# American Institute of Mathematical Sciences

2005, 2005(Special): 605-610. doi: 10.3934/proc.2005.2005.605

## A new regularity estimate for solutions of singular parabolic equations

 1 Department of Mathematics, Iowa State University, Ames, IA 50011

Received  September 2004 Revised  February 2005 Published  September 2005

In 1982, K. Ecker showed that solutions of the parabolic equation $u_t=\operatorname {div} \left( \frac {Du}{(1+|Du|^2)^{1/2}}\right) + H(x,u)$ have a very unusual regularity property: The interior regularity of $u$ is determined only by its initial regularity. In this note, we show that a similar result is true for a general class of equations. The model equation is $u_t=\operatorname {div} \left( |Du|^{p-2} {Du}\right)$ with $1 Citation: Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605  [1] Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721 [2] Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 [3] Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029 [4] Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277 [5] Denis R. Akhmetov, Renato Spigler.$L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051 [6] Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258 [7] Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731 [8] Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010 [9] Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107 [10] Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155 [11] Pengyu Chen, Xuping Zhang. Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020308 [12] D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499 [13] Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013 [14] Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69 [15] Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99. [16] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [17] Tommaso Leonori, Ireneo Peral, Ana Primo, Fernando Soria. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6031-6068. doi: 10.3934/dcds.2015.35.6031 [18] Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313 [19] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020247 [20] 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.

Impact Factor: