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Publications results for "Author=3D(EDEN, A*)" =

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MR2196805 = (2006h:35246)=20
Eden,=20 A.(TR-BOG);=20 Erbay,=20 H. A.; Muslu,=20 G. M.(TR-ISTNT)
Two remarks on a generalized Davey-Stewartson = system. (English summary)
Nonlinear=20 Anal. 64=20 (2006),=20 no.=20 5, 979--986.
35Q55=20 (35B40)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "We present two results on a generalized Davey-Stewartson = system,=20 both following from the pseudo-conformal invariance of its solutions. In = the=20 hyperbolic-elliptic-elliptic case, under some conditions on the physical = parameters, we establish a blow-up profile. These conditions turn out to = be=20 necessary conditions for the existence of a special `radial' solution. = In the=20 elliptic-elliptic-elliptic case, under milder conditions, we show that = the=20 $L^p$-norms of the solutions decay to zero algebraically in time for=20 $2<p<\infty$."


References
  1. C. Babaoglu, A. Eden, S. Erbay, Global existence and nonexistence = results=20 for a generalized Davey-Stewartson equation, J. Phys. A-Math. Gen. 37 = (2004)=20 11531--11546. MR2104302=20
  2. C. Babaoglu, S. Erbay, Two dimensional wave packets in an elastic = solid=20 with couple stresses, Int. J. Nonlinear Mech. 39 (2004) 941--949.=20
  3. J. Bourgain, Global Solutions of the Nonlinear Schr=C3=B6dinger = Equation, vol.=20 46, AMS Colloquium Publications, 1999. MR1691575=20 (2000h:35147)
  4. T. Cazenave, Semilinear Schr=C3=B6dinger Equations, AMS Courant = Lecture Notes=20 in Mathematics, vol. 10, 2003. MR2002047=20 (2004j:35266)
  5. R. Cipolatti, O. Kavian, Existence of pseudo-conformally invariant = solutions to the Davey-Stewartson system, J. Differential Equations = 176 (1)=20 (2001) 223--247. MR1861188=20 (2002j:35277)
  6. P. Constantin, Decay estimates for Schr=C3=B6dinger equations, = Commun. Math.=20 Phys. 127 (1990) 101--108. MR1036116=20 (91c:35137)
  7. J.M. Ghidaglia, J.C. Saut, On the initial value problem for the=20 Davey-Stewartson systems, Nonlinearity 3 (1990) 475--506. MR1054584=20 (91h:35299)
  8. J.M. Ghidaglia, J.C. Saut, Nonelliptic Schr=C3=B6dinger equations, = J. Nonlinear=20 Sci. 3 (1993) 169--195. MR1220173=20 (94c:35158)
  9. H. Nawa, M. Tsutsumi, On blow-up for the pseudo-conformally = invariant=20 nonlinear Schr=C3=B6dinger equation, Funkcialaj Ekvacioj 32 (1989) = 417--428. MR1040169=20 (91e:35193)
  10. T. Ozawa, Exact blow-up solutions to the Cauchy problem for the=20 Davey-Stewartson systems, Proc. R. Soc. London 436 (1992) 345--349. MR1177134=20 (93g:35130)
  11. C. Sulem, P. Sulem, The Nonlinear Schr=C3=B6dinger Equation = Self-Focusing and=20 Wave Collapse, Springer, Toronto, 1999. MR1696311=20 (2000f:35139)
  12. M.I. Weinstein, On the structure and formation of singularities in = solutions to nonlinear dispersive evolution equations, Commun. Partial = Differential Equations 11 (5) (1986) 545--565. MR0829596=20 (87i:35026)
  13. M.I. Weinstein, The nonlinear Schr=C3=B6dinger = equation---singularity=20 formation, stability and dispersion, Contemp. Math. 99 (1989) = 213--232. MR1034501=20 (90m:35188)
This list reflects references listed in the = original paper as=20 accurately as possible with no attempt to correct error.=20

Citations

From References: 0

From Reviews: 0

MR2138979 = (2006a:35148)=20
Eden,=20 A.(TR-BOG);=20 Kalantarov,=20 V. K.
On global behavior of solutions to an = inverse=20 problem for nonlinear parabolic equations. (English=20 summary)
J.=20 Math. Anal. Appl. 307=20 (2005),=20 no.=20 1, 120--133.
35K55=20 (35K20 35R30)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

The authors consider an inverse source problem for a semilinear = parabolic=20 heat equation on a smooth (bounded or unbounded) domain with Dirichlet = boundary=20 conditions. The nonlinearity is of power type plus sublinear = corrections. The=20 source term is assumed to be decomposed into a product of a = time-dependent part=20 and a spatial part $w(x)$. The spatial component is assumed to be known, = and the=20 solution $u$ must satisfy a compatibility condition with $w$. By using a = concavity argument plus energy estimates, the authors prove that if the = initial=20 data is sufficiently large with respect to $w$, then the solution blows = up in=20 finite time. They also show that when the power nonlinearity has the = opposite=20 sign and the domain is bounded, solutions exist globally in time and = exhibit a=20 certain decay for $t\to\infty$.

Reviewed by Anna=20 L. Mazzucato

References
  1. C. Budd, B. Dold, A. Stuart, Blowup in a partial differential = equation=20 with conserved first integral, SIAM J. Appl. Math. 53 (1993) 718--742. = MR1218381=20 (94a:35015)
  2. J.R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, = Menlo=20 Park, CA, 1984. MR0747979=20 (86b:35073)
  3. J.R. Cannon, Y. Lin, Determination of a parameter $p(t)$ in some=20 quasi-linear differential equations, Inverse Problems 4 (1988) 35--45. = MR0940789=20 (89h:35333)
  4. A.F. Guvenilir, V.K. Kalantarov, The asymptotic behavior of = solutions to=20 an inverse problem for differential operator equations, Math. Comput.=20 Modelling 37 (2003) 907--914. MR1988554=20 (2004f:35183)
  5. B. Hu, H.-M. Yin, Semilinear parabolic equations with prescribed = energy,=20 Rend. Circ. Mat. Palermo (2) 44 (1995) 479--505. MR1388759=20 (97c:35094)
  6. D.A. Jones, E.S. Titi, Determining finite volume elements for the = 2D=20 Navier-Stokes equations, Phys. D 60 (1992) 165--174. MR1195597=20 (93j:35133)
  7. V.K. Kalantarov, O.A. Ladyzhenskaya, Formation of collapses in = quasilinear=20 equations of parabolic and hyperbolic types, Zap. Nauchn. Sem. = Leningrad.=20 Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977) 77--102. MR0604036=20 (58 #29269)
  8. H.A. Levine, Instability and nonexistence of global solutions to = nonlinear=20 wave equations of the form $Pu_{tt}=3D-Au+F(u)$, Trans. Amer. Math. = Soc. 192=20 (1974) 1--21. MR0344697=20 (49 #9436)
  9. H.A. Levine, Some nonexistence and instability theorems for = solutions of=20 formally parabolic equations of the form $Pu_{t}=3D-Au+F(u)$, Arch. = Rational=20 Mech. Anal. 51 (1973) 371--386. MR0348216=20 (50 #714)
  10. A.I. Prilepko, D.G. Orlovskii, I.A. Vasin, Methods for Solving = Inverse=20 Problems in Mathematical Physics, Dekker, New York, 2000. MR1748236=20 (2001m:35321)
  11. R. Riganti, E. Savateev, Solution of an inverse problem for the = nonlinear=20 heat equation, Comm. Partial Differential Equations 19 (1994) = 1611--1628. MR1294473=20 (95m:35204)
  12. D.D. Trong, D.D. Ang, Coefficient identification for a parabolic = equation,=20 Inverse Problems 10 (1994) 733--752. MR1278585=20 (95c:35269)
  13. I.A. Vasin, V.L. Kamynin, On the asymptotic behavior of solutions = to=20 inverse problems for parabolic equations, Siberian Math. J. 38 (1997)=20 647--662. MR1474910=20 (2000c:35252)
This list reflects references listed in the = original paper as=20 accurately as possible with no attempt to correct error.=20

