On the asymptotic behavior for a damped oscillator under a sublinear friction. J. I. Díaz and A. Liñán


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1 h corresponds RAC Rev. R. Acad. Cien. Serie A. Mat. VOL. 95 (1), 2001, pp Matemática Aplicada / Applied Mathematics Comunicación Preliminar / Preliminary Communication On the asymptotic behavior for a damped oscillator under a sublinear friction. J. I. Díaz and A. Liñán equation! " where#%$ &" (')* Abstract. We show that there are two curves of initial data for which the solutions of the corresponding Cauchy problem associated to the as+",(."/10 vanish after a finite time. By using asymptotic and methods and comparison arguments we show that for many other initial data the solutions decay to, in an infinite time, Sobre el comportamiento asintótico de un oscilador amortiguado con un término de fricción sublineal iniciales ecuación 2 3 % 45 " Resumen. Mostramos la existencia de dos curvas de datos para las cuales las soluciones correspondientes del problema de Cauchy asociado a la supuesto#$6 &7 (', se anulan como+",(+"/0 identicamente despues de un tiempo finito. Mediante métodos asintóticos y argumentos de comparación mostramos que para muchos otros datos iniciales las soluciones decaen a " en un tiempo infinito, 1. Introduction We study the asymptotic behavior of solutions of the equation 8:93; (1) wherei!jlkg2mnpo and>qmre:stg UWe shall 92; ;D<V?93;)?A2B=C93;W<59:FHG work with the formulation 9 KYOZF\[ C*]^_A2B=C*`9 KbaYO (2) and[gf wherea Fdcce f ikj*lm n.opje mqopj h f UNotice c fe remaining 9 ; ; <ty zb{+ K9 ;OD<!9}~G (3) which is attaint by dividing bye and by introducing the rescaling that the9 rescaling fails fori FrN. In that case there is no well defined scale for9 to9; ;s<t[d92;s<9ufvg and the equation is reduced with[f as a parameter to characterize the dynamics. The limit caseixw G to the Coulomb friction equation Presentado por Gregorio Millán Recibido: 16 de Abril Aceptado: 10 de Octubre Palabras clave / Keywords: sublinear damped oscillator, Coulomb friction, extinction in a finite time. Mathematics Subject Classifications: 34D05, 70K05,70K99 c 2001 Real Academia de Ciencias, España. 155
2 is the andœ corresponding corresponds it andœ G). is, J. I. Díaz and A. Liñán wherey z { of ƒ byy z { is the maximal monotone graph given if if ˆ G2M Š_ N"M N) if FHG MandN K& OQF N, StG UThe limit equation wheniuw N with the linear damping equation We recall that, even if the nonlinear term?9 ;?A BDC9 ; is not a Lipschitz continuous function of9 ; the existence and uniqueness of solutions Œ of the associate Cauchy problem AŽ 99; ; KG+OQF 92 +MW9 <V?9 ;?A2B=C 9 ;KGOQF P ; <!9:FHG Y S G M (and of the limit problemsœ C to the equations (3) and (4) respectively) is well known in the literature: see, e.g. Brezis [1]. An easy application of the results of the above reference yields to a rigorous proof of the convergence of solutions whenixw G andixw N. The asymptotic behavior, fory w Mof solutions of the limit problemsœ C well known (see, 9 š for instance, Jordan Smith [5]). In the first case the decay is exponential. In the second one it is easy to see that given93 anyyz and P there exist a finite time F 4K9 "M* P PO and a number LJ Š NMN) such that for 4K93 +M* P PO. For problemœ (ŒA is wellknown thatk9 (M*9 ;*O wœkg MRG+O asy w (see, e.g. Haraux [4]). For a numerical study of A) see [6]. The main result of this paper is to show that the generic asymptotic behavior above described for the limit caseœ tokg MRG+O caseižjžkg M NqO tokg MGO only exceptional for the sublinear since the generic orbitsk9 ŸM*9 ;O decay in a infinite time and only two uniparametric families of them decay in a finite time: in other words, whenixw G exceptional behavior becomes generic. 9 ; ; <59 ; <!9:FHG U (4) 2. Formal results via asymptotic arguments We can rewrite the equation (2) in as the planar system 9 ; F which, by eliminating the time variable, forg F G, leads to the differential equation of the orbits in the phase plane 4F BDC (6) F š K 9O ifžf K9=O and that allows us to carry out a phase plane description of the dynamics. We remark that the plane phase is antisymmetric since is a solution of (6) then the function is also solution. So, it is enough to describe a semiplane (for instance9 By multiplying by9 andm respectively, we get thatk9 <V O; 9 KNP q9wm Nq P O pointkg2mgo <! S SHN F š?d?a+ sc. On the other hand, it is easy to see that satisfy a system which has the as a spiral unstable critical point. For values of the orbits of the system are given, in first approximation, by9 < FH because? =?A2B=C is small compared with9. The effect of this term is to decrease slowly For[5S[ªš«FV with time giving the trajectory a spiral character. ForI FvN the character of the trayectories close to the origin depends on the parameter[ƒu the origin is a stable mode and for[ ˆ [ª Iuw < is a stable spiral corresponding to underdamped oscillations. It should be noticed that fori S N originkg MGO the origin becomes a stable spiral point. The limit case can be described analytically with twotime scale methods (see [3]). We shall prove that there are two modes of approach to the origin and so that the is a node for the system (5). The lines of zero slope are given by ; F BDC (5) 9~F? =?A2B=CU (7) 156
