ON OSCILLATIONS OF SOLUTIONS OF THE FOURTH-ORDER THIN FILM EQUATION NEAR HETEROCLINIC BIFURCATION POINT

Transcription

1 ON OSCILLATIONS OF SOLUTIONS OF THE FOURTH-ORDER THIN FILM EQUATION NEAR HETEROCLINIC BIFURCATION POINT V.A. GALAKTIONOV arxiv:3.76v [math.ap] Dec 3 Abstract. The behaviour of solutions of the Cauch problem for the D fourth-order thin film equation (the TFE 4) u t = ( u n u xxx ) x in R R +, where, mainl, n ( 3,], is studied. Using the standard mass-preserving (ZKB-tpe) similarit solution of the TFE 4, a third-order ODE for the profile occurs: u s (x,t) = t 4+n, = x = ( f n f ) + t /(4+n) 4+n (f) = = f n f + n+4 f =. For the Cauch problem, the oscillator behaviour of near the interface was studied in [3, 6], etc. It was shown that a periodic oscillator component ϕ(s), describing the actual sign-changing behaviour of u s (x,t) close to finite interfaces, exists up to a critical homoclinic bifurcation exponent n h, i.e., for < n < n h ( 3,n +), where n + = = Careful numerics show that n h = In the present paper, a non-oscillator behaviour of for n [n h,], i.e., above the heteroclinic value, is revealed b a combination of analtical and numerical methods. In particular, this implies that, for n [n h,] (and, at least, up to 3), the Cauch setting coincides with the standard zero-height-angle-flux free-boundar one (with non-negative solutions), studied in detail in man well-known papers since the end of 98s, including Bernis Friedman s seminal paper [] of 99.. Introduction: TFE 4, first similarit solution, and main result.. The TFE 4 and first mass-preserving similarit solution. We stud the first mass-preserving self-similar solution of the fourth-order thin film equation (the TFE 4) in one dimension (.) u t = ( u n u xxx ) x in R R +, where, inthe most of the cases, n ( 3,] isafixed exponent. Equation (.) is written for solutions of changing sign, which is an essential feature for the Cauch problem(cp) to be considered, though, in a supercritical parameter n-range(see below), solutions turn out to be non-negative, as happens for the standard and much more well-studied (at least, since Date: Ma 9, Mathematics Subject Classification. 35K55, 35K65. Ke words and phrases. 4th-order thin film equation, the Cauch problem, oscillator solutions, critical heteroclinic exponent n h, ZKB similarit solution.

2 98s) free-boundar problem (FBP) with zero-height, zero-contact angle, and zero-flux conditions; see references in [3] and in several other papers mentioned below. Thus, we will need well-known similarit solutions of the TFE 4 (.). Source-tpe mass-preserved (i.e., of a ZKB-tpe) similarit solutions of the FBP for (.) for arbitrar n were studied in [6] for N = and in [5] for the equation in R N. More information on similarit and other solutions can be found in [, 3, 4, 7, 9, ]. TFEs admit non-negative solutions constructed b singular parabolic approximations of the degenerate nonlinear coefficients. We refer to the pioneering paper b Bernis and Friedman [] (99) and to various extensions in [,, 8,, 9,, 3, 6] and the references therein. See also [] for mathematics of solutions of the FBP and CP of changing sign (such a class of solutions of the CP will be considered later on). In both the FBP and the CP, the source-tpe solutions of the TFE 4 (.) are (.) u s (x,t) = t 4+n, = x t /(4+n), where, on substitution of (.) into (.), solves a divergent fourth-order ordinar differential equation (ODE): (.3) ( f n f ) + 4+n (f) =. On integration once, b using the continuit of the flux function f n f at the interface, where f =, one obtains a simpler third-order ODE of the form (.4) f n f = 4+n f. In view of existence of evident scaling, we, first, get rid of the multiplier, and, next, 4+n pose the smmetr condition at the origin and the last normalizing one to get (.5) f n f = f for >, with f () =, and f() =. Later on, for asmptotic expansions close to the interface point, assuming that such a solution has its interface as some finite > (so that for ), we perform a reflection, (.6) = f n f = ( )f for small >. From now on, we will mainl concentrate on solutions and the corresponding profiles for the Cauch problem onl, though, for some supercritical range from n ( 3,] and even further, as will turn out, we will conclude that this covers the FBP setting as well.

