REMARKS ON QUASILINEAR EVOLUTIONS EQUATIONS

In this paper we study the existence result of classical solutions for the quasilinear equation utt−Δu−M(∫Ω|∇u|2dx)Δutt=f, with initial data u(0)=u0, u(1)=u1 and homogeneous boundary conditions.

follo the same pattern as Lions's book [2].
Ebihara et al [I] was proved that there est only one classical solution for a selinear del, given by following initial-boundary value pr oblem En C1 D COD .COD u when the followlng hotheses hold.CiD PICA_) E CICO,+oa.),arid tkere eZst pos[t)e constants a, p such that d ere M and for 9 we are denoting the do--in of the operator Ms. e in result of thxs paper is to prove the exxstence resu!t of c!assical solutions for ystem C 1.1-C 1.3] when HI. H is a continuos /urction such that.HCk9 >_ m > 0 ClCx,lZ */z + Ix,0l Fom here t ollo pIying the re!ation above to t have" From the two Iast inequaIities we concIude: t)IZ Finally, from the hotheses, the last inequality nd Gronwll "s inumlity the reEult of Le By the compacity of [O,T], there exist s I, sz' N' satisfying rand from the intwiEe convergence of P conclude thmt there m sitive nur N such that I IPm''s,3 P,..sIZ < . ,m,, N. , N C. [ IP (x,t-x,s3}{=2 /= + [ IP x,s=-P x,s)lZ2 REMARK 2..-UNIQUENESS: If M is 1 ocaz z y Lipschitz, then we have uniqueness.In fact, let and be two solutions, putting w uwe have , ,',w-Mc'.FcIV,alZd.x.),t cMc'J" IVulZd.x.) Me j" IVulZd.x),,,u Multi pl yi ng by Att appl yi ng HI and t he Li pschi t z condi t i on on H we have that there exists a positive constant c such that: only one cLczsEc we conclude that there exists a subsequence of Cu(m).)me which we still denoting of the same way and a function u sati sfyi ng From the last convergences and the Lions-Aubin's theorem Csee Lions's [-], theorem 5.1, chap 1) we conclude in particular that- u u sLron[y n CCEO,2"1; o By .tandardmethods we can prove that u is a strong solution of system

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: (honorZ eigen  functions and eigen value of the Laplacan respectively.Let's denote by V the finite dimensional vector space generated to see that ASP P A s in AS).Moreover'we have lhat m en the aproted problem i defined a follow.utmtAu(m MC nlVu(mIzdx>Au /to prove the main rezult of this paper we will show the fol !owl ng Lemmas-LEMMA 2.1.-Let' s suppose a u, u, uu.C('O, T-izC'fl.)_) (:znd O

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hal I y by C 2.1 ), C .3),(  .4) and the fol I owl ng I nequal i ty IP x. -P x. zlt7 /* .1), (I.E) and CI.33.Remains to how that is a classical solution.