# American Institute of Mathematical Sciences

July  2013, 12(4): 1731-1744. doi: 10.3934/cpaa.2013.12.1731

## Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials

 1 Department of Mathematics, Swansea University, Swansea SA2 8PP, United Kingdom, United Kingdom 2 Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine

Received  April 2011 Revised  March 2012 Published  November 2012

For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
Citation: Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731
##### References:
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show all references

##### References:
 [1] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III), 125 (1957), 25.   Google Scholar [2] E. DiBenedetto, "Degenerate Parabolic Equations,", Springer, (1993).   Google Scholar [3] E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa Cl. Sci, 13 (1986), 487.   Google Scholar [4] E. DiBenedetto, U. Gianazza and V. Vespri, A Harnack inequality for a degenerate parabolic equation,, Acta Mathematica, 200 (2008), 181.   Google Scholar [5] F. Duzaar and G. Mingione, Gradient estimates in non-linear potential theory,, Rend. Lincei - Mat. Appl., 20 (2009), 179.   Google Scholar [6] F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials,, Amer. J. Math., 133 (2011), 1093.   Google Scholar [7] M. de Guzmán, A covering lemma with applications to differentiability of measures and singular integral operators,, Studia Math., 34 (1970), 299.   Google Scholar [8] M. de Guzmán, "Differentiation of Integrals in $R^n$,", Lecture Notes in Math., 481 (1975).   Google Scholar [9] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.   Google Scholar [10] D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.   Google Scholar [11] V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes,, J. Diff. Eq., 247 (2009), 2740.   Google Scholar [12] V. Liskevich, I. I. Skrypnik and Z. Sobol, Potential estimates for quasi-linear parabolic equations,, Advanced Nonlinear Studies, 11 (2011), 905.   Google Scholar [13] J. Malý and W. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 ().   Google Scholar [14] N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.   Google Scholar [15] N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities,, J. Funct. Anal., 256 (2009), 1875.   Google Scholar [16] I. I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations (Russian),, Dokl. Akad. Nauk, 398 (2004), 458.   Google Scholar [17] N. Trudinger and X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory,, Amer. J. Math., 124 (2002), 369.   Google Scholar
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