3 For the Cauch problem, it was formall shown that there exists an oscillator similarit profile of (.3), that are infinitel oscillator as, for not that large exponents (.7) n (,n h ), where n h = ; see [3]. For n (,), existence and uniqueness (up to the mass-scaling) of an oscillator are straightforward consequences of the result in [5] on oscillator solutions for the full divergent fourth-order porous medium equation; see(.4) below. See also some details in [7, 3.7] and interesting oscillator features of similarit dipole solutions of the TFE (.) observed in [8]. Stabilit of such sign changing similarit solutions is quite plausible but was not proved rigorousl being an open problem. Thus, in the present paper, we concentrate on the still unknown parameter range (.8) n [n h,]. Our main goal is to show that, in this parameter range, the above similarit solutions, as solutions of the Cauch problem, are not oscillator, and, moreover, do not change sign in small neingbourhoods of finite interfaces. In addition, we then claim that, for such n s in (.8), similarit (and, most plausibl, not onl those) solutions of the CP and the FBP (see a number of papers mentioned above) coincide.. Heteroclinic subcritical and supecritical ranges We begin with a brief explanation of known results on what happens in the heteroclinic subcritical range (.7), which is important for our final conclusions... Local oscillator structure near interfaces for n (,n h ). Namel, it is known that the asmptotic behaviour of satisfing equation (.4) near the interface point as > is given b the expansion (.) = ( ) µ ϕ(s), s = ln( ), µ = 3 n, where, on substitution to (.5), after scaling, the oscillator component ϕ satisfies the following autonomous ODE, where an exponentiall small as s term (occurring b setting = e s ) is omitted: (.) ϕ +3(µ )ϕ +(3µ 6µ+)ϕ +µ(µ )(µ )ϕ+ ϕ ϕ n =. It was shown that here exists a stable (as s + ) changing sign periodic solution ϕ (s) of (.). The bifurcation value n h was obtained in [3] b some analtic and numerical evidence showing that, asn n h, theode(.)exhibits ausualheteroclinic bifurcation, where a periodic solution is generated from a heteroclinic path of two constant equilibria, [4, Ch. 4]. According to (.), this gives similarit profiles of changing sign, which Not surprisingl and obviousl, this range includes n =, i.e., classic analtic solutions of the CP of the bi-harmonic (parabolic) equation u t = u xxxx obtained from (.) b formall passing to the limit n +. As we have suggested earlier, this natural fact can be used as a proper definition of CPsolutions of (.): these are those that can be continuousl deformed to the bi-harmonic solutions of the CP using this homotopic path. For FBP-solutions, this is not possible [3]: non-negative FBP-solutions for n > converge to similar FBP-ones for n =, which are not oscillator. 3

4 being extended b for > form a compactl supported solution f C α in a neighbourhood of =, with α 3 n. Notice that α + as n +, so the regularit at = improves to C at n = + (and eventuall, to an analtic profile F() for n = for all >, being the Gaussian kernel of the fundamental solution b(x,t) = t 4F(x/t 4) of the linear parabolic operator D t +D 4 x ). Remark: TFE in R N. The above conclusions, at least, formall can be applied to the Cauch problem for the TFE 4 in R N : (.3) u t = ( u n u) in R N R +, with a bounded compactl supported data u (x). Namel, at an point of a sufficientl smooth interface surface, at which there exists a unique tangent hperplane π, the same D oscillator behaviour is expected to occur in the direction of the inward unit normal to π. In other words, the principal phenomenon of existence of oscillator sign-changing behaviour near the smooth interfaces is essentiall D, excluding some special and singular non-generic cases. On the other hand, at possible points of singular cusps at the interfaces (anwa, expected to be non-generic), such an eas approach is not applicable, though almost nothing is known about those singular phenomena for (.3) for an N. Note that the first results on the oscillator behaviour of similarit profiles for fourthorder ODEs related to the source-tpe solutions of the divergent parabolic PDE of the porous medium tpe (the PME 4) (.4) u t = ( u m u) xxxx (m > ) were obtained in [5]. It turns out that these results can be applied to the rescaled TFE (.3), but for n (,) onl. As shown in [3], the corresponding ODEs for source-tpe similarit solutions for (.) and (.4) then coincide after a change of parameters n and m: m = n, n (,). Some results on existence and multiplicit of periodic solutions of ODEs such as (.) are known [3], and often lead to a number of open problems. Therefore, numerical and some analtic evidence remain ke, especiall for sixth and higher-order TFEs; see [4] and [7, Ch. 3]. Returning to the behaviour as s for (.), i.e., approaching the interface, we express the above results as follows: in the heteroclinic subcritical range, there exists a D bundle of asmptotic solutions of (.) in R having the expansion (.) (an exponentiall small term is again omitted) (.5) = ( ) 3 n [ϕ (s+s )+...], with parameters > and s R, where we take into account the phase shift s of the periodic orbit ϕ (s). Here, we should treat as a free parameter, though, in the above setting, it is fixed b conditions at As usual, such a regularit is based of the actual smoothness of the algebraic envelope ( ) µ in (.) and does not involve transversal zeros nearb, at which such a smoothness breaks down. 4

