HETEROCLINIC ORBITS, MOBILITY PARAMETERS AND STABILITY FOR THIN FILM TYPE EQUATIONS

Transcription

1 Electronic Journal of Differential Equations, Vol. (), No. 95, pp. 9. ISSN: URL: ttp://ejde.mat.swt.edu or ttp://ejde.mat.unt.edu ftp ejde.mat.swt.edu (login: ftp) HETEROCLINIC ORBITS, MOBILITY PARAMETERS AND STABILITY FOR THIN FILM TYPE EQUATIONS RICHARD. S. LAUGESEN & MARY C. PUGH Abstract. We study te pase space of te evolution equation t = ( n ) B( m ), were (, t). Te parameters n >, m R, and te Bond number B > are given. We find numerically, for some ranges of n and m, tat perturbing te positive periodic steady state in a certain direction yields a solution tat relaes to te constant steady state. Meanwile perturbing in te opposite direction yields a solution tat appears to touc down or rupture in finite time, apparently approacing a compactly supported droplet steady state. We ten investigate te structural stability of te evolution by canging te mobility coefficients, n and m. We find evidence tat te above eteroclinic orbits between steady states are perturbed but not broken, wen te mobilities are suitably canged. We also investigate touc down singularities, in wic te solution canges from being everywere positive to being zero at isolated points in space. We find tat canges in te mobility eponent n can affect te number of touc down points per period, and affect weter tese singularities occur in finite or infinite time. We study te evolution equation. Introduction t = ( n ) B( m ), (.) were n >, m R, and te Bond number B >. Tis is te one-dimensional version of t = (f() ) (g() ) (.) wit f() = n, g() = B m. Suc equations ave been used to model te dynamics of a tin film of viscous liquid. Te air/liquid interface is at eigt z = (, y, t) and te liquid/solid interface is at z =. Te one dimensional equation (.) applies if te liquid film is uniform in te y direction. Te fourt order term in equation (.) reflects surface tension effects [, 4]. Typically f() n as, for some n 3, and tis motivates our coice of f() = n in (.). Notice also tat te fourt order term ( n ) in (.) is linearly stabilizing around te constant steady state. Matematics Subject Classification. 35K55, 37C9,37L5, 76D8. Key words and prases. Nonlinear PDE of parabolic type, eteroclinic orbits, stability problems, lubrication teory. c Soutwest Teas State University. Submitted February, 8,. Publised November 5,.

2 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE /95 Te second order term in (.) can reflect gravity, van der Waals interactions, termocapillary effects or te geometry of te solid substrate. Typically g() B m as, for some m, B R. Coosing B >, te second order term B( m ) in (.) is linearly destabilizing around te constant steady state (it is like a backwards eat equation term). Te competition between tis destabilizing term and te stabilizing fourt order term generates interesting dynamics. Te dynamics are less interesting wen B, wic we do not consider in tis paper. P P 3P 4P Figure. Four types of steady state. Te -ais is space. Te two steady states on te left etend smootly to be P -periodic. Te two on te rigt ave less regularity and are called droplet steady states. Te tird profile as zero contact angles and te fourt as nonzero contact angles. Bot ave lengt less tan P, and so are possible long time limits of a P -periodic solution of te evolution equation. Definitions. In [7] we studied te four types of steady state sown in Figure. Tere are two smoot types of steady state: constant and (nonconstant) positive periodic. Two oter types of steady state wit lower regularity are te droplet steady states. Tey eiter ave zero contact angles or nonzero contact angles. Positive periodic steady states are classical solutions of te steady state equation (tat is, (.) wit t ). Wen we refer to te period, P, of a nonconstant positive periodic steady state, we mean te sortest period. Te area, A, is ten defined to be P ss d. Constant steady states ave no sortest period, owever we will always discuss tem in te contet of an initial value problem, for wic te period and area are unambiguous. Because constant steady states are periodic steady states of te most trivial kind, in te following wen we refer to positive periodic steady states we implicitly mean nonconstant ones. A droplet steady state ss is by definition positive on some interval (a, b) and zero elsewere, wit ss C [a, b]; ss satisfies te steady state equation classically on (a, b), and as equal acute contact angles: ss(a) = ss(b) <. (Trougout te paper, if a function as only one independent variable ten we use to denote differentiation wit respect to tat variable: ss = ( ss ).) Te area, A, of a droplet is A = b a ss d and te lengt is b a. As suggested by Figure, we are interested in droplet steady states wose lengt is equal to or sorter tan P. Tis is because we want to study te initial value problem wit positive initial data of period P and area A. A steady droplet wit area A wose lengt is less tan or equal to P must be considered as a possible long time limit of te evolution. A configuration of droplet steady states is defined to be a collection of steady droplets wose supports are disjoint. Any suc configuration wose total area is A and wose total lengt is less tan P must also be considered as a possible long time limit of te initial value problem.

