Asymptotic Behavior of Bounded Solutions to a First Order Gradient-like System

In this paper, we prove the long time behavior of bounded solutions to a rst order gradient-like system with low damping and perturbation terms. Our convergence results are obtained under some hypotheses of KurdykaLojasiewicz inequality and the angle and comparability condition.


Introduction
he min gol of this pper is to otin the symptoti ehvior of ounded solutions to the grdientElike system s follows u (t)+γ(t)u(t)+G(u(t)) = f (t), t ∈ [0, ∞), @IA where the unknown u(t) ∈ R n D the dmping term γ ∈ L 1 (R + , R + )D the perturtion term ) is tnE gent vetor (eld on R n F oughly spekingD we study in this pper the e'et of dding low dmping term γ(t)u(t) nd perturtion term f (t) to the eqution u (t) + G(u(t)) = 0, t ∈ [0, ∞), @PA on the long time ehvior of the trjetories uF his type of prolem hve een studied in mny reent ppers with di'erent ssumptions of GF he typil sitution of @PA is the se of grdient system when G = ∇F F his grdiE ent system ws studied y mny uthors suh s ID TD IRD ISD IV or PIF sn the lssiE l resultD they proved tht the ounded soluE tion onverges to n equilirium s t goes to inE (nity if the funtion F is rel nlyti in IVF wore lterD F ghill et al.V estlished n generl result whih gurntees tht the onE vergene result lso holds for the grdientElike system @PAF his onvergene result ws proved under the hypotheses of the vojsiewiz inequlE ity of F nd the ngle ondition of G nd ∇F F sn IW nd PHD the uthors extended the result y uurdykEvojsiewiz inequlityF woreoverD the onvergene rtes ws otined if F stis(es vojsiewiz inequlity nd G, ∇F stisfy ngle nd omprility onditionF eentlyD F ghill nd wF tendoui U or rung nd k IU onsidered the eqution in the non homogeneous seF hey showed tht ny ounded solution of the grdient system u + ∇F (u) = f (t), t ≥ 0, @QA onverges to ritil point of F t in(nity unE der the following ondition for some positive onstnt µF he foring term f (t) quikly deys to zero s t goes to in(nE ity in this seF sn previous work PPD we proved the onvergene result of @QA under low L 1 Eondition of the perturtion termF woreE overD the rte of onvergene ws even otined under vojsiewiz inequlity of the vypunov funtion F F he onvergene results hve een generlized to some seond order systemsD suh s PD RD SD IP or IQ re referenes thereinF woreoverD wF qhisi et.al. hve estimted the dey rtes for solutions of semi liner dissipE tive equtions in W nd IHF wotivted y these worksD we estlish in this pper the onvergene results for the (rst orE der non homogeneous grdientElike system @IA with the e'et of low dmping nd foring termsF wore preiselyD we onsider the equE tion @IA with γ nd f stisfy the following onE dition sn dditionD we lso ssume some key hypotheE ses suh s the ngle nd omprility ondiE tion of G nd ∇F nd the uurdykEvojsiewiz inequlity of F s in mny other rtilesF he nie feture of ngle nd omprility ondiE tion of G nd ∇F is tht G is oinident with the grdient of F with respet to iemnnin metri gF e refer the reder to the rtile of frt et.al. Q for the detilF nder these sE sumptionsD we prove tht the ounded solution u to eqution @IA onverges to ritil point ϕ ∈ ω[u] t in(nity nd u ∈ L 1 (R + )F he pper is orgnized s followsF sn the next setionD we present some ssumptions nd defE initions tht we use through the whole of the pperF e lso rell the existene of iemnE nin metri g suh tht G = ∇ g F in this setionF sn the lst setionD we estlish the symptoti ehvior of ounded solutions to grdientElike system @IAF yur results re divided into three theorems for the onveniene of the rederF 2.
Preliminaries sn this setionD we give the key ssumption of ngle nd omprility ondition to oE tin the onvergene result of grdientElike sysE temF e lso rell some de(nitions out the vypunov funtionD the uurdykEvojsiewiz inE equlity nd the grdient of funtion with reE spet to iemnnin metriF e onsider ontinuous tngent vetor (eld R)F sn this pperD we lwys sE sume the ngle nd omprility ondition of ∇F nd GD iFeD there exists positive onstnt a > 0 suh tht for ny u ∈ R n there holds Moreover, we say that F is a strict Lyapunov function if ∇F (u) = 0 implies G(u) = 0. e remrk tht F is strit vypunov funE tion if the ngle nd omprility ondition @SA holdsF Denition 2. We say that the function F satises a Kurdyka-Lojasiewicz inequality at η if there exist δ > 0 and a non decreasing function where B δ (η) denotes the ball centered at η and radius δ in R n .
his de(nition is relted to the vojsiewiz heorem in IV elowF Theorem 1.If F is real analytic in a neighborhood of η then F satises the Kurdyka-Lojasiewicz inequality @UA at η. e denote y ∇ g(u) F (u) the grdient of F with respet to iemnnin metri g on R n t uD iFeFD for ny v ∈ R n D we hve @VA por simpliityD we write F e denote y • g the indued normF sn QD the uthors showed tht there exists iemnnin metri g whih is equivlent to the iuliden metri suh tht G = ∇ g F F e rell this reE sult in the following theoremF Theorem 2. Assume the angle and comparability condition @SA of ∇F and G holds.Then there exists a Riemannian metric g on Moreover, there exist positive constants α, β such that Denition 3.For any trajectory u belongs to