3 G. On the asymptotic behavior for a damped oscillator 9 ˆ G MTS tokg2mg+o regions +K9WM Oš«9LS G MP ˆ 9 C*]RA= +K9WM*2Oš«So the convergence is only possible through the of9us G K 9=OC]AW. Let us see that the ordinary mode corresponds ˆ Let ŽF to orbits that are very close to the ones corresponding to small effects of the inertia. Due to the symmetry it is enough to describe this behavior for the orbits approaching the origin with values and 6S G UEquation (6) takes the form F²9 C]A < ³ K9=O isuf±9 C]RA The line of zero slope and we search for orbits obeying, forg ˆ 9 ˆ ˆ N, to the expression for some function³ K9O. If we anticipate the conditiong ˆ ³ K9O ˆ ˆ 9 C]RA, equation (6) takes the linearized form C A 9 ^q m B=C*`³µ<9 m ³ I 9 ^ CŸB m `³ŽFgG UThus the first term can be neglected, compared with the last one, and then the solution can be written with as³ K9OQ V u¹)º» ŠIƒ " KN+ I Ob 9 B ji lm n m thatž 9 an arbitrary C]A constant ¼; (which explain the name of ordinary orbits). This type of orbits are given, close to the origin, by the approximate equation ½9 (7), which for the orbits that reach the origin from below implies and so, integrating the simplified equation F 9 < A U (8) ½Y F 9 C]A (9) we get that 9 ¾ ŠKN I OKY I < YCO A]^CŸBA` (10) and so that it takes an infinite time to reach the origin. for#:5 0 Figure 1. The two exceptional orbits and, in the small figure, several ordinary orbits entering to the origin tangentially to the zero slop curve Some different orbits approaching the origin can be found by searching among solutions with large 157
4 G) and we, andî andî of can grows J. I. Díaz and A. Liñán with?9à?c*]ra values of?d?compared. Thus, close to the origin, the orbits with negative are very close to the solutions of the equation found by replacing (8) by the simplified the equation F A (11) ( corresponding to a balance of inertia and damping. The solution ending at the origin KGOšF\G) is given by for5s G Notice that it involves no arbitrary constant. So this curve is unique (a symmetric curve arises and9 9 ˆ ˆ ˆ which justifies the term of extraordinary orbit. The time evolution of this orbit is given, for N, by integrating the equation ½9 and so9 ŽF ^ BA+`Š^ BA`p^ CŸBA` C A KY YO ^ BA+Ầ]^C(BA+`Mwhere in generalãs FÅÄŽÆ7º that " ±Ç ŠK I OD?93? C*]^ BA+` G MÃs. This indicate that the motion (of this approximated solution) ends at a finite time,y, determined by the initial conditions which, by (13) must satisfy We point out that the two exceptional orbits emanating from the origin spiral around the origin when9 < toward infinity and È so G each of them is a separatrix curve in the phase plane. Notice that due to the autonomus nature of the equation, if9 (Œ datak93 +M* P 7O is the solution of the Cauchy problem A) of initial then for any parameter the function 9 «kfž9 KY < È O (Œ coincides with the solution of A) of initial datak9 KÈ O(M*93;KÈ (Œ O*OŸUIn (Œ this way, the above extraordinary orbits give rise to two curves of initial data which the corresponding solutions of A) vanish after a finite time. We end this section by pointing out that the solution of problem A) forg ˆÉIÊˆ ˆ N andi w G % F G takes an asymptotic form which can be easily described. The differential equations of the orbits simplify if is finite to " 6F² 96 HN forls G and 6F² 9Ë<gN at9ufž N at9f\ N or9ufín if6s G forìf² xs G2UThe solutions are circles with center and center9f±n intervalkg MNPO ˆ G if joined. An orbit formed with half circles with centers when it hits the from below it is transformed into an orbit that reaches the origin following very closely that segment, governed by the equation (9) of solution (10). In the limitilw G found that any point J (Œ Š_ N"M N) is an asymptotically stable stationary state of ). 3. Estimates on the decay K9=O¾Fv +K µ I Op9 C*]^ BA+`U (12) ½Y FvŠK Á I O9 C*]^ BA+` (13) In a previous work, by using a fixed point argument, we show Theorem 1 ([2]) There exists two curvesî B datak93 +M* P qo initial for which the solutions9 of the corresponding Cauchy problemkœ A O vanish after a finite time. andžfv It is possible to give some additional results on these two curves: Theorem 2 (i) Near the origin the curvesî B be represented by two functions,ufï K9=O someð4s G U B K9O(Msolutions of the equation (6), where «ŠG2M1Ðq w K MG7 and B «Š_ ÑÐMRGP w ŠG MŸ< O, for (ii) Functions B satisfy that ƒò KGOQF G Óˆ Ô!Õ ½y KyPO andô ½y B Õ In particular, ÀÖ K9OØ Ž when9ž G and QÖB K9=OØÙ < when9ëùág2u B KyPO ˆ < U (14) 158