3 EJDE /95 HETEROCLINIC ORBITS 3 Background, and Overview of Paper. In [8] we proved linear stability results for steady states. Section summarizes tese results by means of a family of bifurcation diagrams, and also presents a weakly nonlinear stability analysis. Our investigations in te rest of te paper are guided by te study of a dissipated energy for te evolution equation. Tis energy is defined for P -periodic functions l on R to be P [ ] E(l) = (l ()) B (m n + )(m n + ) lm n+ () d. (.3) Tis energy is strictly dissipated: if (, t) is a smoot, spatially P -periodic solution of (.) ten d dt E((, t)), wit equality if and only if is a steady state of te evolution (cf. [9,.]). Like te evolution equation, te energy E(l) is insensitive to translation in. Te evolution (.) describes gradient flow for te energy E, wit respect to te following weigted H inner product. Let (, t) be a positive smoot function tat is P -periodic in. For eac t, define an inner product on te space of continuous P -periodic functions wit mean value zero, by u, v := P U ()V ()(, t) n d, (.4) were U and V are P -periodic, ave zero mean, and are determined from u, v, and (, t) via: u = ( n U ) and v = ( n V ). Ten te evolution equation (.) is equivalent to: δe δφ = lim E( + sφ) E() = t, φ s s for all continuous P -periodic φ aving mean value zero. Te variation of te energy is most negative in te direction φ = t, so tat te evolution equation for (, t) describes flow by steepest descent on te energy surface of E, wit respect to te time- and solution-dependent inner product (.4). Te steady states are critical points of tis energy surface, wit unstable steady states being saddle points and asymptotically stable steady states being minima. Te above gradient flow formulation was observed by Fife [] and by Taylor and Can [7]. Gradient flow ideas for related equations ave been used in [, 3, 6]. We refer te readers to Fife s survey article on pattern formation in gradient systems []. Note tat te perturbing function φ is required to ave mean value zero because te equation (.) preserves area, under periodic (or Neumann) boundary conditions. Tat is, given initial data wit area A, only tose points on te energy landscape representing functions wit te same area will be accessible to te solution. In [9] we studied te energy landscape of (.3) under te evolution (.), and conjectured te eistence of eteroclinic orbits from certain ig-energy steady states to certain low-energy steady states. In tis paper we find suc orbits numerically, in Section 3., by computing solutions of te initial value problem for (.). We furter ask ow robust tese orbits are under canges in te mobility coefficients n and m. If we vary m and n wile keeping m n fied, te steady states and teir energy stability are uncanged (since E depends only on m n). We find tat te eteroclinic orbits are perturbed but do not break, wen m and n are varied in tis way.

4 4 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE /95 In Sections 3. and 3.3 we study orbits from a positive periodic steady state to a droplet steady state. Here te long time limit is not strictly positive, raising te question of weter or not te solution will be positive (and ence classical) for all time, or weter tere will be touc down singularities, places and times at wic te solution is zero (and ence weak). We find tat te value of m n can affect weter touc down singularities are present ( 3.), as well as te number of suc singularities tat arise per period ( 3.3). Section 4 presents a detailed numerical study of te evolution equation (.), for initial data close to a steady state and for a number of different eponents n and m. Our stability and energy level results from [8, 9] lead to many predictions for te beavior of te solutions, bot sort and long time, and tese predictions are borne out by our simulations. However tere are also situations for wic we can make no prediction, and were te numerically observed beavior is rater intriguing. In Section 5 we sum up te paper. Appendi A discusses our numerical metods and gives te specific parameters used in te simulations.. Bifurcation diagrams and weakly nonlinear analysis First, we rescale te equation and present te weakly nonlinear analysis near te constant steady state. Ten, we present bifurcation diagrams tat summarize te linear stability of constant steady states and positive periodic steady states... Non-dimensionalizing te equation. A solution of te evolution (.) reflects five free parameters: m, n, B, te period P, and te area A. First, we rescale space, time, and te solution itself: ζ = P, t = P n+4 A n A t, and P η(ζ, t ) = (, t). Te rescaled solution, η, as period and area and satisfies te rescaled evolution equation η t = (η n η ζζζ ) ζ E(η m η ζ ) ζ (.) were E = BA q P 3 q and q := m n +. (.) In te simulations tat follow, in Sections 3 and 4, we always take P = A =. Ten te original evolution equation (.) and te rescaled evolution (.) are identical, and B = E is te bifurcation parameter. So in te following we will refer to, B and equation (.), rater tan to η, E and equation (.)... Weakly nonlinear analysis. For te weakly nonlinear analysis, we consider values of B suc tat te constant steady state as one mode wic is barely linearly stable or is barely linearly unstable. Since all oter modes are strongly damped, tis provides a separation of timescales, allowing one to find a reduced representation of te PDE in terms of an ODE governing te amplitude of te unstable mode (cf. [, 5.]). Linearizing equation (.) about = and considering perturbations ɛ cos(kπ+ φ), one finds a critical value k c = B/(π). If k c ten tere are no unstable modes. If k c > ten tere is a finite collection of linearly unstable modes. We assume B = 4π + Qδ were Q = ±. Here, δ is a small parameter and varying δ results in k c moving troug te wave number. We ten introduce a slow time-scale τ = δ t and epand te solution in orders of δ: (, τ) =

5 EJDE /95 HETEROCLINIC ORBITS 5 + δ (, τ) + δ (, τ) + O(δ 3 ). For simplicity, we assume te solution is even. By te usual arguments, (, τ) = A(τ) cos(π) and (, τ) = B(τ) cos(4π). Putting tis ansatz into te evolution equation (.) and epanding in orders of δ, we find tere are no O() or O(δ) terms. At O(δ ) one determines te amplitude B(τ) in terms of A(τ). At O(δ 3 ) one finds tat A(t) satisfies da dτ = 4π QA(τ) κa(τ) 3, were κ = 8 3 π4 (q )(7/4 q) and we recall q = m n +. Te dynamics of te amplitude A(τ) depend on te signs of te Landau constant κ and of te linear term. If κ > ten for Q = te steady state A (τ) is linearly unstable and A(τ) saturates to te linearly stable steady state A c (τ) σ/κ. Tis corresponds to a supercritical bifurcation. If κ < ten for Q = te steady state A (τ) is linearly stable and te steady state A c (τ) σ/κ is linearly unstable. Tis corresponds to a subcritical bifurcation. And so, < q < 7/4 = supercritical bifurcation, q < or 7/4 < q = subcritical bifurcation. Subcritical bifurcations are often seen in systems tat can ave finite-time pincing (rupture) singularities e.g. [, 3.], [3, IV]. Te above weakly nonlinear analysis was done for q = 3 in [3,.3]..3. Bifurcation diagrams. Figure gives bifurcation diagrams for representative q-values. To construct a bifurcation diagram, we fi a value for q = m n + and ten compute a collection of positive periodic steady states ss all wit period and area. Eac steady state satisfies te steady version of (.) for some value of B. We ten plot te amplitude of ss versus te bifurcation parameter B. We use our linear stability results from [8] to determine weter to plot wit solid lines (linearly stable) or wit dased lines (linearly unstable). Te orizontal aes of tese diagrams sow te linear stability of te (zero-amplitude) constant steady state. Tese stability results are all wit respect to zero-mean perturbations aving te same period as te steady state. Perturbations wit longer period always lead to linear instability [8]. Figure provides nine graps to elp te reader visualize te dependence of te diagram on q. (In wat follows, we do not discuss simulations for q =.76 or.775.) By eamining te bifurcation diagrams near te point (4π, ), one observes te subcritical and supercritical bifurcations predicted by te weakly nonlinear stability analysis. Te bifurcation diagrams also encode eistence information for steady states. Consider Figure b for q = /. It starts at B = 33.7 and ends at B = 4π wit amplitude zero. If B equals 3 or 4, for eample, ten tere is no (nonconstant) positive -periodic steady state wit area. If B = 34 ten tere is a unique nonconstant positive -periodic steady state wit area. Te monotonicity of te bifurcation diagram corresponds to uniqueness of te positive periodic steady state wit specified period and area, if it eists. If te diagram is non-monotonic, tere may be zero, one, or two (nonconstant) positive -periodic steady states wit area, depending on te value of B. Non-monotonicity olds for q (7/4,.794) were.794 approimates a criticial eponent q (see [7, 5.]).