5 =. Recall that, as s, the periodic solution ϕ (s) is thus unstable b the obvious reason: a D unstable manifold is then generated b small perturbations of. Therefore, matching the D bundle (.5) with two smmetr conditions at the origin in (.5) leads to a formall well-posed problem of D matching. However, using this matching approach directl is a difficult open problem (for n (,n h ), where existence of was not properl proved; recall that, for n =, there is a pure algebraic approach to construct such a unique oscillator profile [3]). Note that, in the case of an analtic dependence on parameters involved, such a problem cannot possess more than a countable set of isolated solutions... n ( 3,]: numerical results. Before deriving proper asmptotic expansions in the supercritical range in the next section, we present some numerical evidence that, at n = n h, oscillations of, as a solution of (.5), near the interface disappear. As usual, for numerics, we use the regularization in the third-order operator in (.5) b replacing (.6) f n (ε +f ) n, with ε =, which is sufficient for the accurac required; see below. Note that, for our purposes in the Cauch problem, we principall cannot use the pioneering FBP-regularization, introduced in Bernis Friedman [] (not analtic as in (.6), except n =, to be treated speciall), leading, in man cases, to non-negative solutions. However, we expect that, for n [n h,], both analtic and non-analtic ε-regularizations lead to the same solutions, i.e., the Cauch and FBP settings coincide (though proving that rigorousl is ver difficult). In our numerics b the MatLab, we were often obliged to keep a high accurac, with bothtolerancies, andthetotalnumber ofpointsontheinterval (,3)ofabout 5, though using the simplest ODE solver for the Cauch problem ode45 took, sometimes, 5 minutes for each shooting from = for each fixed value of n. There appears a simple one-parameter shooting problem for the ODE (.5) from = with the onl parameter (.7) f () = µ R. Then, varing µ <, we are tring to approach as close as possible to the corresponding finite interface at some >, and, eventuall, to see whether changes sign or not in a small vicinit of. A few of such results are presented below. Thus, let us begin with the oscillator range (.7). In Figure, our numerics catch a changing sign behaviour as for n =.7 < n h. Our shooting from =, with f() =, even with the above mentioned accurac and with about 5 points, allows us to see just a single zero near. We do not think that, in a few minutes of PC time, as here, the next, to sa nothing of other zeros of, near can be viewed b such a shooting. In the next Figure, we start to approach the heteroclinic bifurcation value n h from below. Namel, we take n =.75, which is still slightl less. We again were able to see the first zero, though to reach that, we increased the number of points, but still observed a non-smooth profile close to the interface. We see that a negative hump in this 5

6 5 x 4 TFE4: shooting for n=.7, oscillator Figure. The first zero, close to =.67, of near = for n =.7. x 7 TFE 4: shooting for n=.75, oscillator Figure. The first zero, close to =.688, of near = for n =.75 < n h. Here, for the last profile, f () = figure, which is alread of the order 7, so one can expect that further approaching n h from below will require a completel different and more powerful computer tools. We next take n ver close to n h, where (.8) n =.75987, f () = µ = , =.697. In this case we enhance our numerics b taking points on the interval (,.7) to get the results in Figure 3. Thus, up to the accurac 7 (and more, as easil seen), is not oscillator for, as our analtical predictions said earlier. 6

7 x 7 TFE 4: shooting for n= n h, non oscillator.8.6 µ= , interface point.4 µ= Figure 3. Non-oscillator near = for n = n h. Here, for the last profile, µ = f () = x 5 TFE 4: shooting for n=.8, non oscillator Figure 4. Non-oscillator for n =.8 > n h. Now, consider the supercritical range above n h. Figure 4, constructed for n =.8 > n h (but rather close to it), shows that, with the prescribed accurac, we do not see an sign changes near the interfaces, unlike the previous figures. In Figure 5, we show a global structure of this, which looks like being non-negative, but, to verif that, one needs more microscopic structure near the interface given in the previous figure. Finall, for completeness, in Figure 6, we present a full view of for the special case n =, where both analtic and non-analtic (for the FBP) regularizations coincide. Again, microscopicall, no sign changes of were observed. 7

