January  2020, 19(1): 145-174. doi: 10.3934/cpaa.2020009

Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received  October 2018 Revised  January 2019 Published  July 2019

Fund Project: This work was supported by the National Natural Science Foundations of China No. 11431008 and No. 11871041.

We show the existence and $ C^{k, \gamma} $ smoothness of local integral manifolds at an equilibrium point for nonautonomous and ill-posed equations with sectorially dichotomous operator, provided that the nonlinearities are $ C^{k, \gamma} $ smooth with respect to the state variable. $ C^{k, \gamma} $ local unstable integral manifold follows from $ C^{k, \gamma} $ local stable integral manifold by reversing time variable directly. As an application, an elliptic PDE in infinite cylindrical domain is discussed.

Citation: Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure and Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009
References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2$^{nd}$ edition, Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

A. CalsinaX. Mora and J. Solá-Morales, The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic singular limits, J. Differential Equations, 102 (1993), 244-304.  doi: 10.1006/jdeq.1993.1030.

[3]

S.-N. Chow and K. Lu, $C^{k}$ centre unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 303-320.  doi: 10.1017/S0308210500014682.

[4]

L. W. Deng and D. M. Xiao, Dichotomous solutions for semilinear ill-posed equations with sectorially dichotomous operator, J. Differential Equations, 267 (2019), 1201-1246.  doi: 10.1016/j.jde.2019.02.006.

[5]

M. S. ElBialy, Locally Lipschitz perturbations of bi-semigroup, Commun. Pure Appl. Anal., 9 (2010), 327-349.  doi: 10.3934/cpaa.2010.9.327.

[6]

M. S. ElBialy, Stable and unstable manifolds for hyperbolic bi-semigroups, J. Funct. Anal., 262 (2012), 2516-2560.  doi: 10.1016/j.jfa.2011.11.031.

[7]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[8]

A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture Notes in Math., Vol.841, Springer, Berlin, 1981. doi: 10.1007/BFb0089606.

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol.840, Springer, Berlin, 1981. doi: 10.1007/BFb0089647.

[10]

K. Kirchgassner, Wave solutions of reversible systems and applications, J. Differential Equations, 45 (1982), 113-127.  doi: 10.1016/0022-0396(82)90058-4.

[11]

O. E. Lanford, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens, in Nonlinear Problems in the Physical Sciences and Biology (eds. I. Stakgold, D. D. Joseph and D. H. Sattinger), Lecture Notes in Mathematics, 322 (1973), 159–192. doi: 10.1007/BFb0060566.

[12]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

[13]

R. de la Llave, A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities, J. Dynam. Differential Equations, 21 (2009), 371-415.  doi: 10.1007/s10884-009-9140-y.

[14]

Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete Contin. Dynam. Systems, 5 (1999), 233-268.  doi: 10.3934/dcds.1999.5.233.

[15]

A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Differential Equations, 65 (1986), 68-88.  doi: 10.1016/0022-0396(86)90042-2.

[16]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66.  doi: 10.1002/mma.1670100105.

[17]

A. Mielke, Normal hyperbolicity of center manifolds and Saint-Venant's principle, Arch. Rational Mech. Anal., 110 (1988), 353-372.  doi: 10.1007/BF00393272.

[18]

A. Mielke, On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds, J. Dynam. Differential Equations, 4 (1992), 419-443.  doi: 10.1007/BF01053805.

[19]

A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations, 110 (1994), 322-355.  doi: 10.1006/jdeq.1994.1070.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

J. C. Robinson, Infinite-Dimensional Dynamical System: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge texts in Applied Mathematics, Cambridge University press, 2001. doi: 10.1007/978-94-010-0732-0.

[22]

K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems, London Math. Soc. Lect. Notes Ser., vol.2233, Cambridge University Press, 1995. doi: 10.1017/CBO9780511662362.

[23]

C. Tretter and C. Wyss, Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations, J. Evol. Equ., 14 (2014), 121-153.  doi: 10.1007/s00028-013-0210-6.

[24]

A. Zamboni, Maximal regularity for evolution problems on the line, Differential Integral Equations, 22 (2009), 519-542. 

show all references

References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, 2$^{nd}$ edition, Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

A. CalsinaX. Mora and J. Solá-Morales, The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic singular limits, J. Differential Equations, 102 (1993), 244-304.  doi: 10.1006/jdeq.1993.1030.

[3]

S.-N. Chow and K. Lu, $C^{k}$ centre unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 303-320.  doi: 10.1017/S0308210500014682.

[4]

L. W. Deng and D. M. Xiao, Dichotomous solutions for semilinear ill-posed equations with sectorially dichotomous operator, J. Differential Equations, 267 (2019), 1201-1246.  doi: 10.1016/j.jde.2019.02.006.

[5]

M. S. ElBialy, Locally Lipschitz perturbations of bi-semigroup, Commun. Pure Appl. Anal., 9 (2010), 327-349.  doi: 10.3934/cpaa.2010.9.327.

[6]

M. S. ElBialy, Stable and unstable manifolds for hyperbolic bi-semigroups, J. Funct. Anal., 262 (2012), 2516-2560.  doi: 10.1016/j.jfa.2011.11.031.

[7]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[8]

A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture Notes in Math., Vol.841, Springer, Berlin, 1981. doi: 10.1007/BFb0089606.

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol.840, Springer, Berlin, 1981. doi: 10.1007/BFb0089647.

[10]

K. Kirchgassner, Wave solutions of reversible systems and applications, J. Differential Equations, 45 (1982), 113-127.  doi: 10.1016/0022-0396(82)90058-4.

[11]

O. E. Lanford, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens, in Nonlinear Problems in the Physical Sciences and Biology (eds. I. Stakgold, D. D. Joseph and D. H. Sattinger), Lecture Notes in Mathematics, 322 (1973), 159–192. doi: 10.1007/BFb0060566.

[12]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

[13]

R. de la Llave, A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities, J. Dynam. Differential Equations, 21 (2009), 371-415.  doi: 10.1007/s10884-009-9140-y.

[14]

Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete Contin. Dynam. Systems, 5 (1999), 233-268.  doi: 10.3934/dcds.1999.5.233.

[15]

A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Differential Equations, 65 (1986), 68-88.  doi: 10.1016/0022-0396(86)90042-2.

[16]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66.  doi: 10.1002/mma.1670100105.

[17]

A. Mielke, Normal hyperbolicity of center manifolds and Saint-Venant's principle, Arch. Rational Mech. Anal., 110 (1988), 353-372.  doi: 10.1007/BF00393272.

[18]

A. Mielke, On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds, J. Dynam. Differential Equations, 4 (1992), 419-443.  doi: 10.1007/BF01053805.

[19]

A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations, 110 (1994), 322-355.  doi: 10.1006/jdeq.1994.1070.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

J. C. Robinson, Infinite-Dimensional Dynamical System: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge texts in Applied Mathematics, Cambridge University press, 2001. doi: 10.1007/978-94-010-0732-0.

[22]

K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems, London Math. Soc. Lect. Notes Ser., vol.2233, Cambridge University Press, 1995. doi: 10.1017/CBO9780511662362.

[23]

C. Tretter and C. Wyss, Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations, J. Evol. Equ., 14 (2014), 121-153.  doi: 10.1007/s00028-013-0210-6.

[24]

A. Zamboni, Maximal regularity for evolution problems on the line, Differential Integral Equations, 22 (2009), 519-542. 

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