6 6 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE /95 q = 3.75 q = /.75 q = B 3 4π π 4 B π 4 B q = 3/.75 q = 7/4.75 q = B 39 4π 4 q = B π 39.6 q = B π 39.6 q = 5/ π 39.6 B π 39.6 B π 4 B Figure. Te orizontal ais is te bifurcation parameter B and te vertical ais is te amplitude of te steady state, ( ma min )/. Dased: linearly unstable; dotted: linearly neutrally stable; solid: linearly stable. Tere is an interval of B values for wic (nonconstant) positive periodic steady states eist. Tey are linearly unstable for q (, ), are linearly stable for q (, 7/4), and are linearly unstable for q (q, ) were q.794. For q (7/4, q ), tere can be two suc steady states, one linearly unstable and one linearly stable. One can prove tat for q, as B te amplitude of te solution converges to 3/4. Te solid and dased lines on te orizontal ais represent te linear stability of te constant (zero amplitude) steady states. 3. Dynamics: te effect of canging te mobility eponents, n and m Here we vary n and m in t = ( n ) B( m ) wile keeping m n fied. We call te eponents n and m mobility parameters, since tey determine te diffusion coefficients of te fourt and second order terms in te equation. Since q = m n + is being kept fied, te energy landscape and steady states of te evolution are uncanged. Te linear stability properties of te constant and positive periodic steady states are also uncanged [8]. We ask tree questions about te effects of canging n and m in tis way: () Can a eteroclinic orbit between steady states be broken, or is it merely perturbed? () Can te type of a singularity be altered (e.g. from finite-time to infinitetime)? (3) Can te number of singularities be altered (e.g. from one to two per period)? In te following, we are interested in eteroclinic orbits between smoot steady states and between smoot steady states and droplet steady states. For tis reason, we take q < or q > 7/4 since for tese q-values one can find linearly unstable positive periodic steady states (see Figure ). For specificity, we present results for q = 5/ simulations. We observed similar penomena for oter values of q.

7 EJDE /95 HETEROCLINIC ORBITS 7 We did not ave to work ard to capture te t limit and so we epect tat our infinite time limits are not saddle points. Te reader wo is curious about infinite time limits tat are saddle points sould refer to [, 6, ]. 3.. Perturbing eteroclinic orbits between smoot steady states. First we fi q = 5/ and find a positive periodic steady state ss aving period and area. It is a steady state of (.) wit B = 35.3 and is linearly unstable, by bifurcation diagram i. We perturb ss, computing te solution of t = ( ) B( 5/ ) wit initial data ss +. ss. We find tat te local minimum of te solution remains fied in space and, after a sort transient, it increases to. Te maimum beaves similarly, decreasing to. Tat is, te solution relaes to te constant steady state as t. Tis suggests tere is a eteroclinic orbit connecting te positive periodic steady state to te constant steady state. We ten vary te mobility coefficients, taking n =,,, and 3, coosing m in turn so tat q = m n + = 5/. Tat is, we compute solutions for te four evolution equations, all wit te same initial data. We find tat all four solutions rela to te constant steady state: te apparent eteroclinic orbit is not broken by tis cange in mobility. But tere are differences attributable to te cange in mobilities: in Figure 3 we plot min (t) and ma (t) versus time for te four solutions. Te larger te eponent n, te longer it takes for te solution to rela to te constant steady state. We eplain tis as follows. Because te mean value of (, t) is, min (t) < for all t and ma (t) > for all t. Tus at eac time, for near te minimum point one as = (, t) > (, t) > (, t) > (, t) 3, suggesting tat te larger n =,,, 3 is, te slower te fourt-order diffusion will be (near te minimum). Similarly, since te maimum is larger tan we ave = (, t) < (, t) < (, t) < (, t) 3 near te maimum, suggesting tat te larger n is, te faster te diffusion will be (near te maimum). Tis conflict of timescales appears to be mediated troug te conservation of mass: te solution moves as slowly as its slowest part. Tus te larger n is, te slower te diffusion. 3.. Canging te type of singularities. Above, we considered a eteroclinic orbit from a positive periodic steady state to a constant steady state. Perturbing in te opposite direction, we find te solution wit initial data ss. ss appears to converge towards a droplet steady state. Tis raises te question of weter te solution will be positive and ence classical its entire time of eistence, or weter tere migt be times at wic te solution is zero at some points, in wic case te solution is weak. Te equation t = ( n ) as been te study of etensive computational work on ow te eponent n affects te spatial structure of singularities and weter tey occur in finite or in infinite time [3]. Simulations suggest tere is a critical eponent < n < suc tat if n > n ten solutions are positive for all time wile solutions can touc down in finite time if n < n. Here, we seek te analogous critical eponent n (q). Te mobility coefficients in t = ( n ) B( m ) can affect weter a positive solution can become zero somewere in finite time. For eample, if 7/ < n m < n + ten it as been proved tat te solution stays positive for all t >, by [4, 4.]. Note tat m < n + means q < 3. For q > 3, Bertozzi and Pug [4] conjecture tat solutions could blow up, wit (, t) H in finite

8 8 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE / ma (t), min (t) time Figure 3. q = 5/. We compute solutions of te evolution equation were te initial data is fied and (m, n) equals (3/, ), (5/, ), (7/, ), and (9/, 3). We ten plot ma (t) and min (t) versus t for te four solutions: dotted n =, dot-dased n =, dased n =, solid n = 3. Te larger n is, te slower ma and min converge to. time. However, te metods of [5] do apply to sow tat tat if n > 7/ ten te solution remains positive for as long as it eists. Tus te critical eponent for touc down singularities satisfies n (q) 7/. For q = and n =, Goldstein et al. [3, 4] presented simulations suggesting a finite-time singularity is possible. Bertozzi and Pug [4] presented numerical simulations for q = and n = 3 in wic te solutions remain positive for all time, toug tey appear to converge to droplets as t. So it seems tat < n () < 3. For q = 5/, we find tat wen n = te solution appears to touc down in finite time, implying n (5/) 7/. To furter approimate n (5/), we performed simulations wit a variety of n-values, all wit te initial data ss. ss. Our findings suggest.65 < n (5/) <.665. Te evidence is presented in Figure 4, were we plot log min (t) versus t. If min (t) is decreasing at an eponential rate ten te grap will be linear. If min (t) is decreasing to zero in finite time wit an algebraic rate ten te grap will drop to at tat time wit a vertical slope. From Figure 4, if n = ten min decreases monotonically in time, eventually decreasing wit an eponential rate. For n =.75,.7,.675, and.665, min decreases and ten increases, ultimately decreasing wit an eponential rate. (We ran te simulations many decades beyond tose sown, to verify te eponential rate of decrease.) Te solutions wit n <.665 appear to be toucing down in finite time. However te n =.665 simulation gives a note of caution; it is possible tat te simulations wit n <.665 would run until min became quite small but would ten increase

9 EJDE /95 HETEROCLINIC ORBITS 9 log ( min (t)) time Figure 4. We fi te initial data and compute te solution of (.) were q = 5/ and n =,.5,.5,.55,.6,.65,.665,.675,.7,.75,and. We plot log ( min (t)) versus t. (In te plot, te graps move rigtwards as n increases.) For te first five values of n te grap appears to go to in finite time. somewat and ultimately decrease eponentially. We stopped our simulations wen te solutions were no longer numerically resolved (see A.4 A.5). To ceck tat te bound on n (5/) is not dependent on our coice of initial data, we cose ten large random perturbations wit v =.78 ssmin. Seven of te resulting solutions relaed to te constant steady state and tree appeared to touc down in finite time. We ten fied tose initial data and varied te value of n. For all tree initial data, we found te above upper and lower bounds on n (5/). Te critical eponent n (q) certainly depends on te value of q. For eample, for q = /, we observe similar penomena to te q = 5/ case presented in Figure 4, and conclude.8 n (/) <.85. To sum up, we find tat canging te mobility coefficients (m, n) wile keeping te energy landscape fied can cange te nature of trajectories across te energy landscape: if n > n (q) ten te solution is smoot and classical at all times, wile if n < n (q) ten tere can be times wen te solution is weak Splitting singularities. Here we demonstrate anoter effect of canging te mobility: a solution tat evolves towards touc down at just one local minimum (per period) can cange into a solution wit two local minima (per period). q = 5/. If n > and q = 5/ ten positive periodic smoot solutions of (.) remain bounded in H for as long as tey eist, since (, t) H M < by [4]. Because te energy E decreases in time, we epect tat tese solutions will converge to a steady state, as t. Te positive periodic steady state is linearly unstable, and so we epect solutions to converge eiter to te constant steady state or to a configuration of steady droplets.

10 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE /95 We take te same initial data as in 3.. Tere, we found tat if n <.665 n (5/) ten solutions appear to touc down in finite time, and if n > n (5/) ten touc down is in infinite time. Here we investigate te nature of te touc down more closely. We find for n = tat te solution touces down in finite time at one point per period, consistent wit a long time limit of one droplet per period (left side of Figure 5). For n = te solution appears to be positive at all times and to touc down at two points per period in te long time limit (rigt side of Figure 5). Tis is consistent wit a long time limit of two steady droplets per period. But it is impossible to contain two zero contact angle steady droplets in an interval of lengt (see A.6.). In fact, we find tat te small proto-droplet flanked by te local minima is actually draining, wit its maimum decreasing to zero like t / / / / Figure 5. q = 5/. Dased line: initial data. Te same initial data is used for bot simulations. Solid lines: te solution at various times. Left: n = ; min (t) occurs at =. As t increases, min (t) decreases and ma (t) increases. At all times tere is one minimum per period. Rigt: n =. Again, as t increases min (t) and ma (t) decrease and increase respectively. Initially tere is only one local minimum but after some time it splits into two. Te minima flank a small proto droplet and suggest a possible long time limit of two droplets per period. But, in fact, te proto droplet drains away as min. Our simulations suggest tat a second critical eponent, wic we call n (q), governs te number of local minima per period. If n < n (q) ten tere is one minimum per period. If n > n (q) ten tere are two local minima per period, wit teir positions moving in time. Tat is, te single local minimum splits into two as n increases troug n (q). Goldstein et al. [3, 4C] observed someting in a similar spirit for t = ( ) B( ), namely a single symmetric singularity tat splits into a pair of asymmetric singularities as B increases past B.35. We find tat < n (5/) <.5. In Figure 6 we plot late time profiles for a range of n. In te top plot, we plot te solutions near = for n = and.5. Te n = profile as only one local minimum, wile te n =.5 profile as two. In te bottom plot, we plot te solutions near = for n =.5,.6 and n =.7. Eac profile as two local minima, wit te distance between te minima increasing wit n. For eac n.5, we

