particular, satisfies the expression of YMAX for θ= 0. In figure 4 below we represent the parametrization for p=1 2. It is possible to obtain an estimate of the time period of the orbits that corresponds to the wave length of microstructure. Using the first integral and the second equation of the system we can write dY dt =√2 εZ=±√2 εr1 2(Y2−1)2−√2ε|H|. Then, if we integrate it between (YMIN , YM AX ), we obtain the spatial period TH= 2εZYMAX YMIN 1 q(Y2−1)2−2√2ε|H| dY = 2εZYM AX YMIN 1 p(Y2−1)2−p2dY, where the integrand does not depend on ε, indeed p=O(1) since |H|= O(ε−1) (see [2] for a discussion of the case ε→0+). We therefore have: Microstructure formation and coarsening phenomena 9 -0.6 -0.4 -0.2 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.2 0.4 0.6 Y Z Figure 4. The parametrization of a periodic solution with p=1 2. Proposition 2.5. For H∈(−1 2√2ε,0), the spatial period satisfies TH=O(ε), (as suggested in [1]). For the energy value H=−1 2√2εthere exists the fixed point P= (0,0,0) and then TH= 0, while if H= 0, there are “heteroclinic connections” and henceforth TH=∞. Finally we obtain the expression of Xas a function of Y: dX dY =Yε √2 ±q1 2(Y2−1)2−√2ε|H| , from which X(Y) = ±ε √2ZY q1 2(Y2−1)2−√2ε|H| dY +C =±ε 2Z1 pQ2−1dQ +C, =±ε √2log |Q+pQ2−1|+ω, where ωis a constant and Q=Y2−1 q2√2ε|H| . We note that max X=O(ε), that is max u=O(ε) (see also [5]). The solution X(Y) is represented in figure 5 below for fixed values of all the involved parameters. 16 Tom´as Caraballo and Renato Colucci in A+). From the second equation we have that ˙x2→+∞which implies that x2→+∞(since x2>1 in A+). A similar result is valid with −∞ replacing +∞and A−replacing A+. The converse is also true since when solution enters A+we have, by the first equation of the system, ˙x1→ ∞ and as a consequence x1→ ∞. From the previous results we have obtained several classes of unbounded solutions. In the next section we will focus on a particular class of such solutions. 4. Approximate Solutions In this section we construct an approximate solution of equation (3.3) with K= 0 in order to qualitatively describe the dynamical features observed in the previous sections. Let us write system (3.5) with c6= 0 in the coordinates (X, Y, Z):   ˙ X ˙ Y ˙ Z =   0 1 0 0 0 √2 ε c √2ε−√2 ε0    X Y Z +  0 0 √2 εY3 .(4.1) In order to use perturbation theory we consider the following change of variable: X=ε¯ X, Y =ε¯ Y , Z =ε¯ Z, c =ε¯c, for any ε6= 0. Then we obtain:      ˙ ¯ X=¯ Y , ˙ ¯ Y=√2 ε¯ Z, ˙ ¯ Z=¯cε √2ε¯ X+√2 ε¯ Y(ε2¯ Y2−1), (4.2) and, in order to simplify the computation, using the cylindrical coordinates, for (X, Y, Z)6= (0,0,0) we have:    ¯ X=¯ X, ¯ Y=Rsin θ, ¯ Z=Rcos θ, we obtain the following system with respect to the variables ( ¯ X, R, θ):      ˙ ¯ X=Rsin θ, ˙ R=cos θ √2ε¯cε ¯ X+ 2R3ε2sin3θ, ˙ θ=√2 ε−sin θ √2Rε ¯cε ¯ X+ 2R3ε2sin3θ. To simplify this system, we use θas an independent variable. The system becomes (d¯ X dθ =√2εR2sin θ 2R−h(ε) sin θ:= F1(ε), dR dθ =h(ε)Rcos θ 2R−h(ε) sin θ:= F2(ε),(4.3) Microstructure formation and coarsening phenomena 17 where we have set h(ε) := h(R, θ, ¯c, ε) = ¯cε ¯ X+ 2ε2R3sin3θ. For the next computations it is worth observing the following properties of the function h(·): h(0) = 0, h0(ε) = ¯c¯ X+ 4εR3sin3θ, h0(0) = ¯c¯ X, h00(ε) = h00(0) = 4R3sin3θ, h000(ε) = h000(0) = 0. Let us consider the Taylor expansion of the functions Fiwith respect to the parameter ε: Fi(ε) = Fi(0) + εF0 i(0) + ε2 2F00 i(0) + O(ε3). We first observe that Fi(0) = 0 for i= 1,2. In details, we have for F1(·): F0 1(ε) = A(θ)·1 2R−h(ε) sin θ+εh0(ε) sin θ [2R−h(ε) sin θ]2, F0 1(0) = A(θ) 2R=Rsin θ √2, F00 1(ε) = A(θ) sin θ·2h0(ε) [2R−h(ε) sin θ]2+εh00(ε) [2R−h(ε) sin θ]2 +2 sin θh02(ε) [2R−h(ε) sin θ]3, F00 1(0) = 2A(θ) sin θh0(0) 4R2=¯c √2sin2θ¯ X, F000 1(ε) = A(θ) sin θ·2h00(ε) [2R−h(ε) sin θ]2+ 4 h02(ε) sin θ [2R−h(ε) sin θ]3 +h00(ε) [2R−h(ε) sin θ]2+ 2 sin θh02(ε) [2R−h(ε) sin θ]3+...., F000 1(0) = 3√2R2sin3θ·Rsin2θ+¯c2¯ X2 4R3, 18 Tom´as Caraballo and Renato Colucci where we have set for simplicity A(θ) = √2R2sin θ. For F2(·) we have: F0 2(ε)=2R B(θ)·h0(ε) [2R−h(ε) sin θ]2, F0 2(0) = B(θ)¯c¯ X 2R=¯c 2¯ Xcos θ, F00 2(ε) = 2RB(θ)h00(ε) [2R−h(ε) sin θ]2+2[h0(ε)]2sin θ [2R−h(ε) sin θ]3, F00 2(0) = 2R3sin2θ+¯c2¯ X2 2Rcos θsin θ, F000 2(ε) = 12Rsin θB(θ)h0(ε)h00(ε) [2R−h(ε) sin θ]3+sin θ(h0(ε))3 [2R−h(ε) sin θ]4, F000 2(0) = 12¯c¯ XR2sin2θcos θsin2θ+¯c2¯ X2 16R4, where we have set for simplicity B(θ) = Rcos θ. Then the system becomes:          d¯ X dθ =εR √2sin θ+ε2 2 ¯c √2sin2θ¯ X+ε3 63√2R2sin3θ·nRsin2θ+¯c2¯ X2 4R3o+O(ε4), dR dθ =ε¯c 2cos θ¯ X+ε2 2n2R3sin2θ+¯c2¯ X2 2Rocos θsin θ +ε3 612¯c¯ XR2sin2θcos θnsin2θ+¯c2¯ X2 16R4o+O(ε4). (4.4) We write, for εsmall, the solution of (4.4) in the following form: (¯ X(θ) = ¯ X0+εH1(θ) + ε2 2L1(θ) + ε3 6M1(θ) + O(ε4), R(θ) = R0+εH2(θ) + ε2 2L2(θ) + ε3 6M2(θ) + O(ε4),(4.5) where Hiand Lifor i= 1,2 are smooth functions of θwhich do not depend on εwith Hi(0) = Li(0) = 0 for i= 1,2. Then, differentiating with respect to θ, we obtain (d¯ X dθ =εdH1 dθ +ε2 2 dL1 dθ +ε3 6M1(θ) + O(ε4), dR dθ =εdH2 dθ +ε2 2 dL2 dθ +ε3 6M2(θ) + O(ε4), while using equations of the system we have:              d¯ X dθ =εR0 √2sin θ+ε2 2n√2 sin θH2(θ) + ¯c √2sin2θX0o +ε3 6n3 √2L2(θ) sin θ+3 √2¯csin2θH1(θ)+3√2R2 0sin3θhR0sin2θ+¯c2¯ X2 0 4R3 0io+O(ε4), dR dθ =ε¯c 2cos θ¯ X0+ε2 2n¯ccos θH1(θ)+2R3 0sin3θcos θ+¯c2¯ X2 0 2R0sin θcos θo ε3 6{...}+O(ε4). Microstructure formation and coarsening phenomena 19 From the two previous expressions we obtain the derivatives of the functions Hi: (dH1 dθ =R0 √2sin θ, dH2 dθ =¯c 2cos θ¯ X0. Integrating the previous equations in [0, θ] we obtain the expression of the functions Hi(·): H1(θ) = R0 √2(1 −cos θ), H2(θ) = ¯c 2¯ X0cos θ. For the functions Li(·) we have the following: (dL1 dθ =¯c √2¯ X0[sin θcos θ+ sin2θ], dL2 dθ =¯cR0 √2(cos θ−cos2θ)+2R3 0sin3θcos θ+¯c2¯ X2 0 2R0sin θcos θ, from which (L1(θ) = ¯c 2√2¯ X0[sin2θ+θ−sin θcos θ], L2(θ) = ¯cR0 √2(sin θ−θ 2−1 