5 implies andî by On the asymptotic behavior for a damped oscillator Ú µ9 K 9O j*lm Û K9O Û 9 m for9 m Û J B K9O Û K 9O j*lm for9 ŠG M1Ðq and J Š ÑÐMRGP pm JÝKG M*Ðq K9Ü OÌFÞ K9ÜO m andk 9ÜO m F B for some \STG. K 9=O «F 9 J ŠG M*9 Ü such K9Ü O. Moreover and K9O Û Û 9 m,ß B «F 9 J Š 9 ÜMG7 m Û Û B K9O +K9sM*2O: K9sM*2O: and are (iii) We have a9ü (iv) There exists regionsß that time invariants for equation (6). the In order to prove that the decay to zero in an infinite time is more generic than the decay to zero in a finite time we need to obtain sharper invariants regions 9 aàtj±kg M9ÜO Theorem 3 (i) There exists small enough such that the regionsß:á «F +K9WM*2O Jâß «J ŠG2M*9 Ü à and 9 m à ¹)º» ŠIØ 7 KN I Op.9 B ji lm n m ˆ Û 9 m «F K9WM2O Jgß B «9 J Š_ 9 Ü < àmrgp andk 9=O m Û Û K 9=O m < à ¹)º» ŠI " KNZ I Ob 9 B jpi lm n m O IfK93 M P qo J are (ii) (respectivelyßëáb (Œ Û Ú some \S G YBA])^ CŸBA` YBA])^ CŸBA`) for B :Fg K9O :Fg B and K9=OŸMis clear and92; Fž K9 O K9 *O F š respectively. A FÅ 9 m and µ9 K9OãS G, with is J KG M1ÐOj*lm M. respectively, ß assured andß B å LetK93 +M* P 7O å å A Û 9 onkg M*93 7O PROOF OF THEOREM 3.. and å K93 PO å K9=OŽFçæ+9 K9O Û å å C]RA <V³ K9O K9O P. Then, onkg2m*93 qo. Such can with³ K9=O FÊ u¹)º» ŠIØ 7 KN I Ob 9 B jpi lm n m with ès²g ³ K9O A 9 j*lm m < 9 æa9 C ˆ G arbitrary for9 JâKG2M*9 ÜO ifæ!sïg ˆ æ+9 is C]A). Finally, for initial data inß á Ú P9 m fory: G m Uä we and Acknowledgement. The research of JID was partially supported by the DGICYT, project REN Both authors thank to Miguel Hermanns for the elaboration of the Figure. time invariants for equation (6). ) then the solution9 of A) satisfies (respectively9 that9 and anyy STG. PROOF OF THEOREM 2. Since the solutions of equation (6) converge to the origin with zero orç (Œ slope the representation of curvesî once we assume?9ø?small enough. Then withy fory ÍY, large enough the solutions9 equation9=;4f² of the corresponding Cauchy problem A) vanish after a finite time.and satisfy the first order ordinary differential Therefore the necessary conditions (14) holds by (easy) well known results. To prove (iii) we call (as in the previous So, for instance,žf a solution of the new orbits equation It is easy to see that this equation admits some explicit sub and supersolutions of the form9 and so the result holds by a comparison argument. Finally, the existence of9 Ü in (iv) can be obtained, for instance, by using the inequalities of (iii) and equation (6). The direction of the flow at the boundaries of the time invariance of these regions.ä let us construct an auxiliary function satisfying that by comparison arguments (forward in the9 direction) we get that necesarily a function be constructed in the form but small enough (use the fact thatéj suitably chosen and that get that9 ; the decay inequality follows from the comparison with the exact solution of9 ;QFv Ú 79 References [1] Brezis, H. (1972). Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, NorthHolland, Amsterdam. [2] Díaz, J. I. and Liñán, A. (2002). On the asymptotic behavior of solutions of a damped oscillator under a sublinear friction term: from the exceptional to the generic behaviors. To appear in Advences in PDE, Lecture Notes in Pure and Applied Mathematics (A. Benkirane and A. Touzani. eds.), Marcel Dekker. 159
6 J. I. Díaz and A. Liñán [3] Díaz, J. I. and Liñán, A., On the dynamics of a constrained oscillator as limit of oscillators under an increasing superlinear friction. To appear. [4] Haraux, A. (1979). Comportement à l infini pour certains systèmes dissipatifs non linéaires, Proc. Roy. Soc. Edinburgh, 84A, [5] Jordan, D. W. and Smith, P. (1979). Nonlinear Ordinary Differential Equations, (Second Edition), Clarendon Press, Oxford. [6] Kuo PenYu and Vazquez, L. (1982). Numerical solution of an ordinary differential equation, Anales de Física, Serie B, 78, J. I. Díaz A. Liñán Facultad de Matemáticas E. T. S. I. Aeronaúticos Universidad Complutense de Madrid Universidad Politécnica de Madrid Madrid, Spain Madrid, Spain 160
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