11 EJDE /95 HETEROCLINIC ORBITS Figure 6. q = 5/. Fied initial data, late time profiles for te solution of (.) computed wit different values of n. Top: n = (single minimum) and.5 (two minima). Bottom: n =.5,.6,.7, were te distance between te two minima increases wit n. find tat te proto-droplet is draining wit its maimum decreasing to zero wit a rate tat depends on n. Te prase splitting singularities is peraps a bit of a misnomer. Te two local minima described above do not appear to touc down in a way tat yields isolated singularities. Rater, te solution between tem appears to be toucing down tey are te endpoints of a developing dry interval. A similar penomenon was observed by Constantin et al. [9, III,IV] wit n = and B = (and wit different boundary conditions): teir proto-droplet decayed like /t. q = /. For tis value of q, we took initial data ss. ss and observed penomena similar to tose in te q = 5/ case, finding < n (/) <.5. We observe beavior similar to tat sown in Figure 5 for q = 5/. Specifically, te long time limit appears to be one droplet. Tis is interesting since, unlike for q = 5/, if q = / ten it is possible to ave two disjoint steady droplets in an interval of lengt (see A.6.). And so te proto-droplet could, in teory, converge to a steady droplet. Noneteless, like for q = 5/, te proto-droplet appears to drain away. We ave not been able to find eamples of q, n, and initial data tat yield a solution wose long time limit is a configuration of more tan one steady droplet per period. 4. Dynamics: q and its effect on eteroclinic orbits We now consider te evolution equation wit a variety of q-values, taking a wide range of initial data near steady states. Te resulting solutions display a diversity of beaviors. Our stability teorems [8, 9] often allow us to predict te observed

12 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE /95 sort time beavior, and our teorems on te energy levels of steady states [9] often allow us to guess te long time limit of te solution. As part of tis work we predict (and find strong numerical evidence for) eteroclinic connections between certain steady states. Interestingly, tere are several cases were we cannot predict te long-time beavior of te solution; in particular see te cases q = and 3/ below. We present results for seven values of q: q = 3, /,, 3/,.768, 5/, 4, cosen from te intervals {(, ], (, ), (, 7/4], (7/4,.794), (.795, 3), [3, )} in wic our teorems from [7, 8, 9] suggest te solutions will display distinct beaviors. (How te above intervals were cosen will become clear in wat follows. Also, we did study oter values of q and found tat te penomena reported ere are robust.) For q = 3 we take n = 3 and m =. For te oter si q-values we take n = and m = q. Our numerical simulations are not greatly affected if we cange n and m in a manner tat keeps q fied, ecept for te features reported in Section q = 3: te van der Waals case. Caracteristic features for q (, ]: positive periodic steady states are linearly unstable and tere are no droplet steady states wit acute contact angles. (See bifurcation diagram a and [7,.].) We study te equation t = ( 3 ) B( ), (4.) for wic q = 3. Te equation was proposed by Williams and Davis [8] to model a tin liquid film wit net repulsive van der Waals interactions, and more recently it as been studied by Zang and Lister [3] and by Witelski and Bernoff [9, 3] q = 3. Perturbing te positive periodic steady state. First, a positive periodic steady state for B =.893 is constructed. It is linearly unstable (see Figure a). Tere is at least one linearly unstable eigenfunction for tis steady state [8] and ence te unstable manifold is at least one-dimensional. Te weakly nonlinear stability analysis suggests tat, at least for nearly-constant positive periodic steady states, te unstable manifold is eactly one dimensional. Even perturbations. Te steady state is even and te evolution equation preserves tis, after an even perturbation. First we perturb ss wit te even, zero-mean perturbation ±ɛ ss. Since te perturbation +ɛ ss lowers te maimum and raises te minimum, one migt ope te resulting solution would converge to te constant steady state. If tis appens for all small ɛ, ten tis would be strong evidence for eistence of a eteroclinic orbit connecting ss to te constant steady state. Perturbing in te direction +. ss, we find tat after a sort transient, te local minimum increases (and maimum decreases) to. Te etremal points remain fied in space, wile te solution relaes to te constant steady state as t ; see Figure 7. (Tis was observed previously in [3, Figure 4b].) We repeated te simulation for smaller values of ɛ and found tat all te perturbations yielded solutions tat relaed to te constant steady state as t. Tis is convincing evidence tat eteroclinic orbits connecting ss to te constant steady state eist. Tere are also teoretical reasons to suspect tese eteroclinic orbits eist: (i) ss is energy unstable in te directions ± ss by [9, Teorem ], (ii) te energy of ss is iger tan tat of te constant steady state by [9,