Citations

From References: 0

From Reviews: 0

MR2117389 = (2005i:35243)=20
Eden,=20 Alp(TR-BOG);=20 Erbay,=20 Saadet
On travelling wave solutions of a = generalized=20 Davey-Stewartson system. (English = summary)=20
IMA=20 J. Appl. Math. 70=20 (2005),=20 no.=20 1, 15--24.
35Q55=20 (35C05 74J30)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "The generalized Davey-Stewartson (GDS) equations, as = derived by C.=20 Babaoglu and S. Erbay \ref[Int. J. Non-Linear Mech. 39 = (2004),=20 no. 6, 941--949], comprise a system of three coupled equations in $2+1$=20 dimensions modelling wave propagation in an infinite elastic medium. The = physical parameters ($\gamma,m_1,m_2,\lambda$, and $n$) of the system = allow one=20 to classify the equations as elliptic-elliptic-elliptic (EEE),=20 elliptic-elliptic-hyperbolic (EEH), elliptic-hyperbolic-hyperbolic = (EHH),=20 hyperbolic-elliptic-elliptic (HEE), hyperbolic-hyperbolic-hyperbolic = (HHH) and=20 hyperbolic-elliptic-hyperbolic (HEH). In this note, we only consider the = EEE and=20 HEE cases and seek travelling wave solutions to GDS systems. By deriving = Pohozaev-type identities we establish some necessary conditions on the=20 parameters for the existence of travelling waves, when solutions satisfy = some=20 integrability conditions. Using the explicit solutions given in \ref[op. = cit.]=20 we also show that the parameter constraints must be weaker in the = absence of=20 such integrability conditions."

Citations

From References: 0

From Reviews: 0

MR2120235 = (2005k:35425)=20
Eden,=20 A.(TR-BOG);=20 Kalantarov,=20 V. K.
On global behavior of solutions to an = inverse=20 problem for semi-linear hyperbolic equations. (English, Russian summary) Zap.=20 Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)=20 318 (2004), Kraev.=20 Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35, 120--134, 310;=20 translation in
J.=20 Math. Sci. (N. Y.) 136 (2006), no.=20 2, 3718--3727
35R30=20 (35L70)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

The paper is devoted to the problem of determination of the function = $F(t)$=20 and the solution $u(t,x)$ of the problem $$u_{tt}-\Delta = u-|u|^pu+b(x,t,u,\nabla=20 u)=3DF(t)w(x),\quad x\in\Omega, \quad t>0,$$ $$u(x,t)=3D0,\quad=20 x\in\partial\Omega,\quad t>0,$$ $$u(x,0)=3Du_0(x),\quad = u_t(x,0)=3Du_1(x),\quad=20 x\in \Omega,$$ $$\int_\Omega u(x,t)w(x)\,dx=3D\phi(t),\quad t>0,$$ = where=20 $\phi(t)$ has a continuous bounded second derivative.

The author = gives=20 two sorts of conditions: conditions providing the absence of a global = solution=20 and conditions for the stability of the solution of the problem. Among = the first=20 type, the key is the condition $\phi(t)\equiv 1$. Among the second type = is the=20 contrary supposition that $\phi(t)$ and its first derivative vanish as = $t\to 0$.=20

The proof is based on a technique of generalized concavity, = developed by=20 the authors, which allow them to prove that a certain function = $\Psi(t)$,=20 satisfying the differential inequality = $$\Psi"\Psi-(1+\alpha)[\Psi']^2\geq=20 -2C_1\Psi\Psi'-C_2\Psi^2,$$ tends to infinity as $t$ tends to a certain = value=20 (which is estimated using the values of constants $C_i$ and initial = conditions).=20

Reviewed by Alexey=20 V. Borovskikh

References
  1. V. Bayrak, M. Can, and F. A. Aliyev, "Nonexistence of global = solutions of=20 a quasilinear hyperbolic equation," Math. Inequal. Appl.,=20 1, 367--374 (1998). MR1629388=20 (2000a:35161)
  2. Ya. Yu. Belov and T. N. Shipina, "The problem of determining the = source=20 function for a system of composite type," J. Inv. Ill-Posed = Problems,=20 6, 287--308 (1988). MR1652101=20 (99k:35188)
  3. M. Can, S. R. Park, and F. A. Aliyev, "Nonexistence of global = solutions of=20 some quasilinear hyperbolic equations," J. Math. Anal. Appl., = 213, 540--553 (1997). MR1470869=20 (98i:35123)
  4. A. Eden and V. K. Kalantarov, "On the global nonexistence of = solutions to=20 an inverse problem for semilinear parabolic equations," J. Math. = Anal.=20 Appl. (submitted).=20
  5. D. Erdem and V. K. Kalantarov, "A remark on the nonexistence of = global=20 solutions to quasilinear hyperbolic and parabolic equations," = Appl. Math.=20 Lett., 15, 521--653 (2002). MR1889508=20 (2003b:35142)
  6. V. Georgiev and G. Todorova, "Existence of a solution of the wave = equation=20 with a nonlinear damping term," J. Diff. Eqs., = 109,=20 295--308 (1994). MR1273304=20 (95b:35141)
  7. A. F. Guvenilir and V. K. Kalantarov, "The asymptotic behavior of=20 solutions to an inverse problem for differential operator equations,"=20 Math. Comp. Modeling, 37, 907--914 (2003). = MR1988554=20 (2004f:35183)
  8. Hu Bei Yin and Hong-Ming, "Semilinear parabolic equations with a=20 prescribed energy," Rend. Circ. Mat. Palermo (2),=20 44, 479--505 (1995). MR1388759=20 (97c:35094)
  9. V. K. Kalantarov, "Collapse of solutions of parabolic and = hyperbolic=20 equations with nonlinear boundary conditions," Zap. Nauchn. Semin. = LOMI, 127, 75--83 (1983). MR0702844=20 (84m:35053)
  10. V. K. Kalantarov, "Blow-up theorems for second-order nonlinear=20 evolutionary equations," in: Turbulence Modeling and Vortex = Dynamics,=20 O. Boratav, A. Eden, and A. Erzan (eds.), Lect. Notes Phys., = Springer=20 Verlag (1997), pp. 169--181.=20
  11. V. K. Kalantarov and O. A. Ladyzhenskaya, "Formation of collapses = in=20 quasilinear equations of parabolic and hyperbolic types," Zap. = Nauchn.=20 Semin. LOMI, 69, 77--102 (1977). MR0604036=20 (58 #29269)
  12. M. Kirane and N. Tatar, "A nonexistence result to a Cauchy problem = in=20 nonlinear one-dimensional thermoelasticity," J. Math. Anal. = Appl.,=20 254, 71--86 (2001). MR1807888=20 (2002a:74042)
  13. R. J. Knops, H. A. Levine, and L. E. Payne, "Nonexistence, = instability,=20 and growth theorems for solutions of a class of abstract nonlinear = equations=20 with applications to nonlinear elastodynamics," Arch. Rat. Mech.=20 Anal., 55, 52--72 (1974). MR0364839=20 (51 #1093)
  14. H. A. Levine, "Instability and nonexistence of global solutions to = nonlinear wave equations of the form $Pu_{tt}=3D-Au+F(u)$," Trans. = Amer.=20 Math. Soc., 192, 1--21 (1974). MR0344697=20 (49 #9436)
  15. H. A. Levine, "Some additional remarks on the nonexistence of = global=20 solutions to nonlinear wave equations," SIAM J. Math. Anal.,=20 5, 138--146 (1974). MR0399682=20 (53 #3525)
  16. H. A. Levine, "A note on a nonexistence theorem for some nonlinear = wave=20 equations," SIAM J. Math. Anal., 5, 644--648 = (1974).=20 MR0361960=20 (50 #14402)
  17. H. A. Levine and L. E. Payne, "Nonexistence of global weak = solutions for=20 classes of nonlinear wave and parabolic equations," J. Math. Anal. = Appl., 55, 329--334 (1976). MR0412575=20 (54 #697)
  18. A. I. Prilepko, D. G. Orlovskii, and I. A. Vasin, Methods for = Solving=20 Inverse Problems in Mathematical Physics, Marcel Dekker, Inc., = New York,=20 Basel (2000). MR1748236=20 (2001m:35321)
  19. Yuming Qin and J. M. Rivera, "Blow-up of solutions to the Cauchy = problem=20 in nonlinear one-dimensional thermoelasticity," J. Math. Anal. = Appl.,=20 292, 160--193 (2004). MR2050223=20 (2005b:35275)
  20. B. Straughan, "Further global nonexistence theorems for abstract = nonlinear=20 wave equations," Proc. Amer. Math. Soc., 48, = 381--390 (1975). MR0365265=20 (51 #1518)
  21. I. A. Vasin and V. L. Kamynin, "On the asymptotic behavior of = solutions to=20 inverse problems for parabolic equations," Sib. Mat. Zh.,=20 38, 750--766 (1997). MR1474910=20 (2000c:35252)
  22. Z. Yang, "Cauchy problem for quasilinear wave equations with a = nonlinear=20 damping and source terms," J. Math. Anal. Appl.,=20 300, 218--243 (2004). MR2100248=20 (2005k:35294)
This list reflects references listed in the = original paper as=20 accurately as possible with no attempt to correct error.=20

Citations

From=20 References: 2

From Reviews: 0

MR2104302
Babaoglu,=20 Ceni(TR-ISTNT);=20 Eden,=20 Alp(TR-BOG);=20 Erbay,=20 Saadet
Global existence and nonexistence = results for a=20 generalized Davey-Stewartson system. (English=20 summary)
J.=20 Phys. A 37=20 (2004),=20 no.=20 48, 11531--11546.
35Q55=20 (37K10)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

{There will be no review of this item.}

References
The reference list for this item has = been=20 removed for technical reasons. =

Citations

From References: 0

From Reviews: 0

MR1911228 = (2003d:37139)=20
Eden,=20 Alp(TR-BOG)
Finite dimensional dynamics on attractors. (English summary) Mathematics & = mathematics=20 education (Bethlehem, 2000), 90--97, World Sci. Publ., River = Edge,=20 NJ, 2002.
37L30=20 (35B41 35K90)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

A finite-dimensional generalized dynamical system is constructed on = the=20 finite-dimensional attractor of a damped hyperbolic equation in a = Hilbert space.=20 The result parallels a result obtained for parabolic equations. The main = ingredient is the H=C3=B6lder-Ma n=C3=A9 projection on a set of finite = fractal dimension.=20

Reviewed by Vittorino=20 Pata

Citations

From References: 0

From Reviews: 0

MR1797526 = (2001i:47063)=20
Eden,=20 Alp(TR-BOG)
A remark on the asymptotic properties of positive = homogeneous maps=20 on homogeneous lattices. (English = summary)=20
Turkish=20 J. Math. 24=20 (2000),=20 no.=20 3, 277--281.
47= B65=20
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "An abstract version of Lyapunov exponents is defined for = positive=20 homogeneous maps on homogeneous lattices and a sufficient condition is = given for=20 the asymptotic stability of the map."

Citations

From References: 0

From Reviews: 0

MR1760257 = (2001b:37111)=20
Eden,=20 A.(TR-BOG);=20 Kalantarov,=20 V.(TR-HCTTS);=20 Miranville,=20 A.(F-POIT-DM)
Finite-dimensional attractors for a general class of = nonautonomous=20 wave equations. (English summary) =
Appl.=20 Math. Lett. 13=20 (2000),=20 no.=20 5, 17--22.
37L30=20 (35B41 35L70)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "Our aim in this note is to construct attactors and = exponential=20 attractors for a general class of nonautonomous semilinear wave = equations.=20 Following the approach described in \ref[V. V. Chepyzhov and M. I. = Vishik, J.=20 Math. Pures Appl. (9) 73 (1994), no. 3, 279--333; MR1273705=20 (95c:34106)], we define a semigroup $S(t)$ associated to an = autonomous=20 system, and then prove, using an energy functional, that $S(t)$ is an=20 $\alpha$-contraction and satisfies the squeezing property." =

Citations

From References: 0

From Reviews: 0

MR1675073 = (2000a:35165)=20
Eden,=20 A.(TR-BOG);=20 Kalantarov,=20 V. K.(TR-HCTTS)
On the discrete squeezing property for semilinear wave=20 equations.
Turkish=20 J. Math. 22=20 (1998),=20 no.=20 3, 335--341.
35L70=20 (34G20 35B40 35L90 47H20)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

The authors consider a class of abstract semilinear, second-order (in = time),=20 dissipative equations in a Hilbert space. This is a natural abstract = framework=20 for the classical semilinear, damped, wave equation. Using energy = estimates,=20 they give sufficient conditions on the nonlinearity and the operator = involved in=20 the evolution equation to guarantee that the so-called squeezing = property holds.=20 Roughly speaking, this property says that the projection of the = semigroup map=20 over the sufficiently large high frequencies is contractive for a = suitable time.=20 The result of this paper improves those in the authors' recent paper = \ref[A.=20 Eden and V. K. Kalantarov, Turkish J. Math. 20 (1996), = no. 3,=20 425--450; MR1482783=20 (98j:34118)].