8 TFE 4: shooting for n=.8, non oscillator Figure 5. The non-oscillator profile for n =.8 > n h TFE 4: shooting for n= Figure 6. The non-negative profile for n =. Wedonotexpect (anddidnotsee) anessential changes bfurtherincreasing n (,3) until the special n = 3 to be treated next..3. n = 3: logarithmicall perturbed linear asmptotics and existence of for the CP. For n = 3, similar to the case n = 3 for the FBP3, we observe a logarithmicall perturbed linear asmptotics (we do not reveal the second term, since n = 3 is 3 Recall that, for the CP, the case n = 3 is not special, and for all n (,n h) the solutions are equall oscillator near the interface, governed b a periodic oscillator component. 8

9 TFE 4: n=3, µ=.4,...,, 3, Figure 7. Non-oscillator for n = 3, the bold-face line. For it, µ = f () =.74..., = currentl out of our main interests): as, (.9) = 3 ( ) ln( ) The corresponding shooting of is presented in Figure 7. Obviousl, such similarit solutions with the expansion(.9) do not satisf the zero contact angle condition(however, other FBPs can be posed), but can be proper solutions of the Cauch problem, so that this behaviour at interfaces can be that of a maximal regularit..4. n = 4: nonexistence of. As a simple illustration of the general results for the TFE 4 in [], saing that solutions of this PDE are strictl positive for all n large enough, consider our ODE (.5) for the next integer n = 4. The results of shooting from = are shown in Figure 8: no solutions for µ = f () from to can reach the zero level f =. It is clear that one does not need an proof (though it is alread available in [], and in much more general PDE fashion). Anwa, as an ODE illustration again, we see from (.6), = b scaling, multipling b f and integrating that, keeping the main singular terms, (.) f f = (f ) + However, the resulting majorising ODE (.) f f = f +... does not admit increasing positive solutions for > small (here, = and f() =, recall). Indeed, multipling b f and integrating ields: (.) 3 (f ) 3 = f f f < for an small f >. 9

10 TFE 4: n=4, nonexistence of ; µ [, ] Figure 8. Nonexistence of with a finite interface for n = Positive expansions near interface for n ( 3,3) Thus, we consider the ODE (.6) close to the interface =, b scaling. Assuming that > for small > ields the following equation: (3.) f n f = +. First of all, we cannot use the standard parabolic expansion from dozens of recent papers, since it is suitable for the FBP for n (, 3 ). Therefore, we are looking for an expansion about a different profile (an explicit solution of (3.) with on the RHS): (3.) f () = B m, where m = 3 n and B n = [ 3 n( 3 n )( 3 n)], which makes sense for an (3.3) n ( 3,3). We next use an algebraic perturbation of (3.) b setting (3.4) = B m +D l +..., where l > m = 3 n and D R is an arbitrar parameter (recall that we need a D bundle to shoot a single smmetr boundar condition f () = ). Substituting into (3.) ields (3.5) D[B n (n )m(m )(m )+B l(l )(l )] m(n )+l 3 = +... One can see that equating both algebraic functions on both sides of (3.5) is no good: this ields a unique orbit, and not a D bundle (see below). Therefore, the onl possible wa is to assume that (3.6) m(n )+l 3 < = 3 n < l < + 3 n,

11 l= (,.5) for n= H n (l) l Figure 9. The graph of H (l) in (3.8) (for n = ): l = and equate to zero the square bracket on the LHS in (3.5). This gives the following cubicall algebraic ( characteristic ) equation for the exponent l: (3.7) l : l(l )(l ) = B n (n ) 3 n( 3 n )( 3 n). It is worth mentioning that a literal balancing m(n )+l 3 = in (3.8) just means that D = in such an (3.4), i.e., this orbit also belongs to the above D bundle. After simple transformations, this is reduced to the cubic equation (3.8) l : H n (l) l(l )(l ) (n ) [ 3 3 n( )( )] 3 n =. n n Here, we are looking for positive real roots l of (3.8) satisfing two conditions in (3.6). We begin our stud of a proper solvabilit of the nonlinear characteristic equation (3.8) with the analtic case n =, as shown in Figure 9, where we present the graph of the algebraic function H (l). Then we have (3.9) n = : l = ( 3, 5 ), so that conditions in (3.6) holds. Note that (3.8) shows that there alwas exists an l >, and, since > 3 n in our parameter range, l > 3 n alwas. For n =.8 (still non-oscillator), the value of l is explained in Figure, where (3.) n =.8 : l =.8..., also satisfing conditions in (3.6). Finall, we have got that such a positive D bundle perfectl exists also in the oscillator range, i.e., for n < n h : see Figure, where (3.) n =.7 < n h : l =

12 l=.8... (,.5) for n= H n (l) l Figure. The graph of H n (l) in (3.8) for n =.8: l =.8... l= (,.5) for n=.7.5 H n (l) l Figure. The graph of H n (l) in (3.8) for n =.7 < n h : l = Concerning a proper justification of the actual existence of the bundle (3.4), (3.8) on a small interval [,δ], on one hand, this can be done b reducing the third-order non-autonomous ODE (3.) to a dnamical sstem in R 4. On the other hand, since the right-hand side of (3.) f = f n is singular at =, there occurs an integrabilit problem. Indeed, a direct first integration of (3.) over a small interval (, ) is impossible since, in the present parameter n-range, f () = +. Therefore, one has to use another representation of the equation.

13 To clarif this, consider, e.g., the case n =. We then integrate (3.) over (,) to get (3.3) ff (f ) = ( ), where all accompaning integrals are convergent at =. Hence, [ (3.4) f = f (f ) ( )]. Note that, in the class of solutions close to (3.4) (with m = 3 for n = ), both terms on the right-hand side of (3.4) are equall involved into the behaviour of as +. Integrating (3.4) two times, with all integrals convergent, since m = 3 = 3 >, so n f () is locall bounded, we arrive at a sstem for {f,f } of a standard form ( ) ( ) f f (3.5) f = M f, with an integral operator M, easil reconstructed from (3.4), being a contraction in C([,δ]) for sufficientl small δ > (meaning C for f) in a properl chosen invariant neighbourhood of each orbit from (3.4) for an fixed D R. As a measure of this neighbourhood, one can use the third term in (3.4), i.e., that one appeared for D = (we have mentioned it above). One can see from (3.8) that a proper characteristic root l exists for all n ( 3,3), so that a positive expansion (3.4), (3.8) close to interfaces for the Cauch problem alwas exists in this whole parameter range. Moreover, one can see that, in this parameter range, (3.6) n ( 3,3) : f() = f () = (f n f )() =, i.e., standard zero-high, zero-contact angle, and zero-flux conditions, which are requirements of the FBP (for the CP, exhibiting, usuall, a smoother behaviour at the interface, those are also valid), take place. We, thus, naturall, come to our: 4. Final conclusions Thus, we have seen that the positive expansion (3.4), (3.8) exists for all n in our range n ( 3,], and also for all n (,3), though this range is out of our current interests. So, the following natural question arises: wh then do we need to take into account an oscillator behaviour for 3 < n < n h? The answer is as follows:. It turns out that the shooting, via the positive D bundle (3.4), (3.8) posed at =, the single smmetr condition at the origin f () = is consistent not for all n s, and onl for n n h.. So, there exists a heteroclinic bifurcation point n = n h, starting with which using positive expansion near the interface becomes inconsistent, so a new oscillator D bundle is necessar, which allows to shoot a proper similarit profile. 3

