A quasi-monotonicity formula for the solution to a semilinear parabolic equation , in with 0-Dirichlet boundary condition is obtained. As an application, it is shown that for some suitable global weak solution u and any compact set there exists a close subset such that u is continuous in and the -dimensional parabolic Hausdorff measure of is finite.
@article{AIHPC_2010__27_6_1333_0, author = {Zheng, Gao-Feng}, title = {A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1333--1360}, publisher = {Elsevier}, volume = {27}, number = {6}, year = {2010}, doi = {10.1016/j.anihpc.2010.07.001}, mrnumber = {2738324}, zbl = {1213.35177}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2010.07.001/} }
TY - JOUR AU - Zheng, Gao-Feng TI - A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 1333 EP - 1360 VL - 27 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2010.07.001/ DO - 10.1016/j.anihpc.2010.07.001 LA - en ID - AIHPC_2010__27_6_1333_0 ER -
%0 Journal Article %A Zheng, Gao-Feng %T A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 1333-1360 %V 27 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2010.07.001/ %R 10.1016/j.anihpc.2010.07.001 %G en %F AIHPC_2010__27_6_1333_0
Zheng, Gao-Feng. A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1333-1360. doi : 10.1016/j.anihpc.2010.07.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.07.001/
[1] Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. 28 (1977), 473-486 | MR | Zbl
,[2] Mathematical Problems from Combustion Theory, Appl. Math. Sci. vol. 83, Springer-Verlag, New York (1989) | MR | Zbl
, ,[3] Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831 | MR | Zbl
, , ,[4] An Introduction to Semilinear Evolution Equations, Oxford Lecture Ser. Math. Appl. vol. 13, The Clarendon Press, Oxford University Press, NY (1998) | MR | Zbl
, ,[5] Some blow-up problems for a semilinear parabolic equation with a potential, J. Differential Equations 244 (2008), 766-802 | MR | Zbl
, ,[6] On partial regularity of the borderline solution of semilinear parabolic problems, Calc. Var. Partial Differential Equations 30 (2007), 251-275 | MR | Zbl
, , ,[7] Local monotonicity formulas for some nonlinear diffusion equations, Calc. Var. Partial Differential Equations 23 (2005), 67-81 | MR | Zbl
,[8] Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447 | MR | Zbl
, ,[9] Immediate regularization after blow-up, SIAM J. Math. Anal. 37 (2005), 752-776 | MR | Zbl
, , ,[10] On the blowing up of solutions of the Cauchy problem for , J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124 | MR | Zbl
,[11] Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1-67 | MR | Zbl
, ,[12] Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319 | MR | Zbl
, ,[13] Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40 | MR | Zbl
, ,[14] Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845-884 | MR | Zbl
, ,[15] Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), 483-514 | MR | Zbl
, , ,[16] On blow-up rate for sign-changing solutions in a convex domain, Math. Methods Appl. Sci. 27 (2004), 1771-1782 | MR | Zbl
, , ,[17] On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305-330 | MR | Zbl
,[18] Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. vol. 23, American Mathematical Society, Providence, RI (1967) | MR
, , ,[19] Some nonexistence and instability theorems for solutions of formally parabolic equations of the form , Arch. Ration. Mech. Anal. 51 (1973), 371-386 | MR
,[20] On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), 1494-1541 | MR | Zbl
, ,[21] Boundedness of global solutions for a supercritical semilinear heat equation and its application, Indiana Univ. Math. J. 54 (2005), 1047-1059 | MR | Zbl
,[22] Multiple blowup of solutions for a semilinear heat equation, Math. Ann. 331 (2005), 461-473 | MR | Zbl
,[23] Various behaviors of solutions for a semilinear heat equation after blowup, J. Funct. Anal. 220 (2005), 214-227 | MR | Zbl
,[24] On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), 97-120 | MR | Zbl
, , ,[25] On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 485-502 | MR | Zbl
,Cité par Sources :