March  2020, 9(1): 61-86. doi: 10.3934/eect.2020017

On quasilinear parabolic equations and continuous maximal regularity

1. 

Department of Mathematics & Computer Science, 221 Richmond Way, University of Richmond, VA 23173, USA

2. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

* Corresponding author: Jeremy LeCrone

Received  August 2018 Revised  August 2019 Published  March 2020 Early access  October 2019

Fund Project: This work was supported by a grant from the Simons Foundation (#426729, Gieri Simonett).

We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.

Citation: Jeremy LeCrone, Gieri Simonett. On quasilinear parabolic equations and continuous maximal regularity. Evolution Equations & Control Theory, 2020, 9 (1) : 61-86. doi: 10.3934/eect.2020017
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[2]

S. B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107.  doi: 10.1017/S0308210500024598.  Google Scholar

[3]

T. Asai, Quasilinear parabolic equation and its applications to fourth order equations with rough initial data, J. Math. Sci. Univ. Tokyo, 19 (2013), 507-532.   Google Scholar

[4]

A. J. BernoffA. L. Bertozzi and T. P. Witelski, Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff, J. Statist. Phys., 93 (1998), 725-776.  doi: 10.1023/B:JOSS.0000033251.81126.af.  Google Scholar

[5]

P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67.  doi: 10.1007/PL00001364.  Google Scholar

[6]

J. EscherU. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419-1433.  doi: 10.1137/S0036141097320675.  Google Scholar

[7]

J. Escher and P. B. Mucha, The surface diffusion flow on rough phase spaces, Discrete Contin. Dyn. Syst., 26 (2010), 431-453.  doi: 10.3934/dcds.2010.26.431.  Google Scholar

[8]

H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math., 16 (2012), 209-235.  doi: 10.4310/AJM.2012.v16.n2.a3.  Google Scholar

[9]

J. LeCroneJ. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ, J. Evol. Equ., 14 (2014), 509-533.  doi: 10.1007/s00028-014-0226-6.  Google Scholar

[10]

J. LeCrone and G. Simonett, Continuous maximal regularity and analytic semigroups, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., 2 (2011), 963–970.  Google Scholar

[11]

J. LeCrone and G. Simonett, On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow, SIAM J. Math. Anal., 45 (2013), 2834-2869.  doi: 10.1137/120883505.  Google Scholar

[12]

J. LeCrone and G. Simonett, On the flow of non-axisymmetric perturbations of cylinders via surface diffusion, J. Differential Equations, 260 (2016), 5510-5531.  doi: 10.1016/j.jde.2015.12.008.  Google Scholar

[13]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547].  Google Scholar

[14]

J. McCoyG. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows, Math. Z., 269 (2011), 147-178.  doi: 10.1007/s00209-010-0720-7.  Google Scholar

[15]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\Bbb{R}^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428.  doi: 10.3934/dcds.2013.33.5407.  Google Scholar

[16]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[17]

J. PrüssG. Simonett and M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications, J. Differential Equations, 264 (2018), 2028-2074.  doi: 10.1016/j.jde.2017.10.010.  Google Scholar

[18]

J. PrüssG. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), 3902-3931.  doi: 10.1016/j.jde.2008.10.034.  Google Scholar

[19]

J. Prüss, G. Simonett and R. Zacher, On normal stability for nonlinear parabolic equations, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl., (2009), 612–621.  Google Scholar

[20]

J. Prüss and M. Wilke, Addendum to the paper "On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ" [MR3250797], J. Evol. Equ., 17 (2017), 1381-1388.  doi: 10.1007/s00028-017-0382-6.  Google Scholar

[21]

Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45-85.  doi: 10.1007/s00030-014-0275-0.  Google Scholar

[22]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 14 (2014), 211-248.  doi: 10.1007/s00028-014-0218-6.  Google Scholar

[23]

G. Wheeler, Lifespan theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698.  doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[24]

G. Wheeler, Surface diffusion flow near spheres, Calc. Var. Partial Differential Equations, 44 (2012), 131-151.  doi: 10.1007/s00526-011-0429-4.  Google Scholar

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[2]