Reviewed by Enrique=20 Zuazua

Citations

From=20 References: 2

From Reviews: 0

MR1607533 = (98m:58125)=20
Eden,=20 A.(TR-BOG);=20 Foias,=20 C.(1-IN);=20 Kalantarov,=20 V.(TR-HCTT)
A remark on two constructions of exponential attractors = for=20 $\alpha$-contractions. (English = summary)=20
J.=20 Dynam. Differential Equations 10=20 (1998),=20 no.=20 1, 37--45.
58F39=20 (34G20 47H09 47H20 58F12)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "An improvement in the original constructions of exponential = attractors is indicated. Namely, when the solution semigroup is=20 $\alpha$-contractive and satisfies the discrete squeezing property, then = even=20 when the invariant set on which the semigroup acts is not compact, the = original=20 construction carries through. We obtain the same conclusion for the = construction=20 with Lyapunov dimension for $\alpha$-constructions."

Reviewed by Igor=20 D. Chueshov

Citations

From=20 References: 4

From=20 Reviews: 1

MR1482783 = (98j:34118)=20
Eden,=20 A.(TR-BOG);=20 Kalantarov,=20 V.(TR-HCTTS)
Finite-dimensional attractors for a class of semilinear = wave=20 equations. (English, Turkish = summary)
Turkish=20 J. Math. 20=20 (1996),=20 no.=20 3, 425--450.
34G20=20 (35L70 35Q53 46N20 47H20 58F39)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "In this paper we give a self-contained survey of results = related to=20 the global attractors for a class of nonlinear wave equations with = damping or=20 viscosity terms. In particular, we prove the existence of a = finite-dimensional=20 attractor and estimate its fractal dimension by imbedding it in an = exponential=20 attractor. Some results on global stability and existence of = finite-dimensional=20 attractors were already partially discussed by Kalantarov \ref["Global = behavior=20 of solutions to nonlinear problems of mathematical physics of classical = and=20 non-classical type", Post Doctoral Thesis, Leningrad, 1988; per bibl.] = and by=20 Eden, A. J. Milani and B. Nicolaenko \ref[J. Math. Anal. Appl.=20 169 (1992), no. 2, 408--419; MR1180900=20 (93f:35200)]; however, we simplify the framework by introducing a = unified=20 approach to both the existence of attractors through = $\alpha$-contractions and=20 the construction of exponential attractors via some Lipschitzianity = condition of=20 the nonlinear operator."

Citations

From References: 0

From Reviews: 0

MR1314810 = (95m:35023)=20
Eden,=20 A.(TR-BOG);=20 Milani,=20 A.(1-WIM)
On the convergence of attractors and exponential = attractors for=20 singularly perturbed hyperbolic equations. (English,=20 Turkish summary)
Turkish=20 J. Math. 19=20 (1995),=20 no.=20 1, 102--117.
35B40=20 (35L70 58F39)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

The authors study the asymptotic behavior as $\epsilon\to 0$ of = attractors=20 for the nonlinear hyperbolic damped equation (1) $U_{tt}+U_t-\Delta = U+g(U)=3Df$,=20 which is a formal singular perturbation of the parabolic equation (2)=20 $V_t-\Delta V+g(V)=3Df$, where $g(U)$ has cubic growth. Both equations = are=20 considered in $Q=3D\Omega\times\bold R^+$, $\Omega\subset\bold R^3$; = $U=3DV=3D0$ on=20 $\partial\Omega$, $t\geq 0$, $U(x,0)=3DU_0(x)$, $U_t(x,0)=3DU_1(x)$,=20 $V(x,0)=3DV_0(x)$.

The authors claim that they give an explicit = and=20 shorter proof of Babin and Vishik's result on weak convergence of = attractors for=20 (1) to attractors for (2) as $\epsilon\to 0$. They also extend a = regularity=20 result of Ghidaglia and Temam for cubic $g(U)$.

Reviewed by Nickolai=20 A. Lar\cprime kin

Citations

From=20 References: 65

From=20 Reviews: 11

MR1335230 = (96i:34148)=20
Eden,=20 A.(1-AZS);=20 Foias,=20 C.(1-IN);=20 Nicolaenko,=20 B.(1-AZS);=20 Temam,=20 R.(F-PARIS11-A)
Exponential attractors for dissipative evolution = equations.=20
RAM:=20 Research in Applied Mathematics, 37. Masson, Paris; John Wiley = &=20 Sons, Ltd., Chichester, 1994. viii+183 pp. ISBN: 2-225-84306-8 =
34G20=20 (34D45 35K99 35Q99 47H20 47N20 58F39)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

FEATURED REVIEW.



1. This book presents an extensive = account of=20 the theory of exponential attractors for infinite-dimensional dynamical = systems=20 defined by dissipative evolution equations in a Hilbert space. = Exponential=20 attractors are "intermediate" objects between inertial manifolds and = global=20 attractors that, like these, help to describe the long-time behavior of = the=20 dynamical system. The authors are mainly concerned with dynamical = systems=20 defined by nonlinear evolution equations that can be written in the = abstract=20 form (1) $u_t+Au+g(u)=3Df$; examples of equations that can be reduced to = this=20 form, and that are treated in various degrees of detail in this book, = are the=20 $2$- and $3$-dimensional Navier-Stokes equations, the = Kuramoto-Sivashinsky=20 equations, the "original" Burgers equations, the Chaffee-Infante=20 reaction-diffusion equations, and the Korteweg-de Vries equations. Among = these,=20 the most important in many respects, and certainly from the historical = point of=20 view, is that of the Navier-Stokes equations for flows in fluid = dynamics, where=20 the dissipative effects are due to a multiplicity of reasons, including=20 friction, diffusion and damping. Indeed, most of the basic notions and = results=20 in the theory of the long-time behavior of infinite-dimensional = dissipative=20 dynamical systems trace their origin in the study of these equations, = and have=20 been inspired by a careful analysis, both theoretical and numerical, of = their=20 solutions.

2. We recall the following basic definitions: (a) A = dynamical=20 system on a Hilbert space $H$ is a family $\scr S=3D\{S(t)\}_{t\geq 0}$ = of=20 (possibly nonlinear) maps $S(t)\colon H\to H$ such that $S(0)=3DI$ and = for any=20 $\theta,t>0$, $S(\theta+t)=3DS(\theta)S(t)$. (b) A subset $A\subset = H$ is a=20 global attractor for $\scr S$ if $A$ is compact, invariant (i.e.\ = $S(t)A=3DA$ for=20 all $t\geq 0$), and attracts all orbits, i.e. (2) for all $h\in H$,=20 $\lim_{t\to+\infty}d_H(S(t)h,A)=3D0$, where $d_H$ is the Hausdorff = distance in=20 $H$, defined by (3) $d_H(A,B)=3D\max\{\delta(A,B),\delta(B,A)\}$, for = any two=20 subsets $A,B\subset H$, and $\delta$ is the semidistance = $\delta(A,B)=3D\sup_{a\in=20 A}\inf_{b\in B}d(a,b)$. (c) A subset $M\subset H$ is an inertial = manifold for=20 $\scr S$ if $M$ is a compact, smooth, finite-dimensional manifold in $H$ = (i.e.=20 $M$ is the graph of a Lipschitz continuous map $\Phi\colon H_m\to = H^\perp_m$,=20 $H_m$ a finite-dimensional subspace of $H$), positively invariant (i.e.\ = $S(t)M\subseteq M$ for all $t\geq 0$), and attracts all orbits at a = uniform=20 exponential rate on bounded sets of $H$, i.e. (4) there exists $k>0$ = such=20 that for any $h\in H$ there exist $t_0\geq 0$, and $c>0$ such that = for all=20 $t\geq t_0$, $d_H(S(t)h,M)\leq ce^{-kt}$, with $c$ independent of $h$ = for $h$ in=20 bounded sets. (d) A subset $E\subset H$ is an exponential attractor for = $\scr S$=20 if $E$ is compact and positively invariant, has finite fractal = dimension, and=20 attracts all orbits at a uniform exponential rate on bounded sets of $H$ = as in=20 (4).

3. The dissipative nature of the evolution equation (1) can = be=20 characterized by the existence of a bounded set $B\subset H$ that = attracts all=20 bounded solutions in a finite time, and that is positively invariant = under $\scr=20 S$, in the sense that orbits originating in $B$ remain in $B$ forever. = Such a=20 set $B$ is called an absorbing set; when the dynamical system admits a = compact=20 absorbing set $B$ (as is the case when (1) is "parabolic", so that there = is a=20 forward regularizing effect for the flow), then a global attractor $A$ = exists,=20 and this coincides with the $\omega$-limit set of $B$, i.e.\ with the = set=20 $\omega(B)=3D\bigcap_{s\geq 0}\overline{\bigcup_{t\geq s}S(t)B}$. In = many cases,=20 including all the examples quoted so far, the global attractor has = finite=20 Hausdorff dimension: this fact would allow us, in principle, to reduce = the study=20 of the long-time behavior of the orbits to that of the solutions of a=20 finite-dimensional system of ODEs (on $A$). In most cases, however, it = is too=20 impractical to pursue this type of analysis, because of several = difficulties=20 associated with attractors; for instance, they may have a quite = complicated and=20 nonsmooth geometrical structure (e.g. when the attractor is a fractal), = or the=20 available estimates on their dimension are too large for computational = purposes.=20 Also, attractors are in most cases not sufficiently robust under = perturbations=20 and numerical approximations (for instance, approximation of attractors = with=20 respect to the Hausdorff distance (3) is in general only upper = semicontinuous).=20

4. One case when inconveniences like these can be overcome is = when the=20 attractor can be imbedded in an inertial manifold $M$: in this case, the = PDE (1)=20 can indeed be reduced to a finite system of ODEs on $M$ (the so-called = inertial=20 form of the system), the smooth structure of $M$ can be fully exploited = for the=20 asymptotic analysis of $\scr S$, and the fact that $M$ attracts the = orbits at a=20 uniform exponential rate once they are in an invariant absorbing set = makes the=20 inertial manifold extremely robust under perturbations and numerical=20 approximations. However, while all the examples cited above do possess a = global=20 attractor, only relatively few are known to possess an inertial = manifold, and in=20 fact there are other examples of systems that are known not to have one. = The=20 main difficulty lies in that essentially all the available methods of=20 constructing an inertial manifold are based on the so-called "spectral = gap=20 condition", which is a quite restrictive condition on the spectrum of = the linear=20 part of the PDE (it must be stressed, though, that this condition is = only a=20 sufficient one for the existence of an inertial manifold). Moreover, = even when=20 an inertial manifold is known to exist, the difficulty of its actual=20 construction is often reflected in the lack of realistic estimates on = its=20 dimension.

5. It is at this juncture that the crucial role of = the=20 exponential attractor emerges in all of its power: indeed, as the = authors of=20 this book remark, "the above-described scenario is not the only possible = one",=20 and in fact it may well happen that, when a finite-dimensional attractor = exists,=20 the long-time behavior of the orbits can still be described by a finite = system=20 of ODEs, having discontinuous coefficients. The set on which these ODEs = describe=20 the dynamical system is precisely the exponential attractor: these are, = as we=20 have said, intermediate objects between attractors and inertial = manifolds, in=20 the sense that while they do not necessarily enjoy a smooth structure, = they do=20 retain two features that attractors do not necessarily have, namely = finite=20 dimensionality and exponential convergence of the orbits. Thus, when an=20 exponential attractor exists, after an "exponentially short" transient = period=20 the dynamics of the system is, indeed, essentially controlled by a = finite system=20 of ODEs. Moreover, all the examples considered by the authors show that = the=20 estimates for the dimension of the exponential attractor either are=20 substantially better than those of the attractor, when this has finite=20 dimension, or can at least be obtained by much less powerful methods.=20 Exponential attractors are also quite robust under perturbations and = numerical=20 approximations; for instance, the authors prove that, in the framework = of=20 standard Galerkin methods, approximate and exact exponential attractors = are=20 essentially continuous with respect to the Hausdorff distance (3). =

6.=20 Clearly, when all three sets $A$, $E$ and $M$ exist, then $A\subseteq = E\subseteq=20 M$; in fact, in this case the simplest way to construct an exponential = attractor=20 is to intersect $M$ with an absorbing ball. As we have seen, however, = the=20 existence of an inertial manifold is not proven for many important = systems, due=20 to the difficulty of verifying the spectral gap condition; on the = contrary, the=20 method presented by the authors to construct the exponential attractor = does not=20 require any kind of gap condition, and thus makes the theory of = exponential=20 attractors applicable to a much wider class of equations. Roughly = speaking, this=20 construction is based on an expansion of the attractor, by subsequently = adding=20 to it a number of points that are not converging exponentially, and in = such a=20 way that the dimension of each newly added cluster of points can be = suitably=20 controlled, so as to ensure that each new set so constructed remains = invariant=20 under $\scr S$. This is achieved by applying the so-called "discrete = squeezing=20 property", which is essentially a dichotomy principle, whereby either = the lower=20 order modes are dominated by the higher order ones, or the system is = already=20 exponentially contracting from the outset. Explicitly, a map $S\colon = H\to H$ is=20 said to satisfy the discrete squeezing property if for all = $\delta\in(0,1)$=20 there are a finite-dimensional closed subspace $H_\delta\subset H$, and = an=20 orthogonal projection $P_\delta\colon H\to H_\delta$ with finite rank=20 $N=3DN(\delta)$, such that if (5)=20 $\|(I-P_\delta)(Su-Sv)\|_H>\|P_\delta(u-v)\|_H$, then=20 $\|Su-Sv\|_H\leq\delta\|u-v\|_H$. In the applications to dynamical = systems=20 defined by equation (1), the typical choice of finite-dimensional = subspace is=20 the span of the eigenvectors corresponding to the first $N$ eigenvalues = of the=20 linear operator $A$, and that of $S$ is $S=3DS(t_*)$, for some = convenient $t_*$=20 chosen in dependence of $N$ so that inequality (1.5) holds with $\delta$ = depending on $N$, $N$ chosen so that $\delta<1$. Thus, whereas = proving the=20 existence of an exponential attractor via the discrete squeezing = property still=20 requires some information on the asymptotic behavior of the eigenvalues, = no kind=20 of spectral gap condition is required.