4sin 2θ) + R3 0 2sin4θ+¯c2¯ X2 0 4R0sin2θ. Finally, since we are only interested in the X component, we need to integrate the following dM1(θ) dθ =3 √2¯cR0 √2(sin θ−θ 2−1 4sin 2θ) + R3 0 2sin4θ+¯c2¯ X2 0 4R0 sin2θsin θ +3 2R0¯csin2θ(1 −cos θ)+3√2R2 0sin3θR0sin2θ+¯c2¯ X2 0 4R3 0, from which M1(θ) = 3 4¯cR0θ{1 + cos θ}− 3 2¯cR0sin θcos θ−3 4¯cR0sin θ−3 4¯cR0sin3θ +15 4√2R3 0−sin4θcos θ 5−4 15 sin2θcos θ+8 15 −8 15 cos θ +9 8√2¯c2¯ X2 0 R0−sin2θcos θ 3+2 3(1 −cos θ) := ¯cR0θP1(θ) + ¯cR0P2(θ) + R3 0P3(θ) + ¯c2¯ X2 0 R0 P4(θ), where Pi(θ), i= 1,2,3,4, are bounded periodic functions. Then we can write the approximated solution as ¯ X(θ) = ¯ X0+εR0 √2[1 −cos θ] + ε2¯c 4√2 ¯ X0[sin2θ+θ−sin θcos θ] +ε3 6[¯cR0θP1(θ) + ¯cR0P2(θ) + R3 0P3(θ) + ¯c2¯ X2 0 R0 P4(θ)] + O(ε4).(4.6) 20 Tom´as Caraballo and Renato Colucci We observe that the coefficient of θis ε2 4 ¯c √2¯ X0+ε √2R0(1 + cos θ). Then, if we choose such that ¯ X0+ε √2R0= 0,(4.7) we have ¯ X(θ) = −εR0 √2cos θ+ε3¯cR0 8[−sin2θ+ sin θcos θ] +ε3 6[+¯cR0P2(θ) + R3 0P3(θ) + ¯c2¯ X2 0 R0 P4(θ)] + ε4 8¯cR0θcos θ+O(ε4).(4.8) we observe that the biggest term (for large time) among the remaining terms of order O(ε4) has the following form ¯c2¯ X0f1(θ)θ2 where f1(θ) is a periodic function. Then we consider the following approximated function: ¯ X◦(θ) = −εR0 √2cos θ+ε3f2(θ) + ε4 8¯cR0θcos θ, (4.9) we observe that the higher order term of the difference between ¯ X◦(θ) and the solution of the system, for large time, can be estimated by ¯ D(θ) = k1(θ2+k2)R0ε5¯c2.(4.10) Coming back to the original coordinates we have X◦(θ) = ε¯ X◦(θ) = −ε2R0 √2cos θ+ε4f2(θ) + ε5 8cR0θcos θ, (4.11) and the error D(θ) = k1(θ2+k2)R0ε7c2.(4.12) Then for θ < ε3the leading term of X◦(θ) is a periodic function: γ(θ) = −ε2R0 √2cos θ, while for time of order θ=O(ε3) the function starts to loose periodicity. Using the third equation of the system in polar coordinates we obtain the relation between θand the “original time” t: θ≈√2 εt. (4.13) The approximate solution qualitatively describes what was obtained by numerical experiments: the solutions appear to be periodic (microstructure formation) for a large interval of time and then start to loose periodicity (in the sense that a coarsening phenomenon occurs). In particular, when θ << ε−3(that is t << ε−2), the function (4.11) appears periodic, while when θ=O(ε−3) (that is t=O(ε2)) the linear term in θcannot be disregard Microstructure formation and coarsening phenomena 21 and the solutions “start to loose periodicity”. We conclude the present work with several numerical simulations to illustrate the results of the present section. In the figures 9 below we compare the solution ¯ X(θ) with the periodic function ¯ X0+ε √2R0[1 −cos θ] in several time scales. θ 012345678910 X -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 θ 0 10 20 30 40 50 60 70 80 90 100 X -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 θ 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 X -0.1 -0.05 0 0.05 