×

\(L^r_{\mathrm{loc}}-L^\infty_{\mathrm{loc}}\) estimates and expansion of positivity for a class of doubly non linear singular parabolic equations. (English) Zbl 1284.35099

Summary: We show some properties regarding the local behaviour of local weak solutions to a class of doubly nonlinear singular parabolic equations.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35K67 Singular parabolic equations
35K55 Nonlinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Bonforte, Super and ultracontractive bounds for doubly nonlinear evolution equations,, Rev. Mat. Iberoamericana, 22, 111 (2006) · Zbl 1103.35021
[2] M. Bonforte, Local smoothing effects, positivity, and Harnack inequalities for the fast \(p\)-Laplacian equation,, Advances in Math., 224, 2151 (2010) · Zbl 1198.35136
[3] M. Bonforte, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations,, Advances in Math., 223, 529 (2010) · Zbl 1184.35083
[4] E. DiBenedetto, <em>Degenerate Parabolic Equations</em>,, Springer-Verlag (1993) · Zbl 0794.35090
[5] E. DiBenedetto, <em>Harnack’s Inequality for Degenerate and Singular Parabolic Equations</em>,, Springer Monographs in Mathematics (2012) · Zbl 1237.35004
[6] S. Fornaro, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations,, Adv. Differential Equations, 13, 139 (2008) · Zbl 1160.35039
[7] S. Fornaro, Energy estimates and integral Harnack inequality for some doubly nonlinear singular parabolic equations,, Contemporary Mathematics, 594, 179 (2013) · Zbl 1322.35082
[8] M. A. Herrero, The Cauchy problem for \(u_t =\Delta u^m\) when \(0< m <1\),, Trans. Amer. Math. Soc., 291, 145 (1985) · Zbl 0583.35052
[9] A. S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations,, Russian Math. Surveys, 42, 169 (1987) · Zbl 0642.35047
[10] A. V. Ivanov, Regularity for doubly nonlinear parabolic equations,, Journal of Mathematical Sciences, 83 (1997) · Zbl 0868.35056
[11] A. V. Ivanov, Existence and uniqueness of a regular solution of the Cauchy-Diriclhet problem for a class of doubly nonlinear parabolic equations,, Journal of Mathematical Sciences, 84 (1997)
[12] J. L. Lions, <em>Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires</em>,, Dunod (1969) · Zbl 0189.40603
[13] M. M. Porzio, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations,, J. Diff. Equations, 103, 146 (1993) · Zbl 0796.35089
[14] D. Stan, Asymptotic behaviour of the doubly nonlinear diffusion equation on bounded domains,, Nonlinear Analysis TMA, 77, 1 (2013) · Zbl 1258.35130
[15] V. Vespri, Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations,, J. Math. Anal. Appl., 181, 104 (1994) · Zbl 0798.35073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.