7. When the dynamical = system=20 defined by equation (1) has a compact absorbing ball, a sufficient = condition for=20 the existence of an exponential attractor is given by a Lipschitz = condition on=20 the restriction of the nonlinearity on a subspace $V$ compactly imbedded = in $H$,=20 namely that (6) there exists $L>0$ such that for all $u,v\in V$,=20 $\|g(u)-g(v)\|_H\leq L\|u-v\|_V$; indeed, one of the major results in = the theory=20 is that if (6) holds, then there exists $t_*>0$ such that $S(t_*)$ = satisfies=20 the discrete squeezing property and, hence, the corresponding dynamical = system=20 $\scr S$ admits an exponential attractor. This result fully = characterizes the=20 asymptotic behavior of solutions of "parabolic" equations like (1), in = terms of=20 exponential attractors. Some extensions of this type of result to = dissipative=20 hyperbolic equations of the form (7) $\epsilon u_{tt}+u_t+Au+g(u)=3Df$, = with=20 $\epsilon>0$, are also discussed in the book: the existence of an = exponential=20 attractor for the orbits $\{u(t),u_t(t)\}$ of (7) is again proven as a=20 consequence of the validity of the discrete squeezing property. This = result is=20 particularly meaningful, because there are explicit examples of damped=20 semilinear wave equations that possess a global attractor but fail to = have a=20 $C^1$ inertial manifold. Examples of equations that can be reduced to = the=20 abstract model (7) are given by the dissipative sine-Gordon and the = Klein-Gordon=20 equations; of particular importance is the quantum mechanics equation = $\epsilon=20 u_{tt}+u_t-\Delta u+u^3-u=3Df$ in three dimensions of space.

8.=20 Specifically, the content of the book is as follows: after a brief but = thorough=20 introduction to the subject (Chapter 1), the exponential attractors are=20 constructed by means of the discrete squeezing property, first for = discrete=20 dynamical systems (Chapter 2), then for continuous dynamical systems = arising=20 from first order evolution equations (Chapter 3). Approximation of such=20 exponential attractors in the context of Galerkin's schemes for first = order=20 equations is briefly discussed in Chapter 4, and applications to = specific=20 examples are discussed in Chapter 5. In Chapter 6 exponential attractors = are=20 constructed for dissipative second order semilinear equations; in = Chapter 7 the=20 authors present an alternative construction of the exponential = attractor, in=20 which the goal of optimal dimension estimates is abandoned in favor of = obtaining=20 a faster rate of convergence in (4). (In some cases, arbitrarily fast=20 convergence is achieved, while retaining good control of at least the = Hausdorff=20 dimension of the exponential attractor.) Exponential attractors and = inertial=20 manifolds are compared in Chapter 8; the authors proceed then with a = more=20 detailed study of the finite-dimensional nature of the dynamics on the=20 exponential attractor and, in particular, address the question whether = it is=20 possible to reconstruct, in a relatively natural way, the dynamics of = the system=20 on the exponential attractor without direct recourse to the underlying = evolution=20 equation. The answer is positive when an inertial manifold exists, as = shown in=20 Chapter 9; when only an exponential attractor is known to exist, a = similar=20 reduction is achieved in Chapter 10, by means of the so-called Ma = n=C3=A9=20 projections. This method is further discussed in Appendix A; the book is = completed by two other appendices, one on the estimate of the = topological=20 entropy of the dynamical system $\scr S$, and the other recalling the = basic=20 definitions of Hausdorff and fractal dimension.

Reviewed by Albert=20 J. Milani

Citations

From=20 References: 7

From Reviews: 0

MR1283060 = (95e:35099)=20
Eden,=20 A.(1-AZS);=20 Rakotoson,=20 J.-M.(F-POIT)
Exponential attractors for a doubly nonlinear = equation.=20
J.=20 Math. Anal. Appl. 185=20 (1994),=20 no.=20 2, 321--339.
35K57=20 (34G20 35B40 47H20 47N20 58F39)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

The aim of this paper is to prove the existence of exponential = attractors for=20 the equation $\partial\sb t\beta(u)-\Delta u+g(x,u)=3D0$, i.e. sets of = finite=20 fractal dimension that attract all the solutions at an exponential rate. = This=20 equation generalizes reaction-diffusion equations and porous medium type = equations. Previous results of the authors have been obtained in = collaboration=20 with B. Michaux \ref[see, e.g., J. Dynam. Differential Equations=20 3 (1991), no. 1, 87--131; MR1094725=20 (91m:35118); Indiana Univ. Math. J. 39 (1990), no. = 3,=20 737--783; MR1078736=20 (91h:35150)]. Regularity of solutions and their decay properties are = used in=20 order to show the discrete squeezing property of the nonlinear semigroup = associated with the initial-boundary value problem for the above = equation.

Reviewed by Piotr=20 Biler

Citations

From=20 References: 1

From Reviews: 0

MR1280141 = (95d:47083)=20
Eden,=20 A.(1-AZS);=20 Foias,=20 C.(1-IN);=20 Nicolaenko,=20 B.(1-AZS)
Exponential attractors of optimal Lyapunov dimension for=20 Navier-Stokes equations. (English = summary)=20
J.=20 Dynam. Differential Equations 6=20 (1994),=20 no.=20 2, 301--323.
47H20=20 (34G20 35Q30 58F39 76D05)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Since its introduction in 1988, the concept of inertial manifolds has = drawn=20 considerable attention from dynamical systems specialists. There were = discovered=20 many equations possessing inertial manifolds. However, it is still = unknown=20 whether inertial manifolds exist for the 2-dimensional Navier-Stokes = equations=20 (originally, these served as the motivating example for the introduction = of the=20 concept). By relaxing the requirement on smoothness in the definition of = the=20 inertial manifold for an abstract dynamical system, the authors = construct=20 compact, exponentially attracting, invariant sets (called exponential=20 attractors) with finite fractal dimension of the same order as the one = for=20 global attractors estimated through Lyapunov exponents. They show that = the 2D=20 Navier-Stokes equations and Boussinesq equations possess such sets.