14 A naive analog in linear spectral non-self-adjoint theor is: a spectral parameter λ, starting at some critical value of a parameter, must become complex, with a nonzero imaginar part (meaning oscillations of eigenfunctions), in order to make consistent algebraic sstems arising. As a consequence and as an answer of a possible (quite reasonable) critics concerning a definite lack of rigorous results in the present paper on the actual global existence of the similarit profiles for the Cauch problem in our range n ( 3,], it is worth mentioning that an kind of shooting 4 using the positive or oscillator bundle at the interface must reveal this critical exponent n h. Recall that n comes from the ODE (.) b using a microscopic blow-up analsis of the ODE (.5) for. We do not think that an king of a global-tpe stud of this ODE (.5) (without a blow-up one as ) can reveal this critical heteroclinic exponent. 3. Finall, bearing in mind (3.6) and the uniqueness of an admissible (positive) expansion, we end up with the following expected conclusion: (4.) for n [n h,3), the FBP and CP have the same solutions, which was alread announced in our earlier papers, but, unfortunatel, without sufficientl convincing details. Acknowledgements. The author would like to thank J.D. Evans for discussions and good questions, essentiall initiated the present research, to be continued for the sixthorder thing film equation (the TFE 6; in lines with our previous papers [4], where all asmptotic/numerical tools get more complicated). References [] J. Becker and G. Grün, The thin-film equation: recent advances and some new perspectives, J. Phs.: Condens. Matter, 7 (5), S9 S37. [] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differ. Equat., 83 (99), [3] F. Bernis, J. Hulshof, and J.R. King, Dipoles and similarit solutions of the thin film equation in the half-line, Nonlinearit, 3 (), [4] F. Bernis, J. Hulshof, and F. Quirós, The linear limit of thin film flows as an obstacle-tpe free boundar problem, SIAM J. Appl. Math., 6 (), [5] F. Bernis and J.B. McLeod, Similarit solutions of a higher order nonlinear diffusion equation, Nonl. Anal., 7 (99), [6] F. Bernis, L.A. Peletier, and S.M. Williams, Source tpe solutions of a fourth order nonlinear degenerate parabolic equation, Nonl. Anal., 8 (99), [7] M. Bowen, J. Hulshof, and J.R. King, Anomaluous exponents and dipole solutions for the thin film equation, SIAM J. Appl. Math., 6 (), [8] M. Bowen and T.P. Witelski, The linear limit of the dipole problem for the thin film equation, SIAM J. Appl. Math., 66 (6), E.g., a naive direct approach using (3.4), (3.8): for D, we have f () >, and, for D, we get f () <, so that, it seems, there must exist a D in between such that f () =. But this does not work, in general, since, for n < n h slightl, no such solutions were observed. Clearl and in addition, the above primitive analsis cannot reveal existence of an n h. 4

15 [9] E.C. Carlen and S. Uluso, Asmptotic equipartition and long-time behaviour of solutions of a thin film eqwuation, J. Differ. Equat., 4 (7), [] J.A. Carrillo and G. Toscani, Long-time asmptotic behaviour for strong solutions of the thin film eqwuations, Comm. Math. Phs., 5 (), [] C.M. Elliott and H. Garcke, On the Cahn Hilliard equation with degenerate mobilit, SIAM J. Math. Anal., 7 (996), [] C. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal., 96 (986), [3] J.D. Evans, V.A. Galaktionov, and J.R. King, Source-tpe solutions of the fourth-order unstable thin film equation, Euro J. Appl. Math., 8 (7), [4] J.D. Evans, V.A. Galaktionov, and J.R. King, Unstable sixth-order thin film equation. I. Blow-up similarit solutions; II. Global similarit patterns, Nonlinearit, (7), , [5] R. Ferreira and F. Bernis, Source-tpe solutions to thin-film equations in higher dimensions, European J. Appl. Math., 8 (997), [6] V.A. Galaktionov and P.J. Harwin, On centre subspace behaviour in thin film equations, SIAM J. Appl. Math., 69 (9), (an earlier preprint in arxiv:9.3995). [7] V.A. Galaktionov and S.R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Phsics, Chapman& Hall/CRC, Talor and Francis Group, Boca Raton, FL, 7. [8] L. Giacomelli, H. Knüpfer, and F. Otto, Smooth zero-contact-angle solutions to a thin film equation around the stead state, J. Differ. Equat., 45 (8), [9] H.P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84 (978), [] L.V. Govor, J. Parisi, G.H. Bauer, and G. Reiter, Instabilit and droplet formation in evaporating thin films of a binar solution, Phs. Rev. E, 7, 563 (5). [] G. Grün, Degenerate parabolic equations of fourth order and a plasticit model with non-local hardening, Z. Anal. Anwendungen, 4 (995), [] R.S. Laugesen and M.C. Pugh, Energ levels of stead states for thin-film-tpe equations, J. Differ. Equat., 8 (), [3] A. Oron, S.H. Davies, and S.G. Bankoff, Long-scale evolution of thin liquids films, Rev. Modern Phs., 69 (997), [4] L. Perko, Differential Equations and Dnamical Sstems, Springer-Verlag, New York, 99. [5] N.F. Smth and J.M. Hill, High-order nonlinear diffusion, IMA J. Appl. Math., 4 (988), [6] T.P. Witelski, A.J. Bernoff, and A.L. Bertozzi, Blow-up and dissipation in a critical-case unstable thin film equation, Euro J. Appl. Math., 5 (4), Department of Math. Sci., Universit of Bath, Bath, BA 7AY address: masvg@bath.ac.uk 5