September  2003, 9(5): 1277-1292. doi: 10.3934/dcds.2003.9.1277

A priori estimates of global solutions of superlinear parabolic problems without variational structure

1. 

Institute of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovak Republic

2. 

Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin, France

Received  December 2001 Revised  February 2003 Published  June 2003

We consider various classes of superlinear parabolic problems, including reaction-diffusion systems and scalar reaction-diffusion equations with convective or dissipative gradient terms. For these problems we prove uniform a priori estimates for all nonnegative global solutions. The existence of an energy functional for these problems is not known, so that traditional methods for a priori estimates do not apply. We use a different approach based on scaling and Fujita-type theorems. In the case of reaction-diffusion systems, we also obtain some universal bounds, i.e. a priori estimates independent of the initial data.
Citation: Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277
[1]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225

[2]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711

[3]

Ryuichi Suzuki. Universal bounds for quasilinear parabolic equations with convection. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 563-586. doi: 10.3934/dcds.2006.16.563

[4]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[5]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

[6]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318

[7]

J.I. Díaz, D. Gómez-Castro. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Conference Publications, 2015, 2015 (special) : 379-386. doi: 10.3934/proc.2015.0379

[8]

Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294

[9]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605

[10]

Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721

[11]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473

[12]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014

[13]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013

[14]

Vincenzo Michael Isaia. Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3459-3481. doi: 10.3934/dcdsb.2017175

[15]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic and Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883

[16]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091

[17]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[18]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[19]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control and Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307

[20]

Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021060

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (161)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]