Main results
sn this setionD we prove the onvergene of ounded solutions to equilirium of the grdientElike system @IAF he min ide of our work is sed on heorem PF epplying heoE rem PD the grdientElike system @IA n e seen s form of grdient systemF Theorem 3. Let u be a bounded solution of @IA and f, γ ∈ L 2 (R + ).Assume that the angle and comparability condition @SA of G and ∇F holds.
Proof.vet us onsider the energy funtion deE (ned y ds. @IHA he funtion Φ is well de(nedF sndeedD pplying heorem P for G nd ∇F under the ngle nd omprility ondition @SAD we hve G = ∇ g F nd the inequlity @WA holdsF gomining with f, γ ∈ L 2 (R + )D we then otin tht fy the ngle nd omprility @SAD eqution @IA nd heorem PD we n estimte the derivtive of the energy funtion Φ s follows Proof.ine u is ounded solution of equE tion @IAD so the ωElimit set ω[u] is non emptyF , whih tends to H s m goes to in(nityF imilrlyD we lso get , where M is n upper ound of the ounded soE lution uF o we n onlude tht the right hnd sides vnish s m goes to in(nityF gomining these estimtions nd equE tion @IAD it follows tht whih tends to H s m goes to in(nityF pinllyD to (nish the proofD we n present the inner produt of ∇F (ϕ) nd v for ny v ∈ R n s follows his equlity shows tht ∇ g F (ϕ) = 0F Theorem 5. Assume that i) the angle and comparability condition @SA of G and ∇F holds; ii) F satises the Kurdyka-Lojasiewicz inequality @UA and the function Θ in @TA-@UA satises for some positive constant k; iii If u is a bounded solution to equation @IA then u ∈ L 1 (R + ) and u(t) converges to an equilibrium point ϕ ∈ ω[u] at innity.
Proof. e onsider gin the energy funtion Φ de(ned y @IHAF st is similr to the proof of heorem QD we lso get the following estimtion for every t ∈ R + F st implies tht the funtion Φ is non inresingF woreoverD sine u is ounded solutionD the ωElimit set ω[u] is non emptyF vet ϕ ∈ ω[u]D we hve Φ(t) onverges to F (ϕ) t in(nityF e remrk tht y sutrting F (ϕ) if needed we my ssume F (ϕ) = 0F st implies tht the energy funtion Φ(t) is non negtive for every t ∈ [0, ∞)F st is esily to see tht if Φ(T ) = 0 for some T ≥ 0 then u is onstnt for ll t ≥ T F here remin nothing to prove in this seF reneD we now ssume tht Φ(t) > 0 for every t ∈ [0, ∞)F ine the solution u is oundedD so there exists positive onstnt M suh tht fy ssumption ii)D there exist δ > 0 nd the funtion Θ stisfying @TA suh tht for ll u ∈ B δ (ϕ)D there holds e now onsider funtion I de(ned y por ny ε ∈ (0, δ)D y the hypotheses @IPA nd @IQAD there exists t * suh tht vet us de(ne sing the de(nition of energy funtion Φ in @IHA nd the hypothesis @IIAD we otin . eording to uurdykEvojsiewiz @UA nd the equivlene etween iemnnin metri g nd iuliden metri in @WAD we hve . epplying the ngle nd omprility ondiE tion @SAD we get tht where . gomining @ITA with @ISAD @IRA nd @WAD it deE dues tht . @IUA fy eqution @IAD we otin his gives sn the other hndD one hs where the seond estimtion is otined y inteE grting @IUA on [t * , T )F his inequlity implies tht T = ∞ nd lso yields u ∈ L 1 (R + )F st folE lows tht the ounded solution u(t) onverges to equilirium point ϕ t in(nityF

,
tends to ϕ s m goes to in(nityF fy heorem QD we hve t+1 t u(s) 2 ds → 0 s t → ∞. whih goes to H s m goes to in(nityF his mens u(t m + s) = u(t m ) + tm+s tm u(r)dr ⇒ ϕ, where ⇒ denotes the uniformly onvergeneF sing the ontinuity of ∇F D we otin tht ∇F (u(t m + s)) uniformly onverges to ∇F (ϕ) in [0, 1]F woreoverD for ny v ∈ R n D we hve tm+1 tm u(s), v g(u(s)) ds