S. B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107.  doi: 10.1017/S0308210500024598.  Google Scholar

[3]

T. Asai, Quasilinear parabolic equation and its applications to fourth order equations with rough initial data, J. Math. Sci. Univ. Tokyo, 19 (2013), 507-532.   Google Scholar

[4]

A. J. BernoffA. L. Bertozzi and T. P. Witelski, Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff, J. Statist. Phys., 93 (1998), 725-776.  doi: 10.1023/B:JOSS.0000033251.81126.af.  Google Scholar

[5]

P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67.  doi: 10.1007/PL00001364.  Google Scholar

[6]

J. EscherU. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419-1433.  doi: 10.1137/S0036141097320675.  Google Scholar

[7]

J. Escher and P. B. Mucha, The surface diffusion flow on rough phase spaces, Discrete Contin. Dyn. Syst., 26 (2010), 431-453.  doi: 10.3934/dcds.2010.26.431.  Google Scholar

[8]

H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math., 16 (2012), 209-235.  doi: 10.4310/AJM.2012.v16.n2.a3.  Google Scholar

[9]

J. LeCroneJ. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ, J. Evol. Equ., 14 (2014), 509-533.  doi: 10.1007/s00028-014-0226-6.  Google Scholar

[10]

J. LeCrone and G. Simonett, Continuous maximal regularity and analytic semigroups, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., 2 (2011), 963–970.  Google Scholar

[11]

J. LeCrone and G. Simonett, On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow, SIAM J. Math. Anal., 45 (2013), 2834-2869.  doi: 10.1137/120883505.  Google Scholar

[12]

J. LeCrone and G. Simonett, On the flow of non-axisymmetric perturbations of cylinders via surface diffusion, J. Differential Equations, 260 (2016), 5510-5531.  doi: 10.1016/j.jde.2015.12.008.  Google Scholar

[13]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547].  Google Scholar

[14]

J. McCoyG. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows, Math. Z., 269 (2011), 147-178.  doi: 10.1007/s00209-010-0720-7.  Google Scholar

[15]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\Bbb{R}^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428.  doi: 10.3934/dcds.2013.33.5407.  Google Scholar

[16]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[17]

J. PrüssG. Simonett and M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications, J. Differential Equations, 264 (2018), 2028-2074.  doi: 10.1016/j.jde.2017.10.010.  Google Scholar

[18]

J. PrüssG. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), 3902-3931.  doi: 10.1016/j.jde.2008.10.034.  Google Scholar

[19]

J. Prüss, G. Simonett and R. Zacher, On normal stability for nonlinear parabolic equations, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl., (2009), 612–621.  Google Scholar

[20]

J. Prüss and M. Wilke, Addendum to the paper "On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ" [MR3250797], J. Evol. Equ., 17 (2017), 1381-1388.  doi: 10.1007/s00028-017-0382-6.  Google Scholar

[21]

Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45-85.  doi: 10.1007/s00030-014-0275-0.  Google Scholar

[22]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 14 (2014), 211-248.  doi: 10.1007/s00028-014-0218-6.  Google Scholar

[23]

G. Wheeler, Lifespan theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698.  doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[24]

G. Wheeler, Surface diffusion flow near spheres, Calc. Var. Partial Differential Equations, 44 (2012), 131-151.  doi: 10.1007/s00526-011-0429-4.  Google Scholar

[1]

Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002

[2]

Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365

[3]

Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021147

[4]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591

[5]

Said Boulite, S. Hadd, L. Maniar. Critical spectrum and stability for population equations with diffusion in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 265-276. doi: 10.3934/dcdsb.2005.5.265

[6]

Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864

[7]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612

[8]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[9]

Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457

[10]

Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416

[11]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

[12]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431

[13]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[14]

Jie Jiang. Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 609-634. doi: 10.3934/dcds.2020025

[15]

Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109

[16]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029

[17]

Norimichi Hirano, Wen Se Kim. Multiplicity and stability result for semilinear parabolic equations. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 271-280. doi: 10.3934/dcds.1996.2.271

[18]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[19]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

[20]

Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (205)
  • HTML views (227)
  • Cited by (0)

Other articles
by authors

[Back to Top]