We study the so-called -superparabolic functions, which are defined as lower semicontinuous supersolutions of a quasilinear parabolic equation. In the linear case, when , we have supercaloric functions and the heat equation. We show that the -superparabolic functions have a spatial Sobolev gradient and a sharp summability exponent is given.
Kinnunen, Juha 1 ; Lindqvist, Peter 2
@article{ASNSP_2005_5_4_1_59_0,
author = {Kinnunen, Juha and Lindqvist, Peter},
title = {Summability of semicontinuous supersolutions to a quasilinear parabolic equation},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {59--78},
year = {2005},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 4},
number = {1},
mrnumber = {2165403},
zbl = {1107.35070},
language = {en},
url = {https://www.numdam.org/item/ASNSP_2005_5_4_1_59_0/}
}
TY - JOUR AU - Kinnunen, Juha AU - Lindqvist, Peter TI - Summability of semicontinuous supersolutions to a quasilinear parabolic equation JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2005 SP - 59 EP - 78 VL - 4 IS - 1 PB - Scuola Normale Superiore, Pisa UR - https://www.numdam.org/item/ASNSP_2005_5_4_1_59_0/ LA - en ID - ASNSP_2005_5_4_1_59_0 ER -
%0 Journal Article %A Kinnunen, Juha %A Lindqvist, Peter %T Summability of semicontinuous supersolutions to a quasilinear parabolic equation %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2005 %P 59-78 %V 4 %N 1 %I Scuola Normale Superiore, Pisa %U https://www.numdam.org/item/ASNSP_2005_5_4_1_59_0/ %G en %F ASNSP_2005_5_4_1_59_0
Kinnunen, Juha; Lindqvist, Peter. Summability of semicontinuous supersolutions to a quasilinear parabolic equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 59-78. https://www.numdam.org/item/ASNSP_2005_5_4_1_59_0/
[1] and , Local behavior of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25 (1967), 81-122. | Zbl | MR
[2] , On selfsimilar motions of compressible fluids in a porous medium (in Russian), Prikl. Mat. Mekh. 16 (1952), 679-698. | Zbl | MR
[3] , “Degenerate Parabolic Equations”, Springer-Verlag, Berlin-Heidelberg-New York, 1993. | Zbl | MR
[4] , and , Current issues on sigular and degenerate evolution equations, in: “Handbook of Differential Equations”, C. Dafermos and E. Feireisl (eds.) in press, Elsevier Publ. | Zbl
[5] , and , On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 (2001), 699-717. | Zbl | MR
[6] and , Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl. (4) to appear. | MR
[7] and , On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM. J. Math. Anal. 27 (1996), 661-683. | Zbl | MR
[8] and , Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 591-613. | Zbl | MR | Numdam
[9] , and , “Linear and Quasilinear Equations of Parabolic Type”, AMS, Providence R. I., 1968.
[10] , On the definition and properties of -superharmonic functions, J. Reine Angew. Math. 365 (1986), 67-79. | Zbl | MR
[11] , “Second Order Parabolic Equations”, World Scientific, Singapore, 1996. | Zbl | MR
[12] , A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. | Zbl | MR
[13] , Correction to: “A Harnack inequality for parabolic differential equations”, Comm. Pure Appl. Math. 20 (1967), 231-236. | Zbl | MR
[14] , Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205-226. | Zbl | MR
[15] , Green functions, potentials, and the Dirichlet problem for the heat equation, Proc. London Math. Soc. 33 (1976), 251-298. | Zbl | MR
[16] , Corrigendum, Proc. London Math. Soc. 37 (1977), 32-34. | Zbl | MR
[17] , , and , “Nonlinear Diffusion Equations”, World Scientific, Singapore, 2001. | Zbl | MR