Reviewed by Vadim=20 Bondarevsky

Citations

From=20 References: 6

From=20 Reviews: 2

MR1231724 = (94d:58091)=20
Ben-Artzi,=20 A.(1-IN);=20 Eden,=20 A.(1-AZS);=20 Foias,=20 C.(1-IN);=20 Nicolaenko,=20 B.(1-AZS)
H=C3=B6lder continuity for the inverse of Ma=C3=B1=C3=A9's = projection. (English summary)
J.=20 Math. Anal. Appl. 178=20 (1993),=20 no.=20 1, 22--29.
58F12=20 (34G20 47N20 58F13)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Let $P_0$ be an orthogonal projection of a compact subset $X$ of=20 $R^m$ to $R^n$, and assume = $n>2d_0$, where=20 $d_0$ is the box-counting dimension of $X$. The main theorem states that = for=20 each positive $\delta$ and positive $\theta < 2d_0/n$, there exists = an=20 orthogonal projection $P$ such that $||P-P_0||<\delta$, $P$ is = one-to-one on=20 $X$, and such that the inverse $P^{-1}$ defined on $P(X)$ is H=C3=B6lder = continuous=20 with exponent $\theta$. The application is to the projection of a = complicated=20 limit set (such as a compact attractor within an inertial manifold) to a = space=20 of lower dimension, in which analysis might be easier to perform.

Reviewed by Timothy=20 Sauer

Citations

From=20 References: 6

From=20 Reviews: 1

MR1223743 = (94h:35250)=20
Eden,=20 A.(1-AZS);=20 Milani,=20 A. J.(1-WIM)
Exponential attractors for extensible beam = equations. (English summary)
Nonlinearity=20 6=20 (1993),=20 no.=20 3, 457--479.
35Q72=20 (58F39 73D35 73K05)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "In this paper we establish a global fast dynamics for a = class of=20 equations that includes the beam equations as studied by Ball and von = K=C3=A1rm=C3=A1n=20 equations for a thin plate. We introduce various energy functionals and = show=20 that they decay exponentially. Using the absorbing sets obtained through = these=20 energy functionals, we present Hale's theory of $\alpha$-contractions = and how it=20 applies to this general framework and deduce the existence of a compact=20 attractor, in parallel to his proof. We also establish the smoothness of = this=20 attractor when the damping is large. Finally, by proving the discrete = squeezing=20 property for these equations, the existence of a compact, = finite-dimensional=20 exponentially attracting set is demonstrated. The use of energy methods=20 throughout allows considerable simplification even when a natural = Lyapunov=20 functional is hard to exhibit. In closing, we also exhibit a simple = alternative=20 proof for Titi's theorem on the existence of inertial manifolds for beam = equations under suitable forces."

Citations

From References: 0

From Reviews: 0

MR1221651 = (94c:58185)=20
Eden,=20 Alp(1-AZS);=20 Foias,=20 Ciprian(1-IN);=20 Nicolaenko,=20 Basil(1-AZS)
Exponential attractors of optimal Lyapunov dimension for=20 Navier-Stokes equations. (English, French=20 summary)
C.=20 R. Acad. Sci. Paris S=C3=A9r. I Math. 316=20 (1993),=20 no.=20 11, 1211--1215.
58F39=20 (35B35 35B40 35Q30 47H20 47N20 58F12 76D05)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

The paper deals with the construction of exponential attractors = (inertial=20 sets) for a class of dissipative dynamical systems generated by = nonlinear=20 parabolic equations. Recall that a compact invariant set is called an=20 exponential attractor if it has finite fractal dimension, contains the = global=20 attractor and attracts exponentially all the trajectories. The existence = of an=20 exponential attractor with fractal dimension estimate of the same order = as the=20 one for the global attractor is proved for the class of dynamical = systems under=20 consideration.

Reviewed by Igor=20 D. Chueshov

Citations

From=20 References: 1

From Reviews: 0

MR1210011 = (94a:47110)=20
Eden,=20 A.(1-AZS);=20 Foias,=20 C.(1-IN);=20 Nicolaenko,=20 B.(1-AZS);=20 She,=20 Z.-S.(1-AZ)
Exponential attractors and their relevance to fluid = dynamics=20 systems. (English summary)
Phys.=20 D 63=20 (1993),=20 no.=20 3-4, 350--360.
47N20=20 (34C35 34G20 35Q30 47H20 58F12 76D05)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

The authors present, in this review paper, some of the recent results = in the=20 mathematical theory of infinite-dimensional dynamical systems. They = start with=20 the abstract set-up of a nonlinear dissipative evolution equation in an=20 appropriate Hilbert space. They recall the definition of some useful=20 mathematical objects in the context: global attractor, inertial = manifold,=20 spectral barrier.

Next the authors emphasize the notion of an=20 exponential attractor, i.e., an invariant (under the flow) set of finite = fractal=20 dimension attracting all solutions at a uniform exponential rate. Here = the key=20 property is the so-called "squeezing property" of the flow, allowing = them to=20 prove the existence of exponential attractors. Finally, the physical = relevance=20 of such attractors is illustrated and discussed in the context of the 2D = Navier-Stokes equations.

Reviewed by Bruno=20 Scheurer

Citations

From=20 References: 1

From Reviews: 0

MR1203055 = (93j:35139)=20
Eden,=20 A.(1-AZS);=20 Milani,=20 A.(1-WIM);=20 Nicolaenko,=20 B.(1-AZS)
Local exponential attractors for models of phase change = for=20 compressible gas dynamics. (English = summary)=20
Nonlinearity=20 6=20 (1993),=20 no.=20 1, 93--117.
35Q35=20 (58F39 76N15)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "We investigate a model for dynamic phase transitions in a = van der=20 Waals compressible fluid. As the pressure is given by a nonconvex = equation of=20 state, which also blows up for a finite volume, the corresponding = initial value=20 problem is of mixed hyperbolic-elliptic type. Therefore, it generates = nontrivial=20 dynamics. The system of conservation laws when regularized with = capillarity=20 terms excludes the appearance of shocks but keeps most of the = interesting=20 dynamics. By introducing the appropriate Hamiltonian function, local = invariant=20 domains are constructed that avoid the blow-up and at the same time = allow=20 solutions of mixed type. We show that, even in regions of mixed type, = the=20 initial value problem exhibits finite-dimensional dynamical behaviour by = establishing the existence of local attractors and of exponential = attractors of=20 finite fractal dimension."

