Abstract
On a cylindrical domain \(E_T\), we consider doubly nonlinear parabolic equations, whose prototype is \(\partial _t u - \mathrm{div}(|u|^{m-1}|Du|^{p-2}Du) = \mu ,\) where \(\mu \) is a non-negative Radon measure having finite total mass \(\mu (E_T)\). The central objective is to establish pointwise estimates for weak solutions in terms of nonlinear parabolic potentials in the doubly degenerate case \((p\ge 2, m>1)\). Moreover, we will prove the sharpness of the estimates by giving an optimal Lorentz space criterion regarding the local uniform boundedness of weak solutions and by comparing them to the decay of the Barenblatt solution.
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References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces (Pure and Applied Mathematics), vol. 140, 2nd edn. Academic Press, Amsterdam (2003)
Andreucci, D.: L\(^{\infty }_{loc}\)-estimates for local solutions of degenerate parabolic equations. SIAM J. Math. Anal. 22, 138–145 (1991)
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Amsterdam (1988)
Bögelein, V., Duzaar, F., Gianazza, U.: Porous medium type equations with measure data and potential estimates. SIAM J. Math. Anal. 45, 3283–3330 (2013)
Bögelein, V., Duzaar, F., Gianazza, U.: Sharp boundedness and continuity results for the singular porous medium equation. Isr. J. Math. 214, 259–314 (2016)
Bögelein, V., Duzaar, F., Gianazza, U.: Very weak solutions of singular porous medium equations with measure data. Commun. Pure Appl. Anal. 14, 23–49 (2015)
Bögelein, V., Duzaar, F., Marcellini, P.: Parabolic systems with p, q-growth: a variational approach. Arch. Ration. Mech. Anal. 210, 219–267 (2013)
Cianchi, A.: Nonlinear potentials, local solutions to elliptic equations and rearrangements. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 10, 335–361 (2011)
DiBenedetto, E.: Degenerate Parabolic Equations (Springer Universitext). Springer, New York (1993)
DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack’s Inequality for Degenerate and Singular Parabolic Equations (Springer Monographs in Mathematics). Springer, New York (2012)
Duzaar, F., Mingione, G.: Gradient continuity estimates. Calc. Var. 39, 379–418 (2010)
Duzaar, F., Mingione, G.: Gradient estimates via linear and nonlinear potentials. J. Funct. Anal. 259, 2961–2998 (2010)
Duzaar, F., Mingione, G.: Gradient estimates via non-linear potentials. Am. J. Math. 133, 1093–1149 (2011)
Ebmeyer, C., Urbano, J.M.: The smoothing property for a class of doubly nonlinear parabolic equations. Trans. Am. Math. Soc. 357, 3239–3253 (2005)
Fornaro, S., Sosio, M., Vespri, V.: \(L^{r}_{loc}-L^{\infty }_{loc}\) estimates and expansion of positivity for a class of doubly non linear singular parabolic equations. Discrete Contin. Dyn. Syst. Ser. S 7, 737–760 (2014)
Ivanov, A.V.: Hölder estimates for a natural class of equations of the type of fast diffusion. J. Math. Sci. (N. Y.) 89, 1607–1630 (1998)
Ivanov, A.V.: Regularity for doubly nonlinear parabolic equations. J. Math. Sci. (N. Y.) 83, 22–37 (1997)
Ivanov, A.V., Mkrtychyan, P.Z.: Existence of Hölder continuous generalized solutions of the first boundary value problem for quasilinear doubly degenerate parabolic equations. J. Sov. Math. 62, 2725–2740 (1992)
Ivanov, A.V., Mkrtychyan, P.Z., Jäger, W.: Existence and uniqueness of a regular solution of the Cauchy–Dirichlet problem for a class of doubly nonlinear parabolic equations. J. Math. Sci. (N. Y.) 84, 845–855 (1997)
Ivanov, A.V., Mkrtychyan, P.Z., Jäger, W.: Erratum to: Existence and uniqueness of a regular solution of the Cauchy–Dirichlet problem for a class of doubly nonlinear parabolic equations. J. Math. Sci. (N. Y.) 184, 786–787 (2012)
Kalashnikov, A.S.: Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russ. Math. Surv. 42, 169–222 (1987)
Kilpeläinen, T., Malý, J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 19, 591–613 (1992)
Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)
Kinnunen, J., Kuusi, T.: Local behaviour of solutions to doubly nonlinear parabolic equations. Math. Ann. 337, 705–728 (2007)
Kuusi, T., Mingione, G.: Gradient regularity for nonlinear parabolic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12, 755–822 (2013)
Kuusi, T., Mingione, G.: Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoam. 28, 535–576 (2012)
Kuusi, T., Mingione, G.: The Wolff gradient bound for degenerate paraoblic equations. J. Eur. Math. Soc. 16, 835–892 (2014)
Kuusi, T., Mingione, G.: Riesz potentials and nonlinear parabolic equations. Arch. Ration. Mech. Anal. 212, 727–780 (2014)
Liskevich, V., Skrypnik, I.I.: Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete Contin. Dyn. Syst. 6, 1029–1042 (2013)
Liskevich, V., Skrypnik, I.I.: Pointwise estimates for solutions to the porous medium equation with measure as a forcing term. Isr. J. Math. 194, 259–275 (2013)
Liskevich, V., Skrypnik, I.I., Sobol, Z.: Estimates of solutions for the parabolic \(p\)-Laplacian equation with measure via parabolic nonlinear potentials. Commun. Pure Appl. Anal. 12, 1731–1744 (2013)
Liskevich, V., Skrypnik, I.I., Sobol, Z.: Potential estimates for quasi-linear parabolic equations. Adv. Nonlinear Stud. 11, 905–915 (2011)
Manfredi, J.J., Vespri, V.: Large time behavior of solutions to a class of doubly nonlinear parabolic equations. Electron. J. Differ. Equ. 1994, 1–17 (1994)
Porzio, M.M., Vespri, V.: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103, 146–178 (1993)
Savaré, G., Vespri, V.: The asymptotic profile of solutions of a class of doubly nonlinear equations. Nonlinear Anal. 22, 1553–1565 (1994)
Siljander, J.: Boundedness of the gradient for a doubly nonlinear parabolic equation. J. Math. Anal. Appl. 371, 158–167 (2010)
Sturm, S.: Pointwise estimates for porous medium type equations with low order terms and measure data. Electron. J. Differ. Equ. 2015, 1–25 (2015)
Sturm, S.: Existence of weak solutions of doubly nonlinear parabolic equations. J. Math. Anal. Appl. 455, 842–863 (2017)
Sturm, S.: Existence of very weak solutions of doubly nonlinear parabolic equations with measure data. Ann. Acad. Sci. Fenn. Math. 42(2), 931–962 (2017)
Tedeev, A., Vespri, V.: Optimal behavior of the support of the solutions to a class of degenerate parabolic systems. Interfaces Free Bound. 17, 143–156 (2015)
Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations (Oxford Lecture Series in Mathematics and Its Applications). Oxford University Press, Oxford (2006)
Vázquez, J.L.: The Porous Medium Equation (Oxford Mathematical Monographs). Oxford University Press, Oxford (2007)
Vespri, V.: Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations. J. Math. Anal. Appl. 181, 104–131 (1994)
Vespri, V.: On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations. Manuscr. Math. 75, 65–80 (1992)
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Sturm, S. Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data. manuscripta math. 157, 295–322 (2018). https://doi.org/10.1007/s00229-018-1014-3
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DOI: https://doi.org/10.1007/s00229-018-1014-3