13 EJDE /95 HETEROCLINIC ORBITS / / Figure 7. q = 3, n = 3. Dased: initial data ss +. ss. Solid: te solution at a number of times. Te local etrema are fied in space and, after a sort transient, min and ma converge monotonically to and te solution relaes to te constant steady state. Teorem 6] (also observed numerically by Witelski and Bernoff [3, 3]), and (iii) te constant steady state is a local minimum of te energy E by [9, Teorem ]. Net we perturb in te opposite direction wit. ss, so tat te maimum is raised and te minimum lowered. Since te perturbation decreases te energy E, we migt epect te solution to subsequently converge to a droplet steady state or to a configuration of droplet steady states. If suc a droplet eists it must ave 9 contact angles, by [7,.]. We find tat after a sort transient, te minimum eigt of te solution decreases in time, appearing to decrease to zero in finite time. Te top left plot in Figure 8 presents te evolution of te solution near =. Te local etrema are fied in space, wit te solution appearing to touc down at one point per period. (Tis was sown previously in [3, Figure 4c].) Computing te derivative of te solution, we find tat its maimum and minimum values grow as time passes, as in te bottom left plot of Figure 8. Tese etremum points of move in time, moving toward = as te singular time approaces. Tis is consistent wit a solution tat touces down wit 9 contact angles in finite time. Te rigt plots in Figure 8 sow a late time profile of. Te work of Zang and Lister [3, 5] on similarity solutions suggests tat (, t) (t c t) /5 H(/(t c t) /5 ) as toucdown approaces; ere t c is te time of toucdown and H is a particular positive function wit H(η) (.87) B /4 η / for large η. Our computations are consistent wit tis ansatz. See [9, 3] for more on te similarity solutions of (4.). After te singular time, one possible beavior of te solution sown in Figure 8 is tat te solution becomes a nonnegative weak solution. Ten it migt rela, as a weak solution, to a droplet steady state wit 9 contact angles. Alternatively, te solution migt, at some later time, become positive and classical again, ultimately

14 4 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE / / / Figure 8. Left top: dased line, initial data ss. ss; solid lines, solution at later times. Te local minimum is fied in space and, after a sort transient, decrease monotonically to zero. Left bottom: dased line, initial slope; solid lines, at same times as above. As t passes, increases and te positions of te etrema move in toward =. Tis suggests 9 contact angles are forming as te singular time approaces. Rigt top: solution at a late time. Rigt bottom: a close up near = at te same time. relaing to te constant steady state. Tis is certainly possible since eac profile sown in Figure 8 as iger energy E tan te constant steady state. We note tat pursuit of tis question will require furter analysis, because te current weak eistence teory for te evolution equation requires n m, wereas ere we ave m < n. Furter, te current weak solution teory does not admit 9 contact angles unless tey occur for a set of times of zero measure. It may be tat completely new tecniques will ave to be developed, to study tis equation. Oter perturbations. To ceck te degree to wic te beaviors described above depend on te coice of perturbation, we performed a number of runs wit random perturbations. (See A.7.) We found tat all te solutions eiter relaed to te constant steady state or else appeared to touc down in finite time. (Te gross dynamics are as in Figures 7 and 8; te finer dynamics concern te positions of te local etrema as a function of time.) 4... q = 3. Perturbing te constant steady state. Suppose B < 4π, so tat te constant steady state is linearly stable wit respect to zero mean perturbations of period. By [9, Teorem ], te constant steady state is a strict local minimum of te energy, and is dynamically stable. Tis is te uninteresting case. Now suppose B > 4π. Ten te constant steady state is a saddle point for te diagram a), we suspect tat a perturbation of te constant steady state will yield a solution tat touces down in finite or infinite time. To investigate, we first take B =.467 < 4π. For all initial data we considered, we found tat te resulting solutions appear to rela to te constant steady state. Tis is as predicted. We ten took B = 63.7 > 4π. Here, te constant steady state as 39 linearly unstable eigenmodes and is a saddle point of te energy. For initial data. cos(π), te solution appears to touc down in finite time, wit one toucdown per period. Figure 9 sows tis evolution over two periods.

15 EJDE /95 HETEROCLINIC ORBITS / / / / Figure 9. q = 3, n = 3. Dased: initial data. cos(π). Solid: solution at various times. Top: te sort time dynamics. Te local minimum decreases, wile te local maimum increases for a wile. Te top ten flattens and two local maima form one to eac side of te flat region. Bottom: later-time dynamics. Te solution appears to touc down at one point per period and continues to ave two local maima per period. Te local minimum is at = and, after a sort transient, decreases monotonically to zero. Remarks. In our studies of positive periodic steady states and constant steady states, we cecked tat te observed penomena persist wit smaller perturbations: te beaviors are robust. For tis reason, in te remainder of te article we will not discuss smaller perturbations. Furter, we will not discuss random perturbations or odd perturbations, since we found tat te observed penomena were like tose observed for even perturbations. (Ecept tat for even perturbations te local etrema are fied in space wile for oter perturbations tey move sligtly as te solution evolves.) We also will not present any furter discussions of perturbations of te constant steady state. We found tat te beaviors were always tose predicted by te bifurcation diagram if te constant steady state is linearly stable ten small perturbations converge back to it, wile if te constant steady state is linearly unstable ten perturbing it yields a solution tat converges to a stable positive periodic steady state, if one eists. If none eists, ten we found tat te solution eiter touces down in finite or infinite time (q < 3) or else it blows up in finite time (q 3). 4.. q = /. Caracteristic features of q (, ): positive periodic steady states are linearly unstable. A Mountain pass scenario can occur te energy of te non-constant positive periodic steady state is iger tan te energies of te constant steady state

16 6 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE /95 and te zero-contact angle droplet steady state. (See bifurcation diagram b and remarks after [9, Teorem ].) We compute solutions of t = ( ) B( / ). First, a positive periodic steady state for B = is constructed. It is linearly unstable (see Figure b). As in 4.., we perturb ss wit ±. ss. For te initial data ss +. ss, we see a eteroclinic connection to te constant steady state, very muc like in te q = 3 case sown in Figure 7. For te initial data ss. ss we find te solution appears to touc down in finite time. Like te q = 3 simulation in Figure 8, min (t) is located at =, and after a sort transient te minimum decreases monotonically in time. But unlike te q = 3 simulation, te profile seems to touc down wit zero contact angles. Tere does eist a zero-angle droplet steady state ĥss tat as te same area as ss, as lengt less tan, and as lower energy tan ss, by [9, Teorem 7]. Presumably tis droplet steady state is te intended long-time limit of te solution, up to translation. But tis cannot currently be proved, because te known zeroangle weak eistence teory requires < n < 3 and q. One migt suspect, based on our simulations, tat tis weak eistence teory sould be etendable to q > q =. Caracteristic features for q = : all positive periodic steady states are linearly neutrally stable. (See bifurcation diagram c and [9, Lemma 4].) Te non-constant positive periodic steady states are neutrally stable, wen q =. We take n = m = and compute solutions of t = ( ) B( ). Numerical simulations for q = ave been presented before by oter autors: wit m = n = in [3], wit m = n = in [4, 8], and wit m = n = 3 in [4]. But te latter two articles do not consider Bond numbers for wic periodic steady states migt be observed. In te first article, Goldstein et al. [3, Fig. 3a] found tat fairly large multi-modal perturbations of positive periodic steady states yield relaation to (generally different) steady states. First, we constructed a positive periodic steady state for B = In te left plot of Figure, we present two simulations confirming tat tis steady state is dynamically stable, wit a small perturbation yielding convergence to a nearby positive periodic steady state. In bot cases, te solution relaes to a positive periodic steady state wit a local minimum close to = and an amplitude close to.8. We ave no rule for predicting te amplitude of te long time limit and, unless te perturbation is even, we ave no way of predicting te position of te local minimum. Te long time dynamics will be especially difficult to predict wen q = since tere are infinitely many -periodic steady states all aving area. (In te q case tere are at most two suc steady states.) Since tese simulations suggest tat te positive periodic steady states are dynamically stable, one migt guess tat solutions cannot touc down in finite time. (Tis is wat we observe later for q = 3/ and q =.768.) And as te bottom left plot of Figure suggests, initial data tat as a sarp local minimum will likely not evolve towards touc down; te local minimum retracts in time, as epected

17 EJDE /95 HETEROCLINIC ORBITS / / / /.6.5 / / Figure. q =, n = ; dased line: te initial data. Left: te solid line is a late time profile of te solution. We found tat te solution relaes to a steady state close to te original steady state.8 cos(π). Top left: () =.8 cos(π) +.3v() were v is a random zero mean perturbation. Bottom left: () =.8 cos(π).9 ep( sin (π)) +.9 ep( sin (π( /))). Rigt: () =.43 cos(π) +.74 cos(4π). Solid lines are at various times. Te etrema are fied in space and, after a sort transient, decrease/increase monotonically wit min (t) toucing down in finite time. for a solution of a surface tension driven flow. On te oter and, initial data tat is very flat near its local minimum does appear to lead to touc down in finite time, as sown in te top rigt plot of Figure. Te bottom rigt plot sows tis evolution near te touc down point q = 3/. Caracteristic features for q (, 7/4]: positive periodic steady states are linearly stable. (See bifurcation diagrams d e.) We compute solutions of t = ( ) B( 3/ ). We constructed a positive periodic steady state ss for B = 4.7. It is linearly stable (see Figure d). Also, every perturbation of te same and sorter period increases te energy, by [9, Teorem 5], and so we epect to observe relaation back to a translate of ss. Tis is precisely wat our simulations sow. We ave not been able to predict te amount of translation tat occurs, but tere is some ope of progress ere, since impressive results on a similar translation problem ave been obtained in [7, 8] for te Can Hilliard equation t = (( 3 ) ) on te wole real line. Net consider zero-mean perturbations of longer period, to wic ss is linearly unstable by [8, Teorem ]. We ask, to wat long-time beavior does tis instability give rise? Te perturbation. cos(π), for eample, raises te local minimum of ss at = and lowers it at =. Te top plot of Figure presents te resulting evolution of te -periodic solution. Te solution appears to touc down in finite time, toug interestingly, it does not do so at =. Instead te toucdown is driven by a dramatic increase in te solution near =. Te bottom plot of Figure sows a close up of te final resolved solution. Te smaller droplet, centered on =, is not close to a steady droplet since it contains a

18 8 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE / / 3/..5 / 3/ Figure. q = 3/, n =. Top plot: dased line is initial data ss. cos(π); eavy solid line is final resolved solution; ligt solid lines are at various times. After a sort transient, (, t) increases monotonically. At late time, a pair of local minima form to eiter side of = and touc down in finite time. Bottom plot: close up of te final resolved solution. local minimum witin itself an impossibility for a steady droplet. Terefore we epect te solution would continue to evolve as a nonnegative weak solution, relaing eiter to a single steady droplet or to some (unknown) configuration of steady droplets q =.768. Caracteristic features for q (7/4,.794): some positive periodic steady states are linearly stable, wile oters are linearly unstable; and tere can be more tan one positive periodic steady state wit te same period and area. (See bifurcation diagrams f, and [7, 5.].) We compute solutions of t = ( ) B(.768 ). Now te possibility arises of a eteroclinic connection between two fundamentally different positive periodic steady states. For Bond number B = we found two distinct positive -periodic steady states, ss and ss, tat ave area. We denote te steady state tat as larger amplitude by ss, and te oter by ss. We epect ss to be linearly stable and ss to be unstable, by [9, Teorem 9], wit ss aving lower energy. Tat is, ss lies on te stable branc of te bifurcation diagram g and ss lies on te unstable branc. Te constant steady state is linearly stable because B < 4π. Our simulations confirmed tese predictions. First, all small perturbations of ss resulted in solutions tat relaed back to ss. All perturbations of ss yielded solutions tat eiter connect to te constant steady state or to ss. Figure sows a typical pair of solutions.