0.1 0.15 θ 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 X -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 9. The comparison between the numerical solution ¯ X(θ) (in red) computed using the Software Matlab and the function ¯ X0+ε √2R0[1−cos θ] (in green) with ε=1 10 ,R0= 1, ¯ X0=−R0 2εand ¯c= 0.1. In the the first two simulations the time intervals are of order O(ε−1) and O(ε−2) respectively, in particular [0,10] and [0,100] respectively. The numerical solution and the function γ(θ) cannot be distinguished. In the remaining simulations, we consider a time of order O(ε−3), [0,5000] and O(ε−4) and [0,10000]. We observe that in the latter case the solution is no more periodic while for a time of order O(ε−3) the solution becomes to move from the periodic approximation. In figure 10 below we compare the solution ¯ X(t) of (4.2) with the function ¯ X0+ε √2R0h1−cos(√2 εt)iin a time scale of order t=O(ε). 22 Tom´as Caraballo and Renato Colucci t 012345678910 X -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Figure 10. The solution ¯ X(t) of (4.2) (in red) and the function ¯ X0+ε √2R0h1−cos(√2 εt)iin a time scale of order t= O(ε). 5. Conclusion In this work we have presented a qualitative description of microstructure formation and coarsening phenomena for the solutions of a nonlinear equation presenting three different time scales with different dynamical features. Even if it can be considered only a preliminary work we consider that the present analysis could represent a good starting point for future research We have related the microstructure formation to the existence of periodic stationary solutions (analyzed in Section 2) while the coarsening phenomenon has been described by studying a class of unbounded solutions (Section 3 and Section 4). We leave for future research the problem of the complete description of the set of stationary solutions and the existence/non existence of solitons. Moreover, we consider that the coarsening phenomenon could be analyzed with different methods both numerically and analytically. Acknowledgements We would like to thank the anonymous referees for their interesting and very helpful suggestions that contributed to clarify the problem under consideration and to improve the quality of the paper. References [1] Bellettini, G.; Fusco, G.; Guglielmi, N. 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Cambridge University Press, Cambridge, (2001). [17] Robinson, J.C. Dimensions, Embeddings, and Attractors. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge. [18] Slemrod, M. Dynamics of measured valued solutions to a backward-forward heat equation. Journal of Dynamics and Differential Equations Volume 3, Number 1, (1991), 1-28. [19] Temam, R. Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer; 2nd edition, (1997). 24 Tom´as Caraballo and Renato Colucci Tom´as Caraballo and Renato Colucci Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain e-mail: [email protected], [email protected]