April  2021, 20(4): 1413-1429. doi: 10.3934/cpaa.2021026

Expanding solutions of quasilinear parabolic equations

Department of Mathematics, University of Patras, 26504 Rio Patras, Greece

Received  February 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Fund Project: The author was supported by Deutsche Forschungsgemeinschaft, grant SCHR 319/9-1

By using the theory of maximal $ L^{q} $-regularity and methods of singular analysis, we show a Taylor's type expansion–with respect to the geodesic distance around an arbitrary point–for solutions of quasilinear parabolic equations on closed manifolds. The powers of the expansion are determined explicitly by the local geometry, whose reflection to the solutions is established through the local space asymptotics.

Citation: Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1413-1429. doi: 10.3934/cpaa.2021026
References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919.  doi: 10.1016/0362-546X(88)90073-9.

[2]

H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I Abstract Linear Theory, Monographs in Mathematics, Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9221-6.

[3]

W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, Birkhäuser Verlag, 2001. doi: 10.1007/978-3-0348-5075-9.

[4]

P. Clément and S. Li, Abstract parabolic quasilinear equations and application to a groundwater flow problem, Adv. Math. Sci. Appl., 3 (1993/94), 17-32. 

[5]

G. Da Prato and P. Grisvard, Sommes d' opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl., 54 (1975), 305-387. 

[6]

L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. Sem. Mat. Univ. Padova, 34 (1964), 205-223. 

[7]

G. Dore, $L^{p}$ regularity for abstract differential equations, in Functional Analysis and Related Topics, Springer Verlag, 1993. doi: 10.1007/BFb0085472.

[8]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.  doi: 10.1007/BF01163654.

[9]

N. Kalton and L. Weis, The $H^{\infty}$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345.  doi: 10.1007/s002080100231.

[10]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9234-6.

[11]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, Springer Verlag, 2016. doi: 10.1007/978-3-319-26654-1.

[12]

Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Lett. A, 75 (1980), 296-298. 

[13]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, Birkhäuser Verlag, 2016. doi: 10.1007/978-3-319-27698-4.

[14]

N. Roidos, Complex powers for cone differential operators and the heat equation on manifolds with conical singularities, Proceedings of the Amer. Math. Soc., 146 (2018), 2995-3007.  doi: 10.1090/proc/13986.

[15]

N. Roidos, The Swift-Hohenberg equation on conic manifolds, J. Math. Anal. Appl., 481 (2020), 123491. doi: 10.1016/j.jmaa.2019.123491.

[16]

N. Roidos and E. Schrohe, Existence and maximal $L^p$-regularity of solutions for the porous medium equation on manifolds with conical singularities, Commun. Partial Differ. Equ., 41 (2016), 1441-1471.  doi: 10.1080/03605302.2016.1219745.

[17]

N. Roidos and E. Schrohe, Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities, Commun. Partial Differ. Equ., 43 (2018), 1456-1484.  doi: 10.1080/03605302.2018.1517788.

[18]

N. Roidos, E. Schrohe and J. Seiler, Bounded $H_{\infty}$-calculus for boundary value problems on manifolds with conical singularities, preprint, arXiv: 1906.03701.

[19]

E. Schrohe and J. Seiler, Bounded $H_{\infty}$-calculus for cone differential operators, J. Evol. Equ., 18 (2018), 1395-1425.  doi: 10.1007/s00028-018-0447-1.

[20]

E. Schrohe and J. Seiler, The resolvent of closed extensions of cone differential operators, Canad. J. Math., 57 (2005), 771-811.  doi: 10.4153/CJM-2005-031-1.

[21]

J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. 

[22]

H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, Pitman Publishing, 1979.

[23]

M. TlidiM. Georgiou and P. Mandel, Transverse patterns in nascent optical bistability, Phys. Rev. A, 48 (1993), 4506-4609. 

[24] J. L. Vázquez, The Porous Medium Equation, Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, 2007. 
[25]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_{p}$-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919.  doi: 10.1016/0362-546X(88)90073-9.

[2]

H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I Abstract Linear Theory, Monographs in Mathematics, Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9221-6.

[3]

W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, Birkhäuser Verlag, 2001. doi: 10.1007/978-3-0348-5075-9.

[4]

P. Clément and S. Li, Abstract parabolic quasilinear equations and application to a groundwater flow problem, Adv. Math. Sci. Appl., 3 (1993/94), 17-32. 

[5]

G. Da Prato and P. Grisvard, Sommes d' opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl., 54 (1975), 305-387. 

[6]

L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. Sem. Mat. Univ. Padova, 34 (1964), 205-223. 

[7]

G. Dore, $L^{p}$ regularity for abstract differential equations, in Functional Analysis and Related Topics, Springer Verlag, 1993. doi: 10.1007/BFb0085472.

[8]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.  doi: 10.1007/BF01163654.

[9]

N. Kalton and L. Weis, The $H^{\infty}$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345.  doi: 10.1007/s002080100231.

[10]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9234-6.

[11]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, Springer Verlag, 2016. doi: 10.1007/978-3-319-26654-1.

[12]

Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Lett. A, 75 (1980), 296-298. 

[13]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, Birkhäuser Verlag, 2016. doi: 10.1007/978-3-319-27698-4.

[14]

N. Roidos, Complex powers for cone differential operators and the heat equation on manifolds with conical singularities, Proceedings of the Amer. Math. Soc., 146 (2018), 2995-3007.  doi: 10.1090/proc/13986.

[15]

N. Roidos, The Swift-Hohenberg equation on conic manifolds, J. Math. Anal. Appl., 481 (2020), 123491. doi: 10.1016/j.jmaa.2019.123491.

[16]

N. Roidos and E. Schrohe, Existence and maximal $L^p$-regularity of solutions for the porous medium equation on manifolds with conical singularities, Commun. Partial Differ. Equ., 41 (2016), 1441-1471.  doi: 10.1080/03605302.2016.1219745.

[17]

N. Roidos and E. Schrohe, Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities, Commun. Partial Differ. Equ., 43 (2018), 1456-1484.  doi: 10.1080/03605302.2018.1517788.

[18]

N. Roidos, E. Schrohe and J. Seiler, Bounded $H_{\infty}$-calculus for boundary value problems on manifolds with conical singularities, preprint, arXiv: 1906.03701.

[19]

E. Schrohe and J. Seiler, Bounded $H_{\infty}$-calculus for cone differential operators, J. Evol. Equ., 18 (2018), 1395-1425.  doi: 10.1007/s00028-018-0447-1.

[20]

E. Schrohe and J. Seiler, The resolvent of closed extensions of cone differential operators, Canad. J. Math., 57 (2005), 771-811.  doi: 10.4153/CJM-2005-031-1.

[21]

J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. 

[22]

H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, Pitman Publishing, 1979.

[23]

M. TlidiM. Georgiou and P. Mandel, Transverse patterns in nascent optical bistability, Phys. Rev. A, 48 (1993), 4506-4609. 

[24] J. L. Vázquez, The Porous Medium Equation, Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, 2007. 
[25]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_{p}$-regularity, Math. Ann., 319 (2001), 735-758.  doi: 10.1007/PL00004457.

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