19 EJDE /95 HETEROCLINIC ORBITS /.5.5 / Figure. q =.768 and n =. Te ligt solid lines are at a sequence of times. Te etrema are fied in space. (a) Dased: initial data ss +. ss; eavy solid: constant steady state. Solution relaes to constant. After a sort transient, ma and min converge monotonically to and relaes to te constant steady state. (b) Dased: initial data ss. ss; eavy solid: ss. After a sort transient, ma increases and min decreases monotonically, wit relaing to ss q = 5/. Caracteristic features of q (.795, 3): positive periodic steady states are linearly unstable. Mountain pass scenario can occur. (See bifurcation diagram i and remarks after [9, Teorem ].) We compute solutions of t = ( ) B( 5/ ). We ran one simulation, for B = 35.3, and found beaviors tat were qualitatively te same as for q = /, in q = 4. Caracteristic features of q [3, ): positive periodic steady states are linearly unstable, and if a positive periodic steady state and a zero-angle droplet steady state ave te same area, ten te period of te former is less tan te lengt of te latter. (See [8, Teorem 7] and te proof of [9, Teorem 7].) We compute solutions of t = ( ) B( 4 ). (4.) Tis equation is super-critical in te sense of Bertozzi and Pug [4], since m > n+ (i.e. q > 3). According to teir conjecture in [4], positive periodic solutions can blow up in finite time, wit (, t). Bertozzi and Pug made te same conjecture for compactly supported weak solutions on te line, and proved in [5] tat blow-up can occur in finite time wen n = and m n + = 3. Specifically, tey proved for suc cases tat if te compactly supported initial data as

20 RICHARD. S. LAUGESEN & MARY C. PUGH EJDE / / Figure 3. q = 4, n =. Dased: initial data ss. ss. Solid: at a sequence of times. Te etrema are fied in space and, after a sort transient, ma increases monotonically towards infinity. After a sort transient, min decreases monotonically to a positive value. negative energy E( ) <, ten te compactly supported weak solution blows up in finite time wit its L and H norms going to infinity. Here we present computational evidence tat positive periodic solutions of (4.) can also blow up in finite time. Furter, we find initial data tat as positive energy yet still appears to yield finite time blow-up, suggesting tat E( ) < is not necessary for blow-up, in te periodic case. We took B =.6, and found a linearly unstable positive periodic steady state. We considered initial data ss ±. ss. Te initial data ss +. ss yielded a solution tat relaed to te constant steady state. Te initial data ss. ss yielded a solution tat appears to blow up in finite time (see Figure 3). A self-similarity ansatz suggests tat (, t) (t c t) /7 H(( /)/(t c t) 3/4 ) as blowup approaces. Here t c is te time of blowup and H is a positive function wit H(η) C η /3 for large η. Our simulations are consistent wit tis ansatz. Self similar blow-up for super-critical eponents as also been found for t = ( 3 ) B( 6 ) (presented by Bertozzi and Pug at te APS Division of Fluid Dynamics meeting, November 997). 5. Conclusions and Future Directions We ave numerically studied te evolution equation t = ( n ) B( m ). Our work suggests tat te energy landscape troug wic solutions travel is fairly simple and tat understanding te relative energy levels of te steady states gives considerable insigt into te dynamics of te solutions. In particular we ave found strong evidence for te eistence of eteroclinic connections between steady states,

21 EJDE /95 HETEROCLINIC ORBITS connections we conjecture to eist based on our teorems [9] on te relative energy levels of steady states. It is an open problem to prove analytically tat tese connections eist. In 3 we presented numerical results on te persistence of eteroclinic connections under canges in te mobility parameters n and m. We canged n and m in a way tat preserved q = m n +, preserving te energy E and ence te steady states and energy landscape. But te timescale of te dynamic solution did cange noticeably in response to canges in te mobilities, even toug te sape of te solution canged little. We would epect tese structural stability observations to continue to old if te mobilities were canged in a way tat, wile not fiing q, perturbed it only a little (so tat te energy landscape is also perturbed only a little). In 3 we furter investigated critical mobility eponents, suc as te critical n above wic solutions remain positive for all time (in oter words, te critical eponent for film rupture or pinc-off). An interesting question for te future is to find formulas for te critical mobility eponents. Tese critical eponents determine important qualitative features of te evolution, and determining tem would sed considerable ligt not only on te equation studied ere but also on related equations tat arise from pysical models. Lastly, in 4 we demonstrated tat te pysical quantities P, A, B, and m n appear to fully determine te large-scale features of te evolution, since tey determine te steady states and te energy landscape via te bifurcation parameter E = BA q P 3 q. Finer details, suc as te motion of te etrema in time, depend on te initial data. Also, we ave not been able to predict te amount of translation of a solution tat migt occur in te long-time limit. Anoter open problem is to determine precisely te long-time limit of an evolution tat approaces a steady droplet configuration. In part te difficulty is tat translates of steady droplets are also steady droplets, so tat even if one knows te lengt and area of eac droplet in te configuration, teir locations relative to eac oter must still be determined. Tus droplet attractors migt ave rater ig dimension. In conclusion, we ope our numerical investigations of te power law evolution (.) will provide resources, ideas and motivation for researcers studying t = (f() ) (g() ) wit non-power law coefficient functions f and g. Some suc numerical studies eist already. For eample, te papers [4, 3] consider an f tat is degenerate (f() = ) and g s tat are not power laws, and tere is of course a large literature on te (non-degenerate) Can Hilliard equation. Acknowledgments. R. S. Laugesen was partially supported by NSF grant number DMS-9978, by a grant from te University of Illinois Researc Board, and by a fellowsip from te University of Illinois Center for Advanced Study. He is grateful for te ospitality of te Department of Matematics at Wasington University in St. Louis. M. Pug was partially supported by NSF grant number DMS-99739, by te MRSEC Program of te NSF under Award Number DMR , by te ASCI Flas Center at te University of Cicago under DOE contract B34495, and by an Alfred P. Sloan fellowsip. Te computations were done using a network of workstations paid for by an NSF SCREMS grant, DMS Part of te researc was conducted wile enjoying te ospitality of te Matematics Department and