Citations

From=20 References: 11

From=20 Reviews: 3

MR1180900 = (93f:35200)=20
Eden,=20 A.(1-AZS);=20 Milani,=20 A. J.(1-WIM);=20 Nicolaenko,=20 B.(1-AZS)
Finite-dimensional exponential attractors for semilinear = wave=20 equations with damping. (English = summary)=20
J.=20 Math. Anal. Appl. 169=20 (1992),=20 no.=20 2, 408--419.
35Q53=20 (34D45 34G20 58F39)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "We consider the initial value problem for a class of = second-order=20 evolution equations that includes, among others, the 3D sine-Gordon = equation=20 with damping and the 3D Klein-Gordon type equations with damping. We = show the=20 existence of a set with finite fractal dimension that contains the = global=20 attractor and attracts all smooth solutions at an exponential rate." =

Citations

From=20 References: 2

From Reviews: 0

MR1135275 = (92j:46056)=20
Eden,=20 A.(1-AZS);=20 Foias,=20 C.(1-IN)
A simple proof of the generalized Lieb-Thirring = inequalities in=20 one-space dimension.
J.=20 Math. Anal. Appl. 162=20 (1991),=20 no.=20 1, 250--254.
46= E35=20
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "The Lieb-Thirring inequalities give a sharp upper bound for = the=20 $L^p$-norm of a function which is the pointwise sum of the squares of a = finite=20 orthonormal sequence of functions that are elements of a suitable = Sobolev space=20 \ref[E. Lieb and W. Thirring, in Studies in mathematical = physics,=20 269--303, Princeton Univ. Press, Princeton, NJ, 1976; MR Zbl 342:35044]. = Originally proven for functions defined on the whole $n$-dimensional = Euclidean=20 space, they were later extended to bounded domains and to suborthogonal=20 sequences of functions \ref[\cita MR0920485=20 (89d:35137) \endcit J.-M. Ghidaglia, M. Marion and R. Temam, = Differential=20 Integral Equations 1 (1988), no. 1, 1--21; MR0920485=20 (89d:35137)]. Here, we present a simple proof of these inequalities = for=20 bounded intervals in one space dimension utilizing simple Sobolev = inequalities=20 and standard results from Hilbert space theory."

Citations

From=20 References: 5

From=20 Reviews: 1

MR1094726 = (92d:58118)=20
Eden,=20 A.(1-IN);=20 Foias,=20 C.(1-IN);=20 Temam,=20 R.(F-PARIS11-A)
Local and global Lyapunov exponents.
J.=20 Dynam. Differential Equations 3=20 (1991),=20 no.=20 1, 133--177.
58F12=20 (34C35 34D08 34D45 34G20 47H20 58F39)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Let $\{S(t)\}$, $t\geq 0$, be a continuous semigroup acting on a = compact=20 invariant subset of a Hilbert space. The authors establish the = connection=20 between the (local and global) Lyapunov exponents of the semiflow = $\{S(t)\}$ and=20 the spectral radii of some positive linear operators on a space of = continuous=20 functions. These results enable them to estimate the Hausdorff and the = fractal=20 dimension of a global attractor of $\{S(t)\}$ by means of the local = Lyapunov=20 exponents. Assuming additionally that the maps $S(t)$ are Lipschitz, = they obtain=20 a relation between topological entropy and the fractal dimension. = General=20 results are applied to the Lorenz attractor, for which they show that=20 $d_H(X)\leq 2.409$. Moreover, they estimate the Hausdorff dimension of = an=20 attractor of a nonlinear evolution equation with a dissipativity = condition on=20 the nonlinear term.

Reviewed by Ryszard=20 Rudnicki

Citations

From=20 References: 7

From=20 Reviews: 1

MR1094725 = (91m:35118)=20
Eden,=20 A.(1-IN);=20 Michaux,=20 B.(1-IN);=20 Rakotoson,=20 J.-M.(F-POIT)
Doubly nonlinear parabolic-type equations as dynamical=20 systems.
J.=20 Dynam. Differential Equations 3=20 (1991),=20 no.=20 1, 87--131.
35K55=20 (34G20 35Qxx 58F12 58F39 76D99 76S05)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

One of the most important goals of dynamical systems theory is to = approach a=20 partial differential equation (PDE) through a system of infinite- or=20 finite-dimensional ordinary differential equations (ODE) in order to use = the=20 theory of the latter to obtain results for the PDE. In the case of = infinite=20 dimension, the general method is to work with semigroup theory. For = finite=20 dimensions one exploits an attractor set with finite fractal dimension, = and then=20 finds a finite-dimensional invariant manifold for the flow that contains = the=20 attractor.

The authors use this second technique for studying a = porous=20 medium type nonlinearity (inside the time derivative) combined with a = weak=20 nonlinearity in the reaction term. Specifically, they study the equation = $(*)$=20 $\partial_t \beta(u)=3D\Delta u+g(x,t,u)$ for $(x,t)\in (0,T)\times = \Omega$, $u=3D0$=20 on $(0,T)\times \partial\Omega$, $\beta(u)|_{t=3D0}=3D\beta(u_0)$, with=20 $\Omega\subseteq R^d$, $d\geq 1$, $\beta\colon=20 R\toR$ nondecreasing, $\beta(0)=3D0$,=20 $|\beta(t)|\leq a|t|+b$, $t\inR$ and $a$, $b$ constant; = $g$ has=20 unbounded growth with respect to the solutions. They prove the existence = and=20 uniqueness of the solution of $(*)$. Under some assumptions on $g$, the=20 existence and the finiteness of the fractal dimension of the compact = attractor=20 are shown. Furthermore, they obtain a lower estimate of the dimension of = the=20 attractor.

Reviewed by Elias=20 Tuma

Citations

From=20 References: 3

From=20 Reviews: 1

MR1078736 = (91h:35150)=20
Eden,=20 A.(1-AZS);=20 Michaux,=20 B.(1-IN);=20 Rakotoson,=20 J.-M.(F-POIT)
Semi-discretized nonlinear evolution equations as discrete = dynamical=20 systems and error analysis.
Indiana=20 Univ. Math. J. 39=20 (1990),=20 no.=20 3, 737--783.
35K55=20 (65M06)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

From the summary: "This paper is a continuation of our study on = doubly=20 nonlinear parabolic type equations. We investigate a time discretization = of the=20 continuous problem by the Euler forward scheme where in addition to the = standard=20 existence, uniqueness and stability questions, we also address the = problem of=20 error estimation and the long-time behavior of the solutions to the = discrete=20 problem. The existence of a compact attractor is proven and its = Hausdorff=20 dimension is estimated using CFT theory."

Citations

From References: 0

From Reviews: 0

MR1077868 = (91h:35169)=20
Eden,=20 A.(1-IN);=20 Michaux,=20 B.(1-IN);=20 Rakotoson,=20 J.-M.(F-POIT)
Error analysis of nonlinear evolution equations and = associated=20 dynamical systems.
Appl.=20 Math. Lett. 3=20 (1990),=20 no.=20 3, 31--34.
35K60=20 (65P05)
PDF=20 Doc=20 Del Clipboar= d Journal Article Make = Link

Summary: "As a continuation of our study on doubly nonlinear = parabolic type=20 equations, we investigate a time discretization of these equations by = the Euler=20 forward scheme. In addition to the standard existence, uniqueness and = stability=20 questions, we also address the problem of error estimation and the long = time=20 behavior of the solutions of the discrete problem. The existence of a = compact=20 attractor is proven and its Hausdorff dimension is estimated using CFT = theory."=20

Citations

From=20 References: 1