A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation

Zheng, Gao-Feng

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1333-1360.

A quasi-monotonicity formula for the solution to a semilinear parabolic equation $u_t=\Delta u+V\left(x\right){\left|u\right|}^{p-1}u$, $p>(N+2)/(N-2)$ in $\Omega \times (0,T)$ 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 $Q\subset \Omega \times (0,T)$ there exists a close subset $Q^{\text{'}}\subset Q$ such that u is continuous in $Q^{\text{'}}$ and the $(N-\frac{4}{p-1})$-dimensional parabolic Hausdorff measure ${\mathscr{H}}^{(N-\frac{4}{p-1})}(Q\setminus Q^{\text{'}})$ of $Q\setminus Q^{\text{'}}$ is finite.

Mots clés : Quasi-monotonicity formula, Partial regularity, Borderline solutions, Semilinear parabolic equations, Potential

[1] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. 28 (1977), 473-486 | MR | Zbl

[2] J. Bebernes, D. Eberly, Mathematical Problems from Combustion Theory, Appl. Math. Sci. vol. 83, Springer-Verlag, New York (1989) | MR | Zbl

[3] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831 | MR | Zbl

[4] T. Cazenave, A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Ser. Math. Appl. vol. 13, The Clarendon Press, Oxford University Press, NY (1998) | MR | Zbl

[5] T. Cheng, G.-F. Zheng, Some blow-up problems for a semilinear parabolic equation with a potential, J. Differential Equations 244 (2008), 766-802 | MR | Zbl

[6] K.-S. Chou, S.-Z. Du, G.-F. Zheng, On partial regularity of the borderline solution of semilinear parabolic problems, Calc. Var. Partial Differential Equations 30 (2007), 251-275 | MR | Zbl

[7] K. Ecker, Local monotonicity formulas for some nonlinear diffusion equations, Calc. Var. Partial Differential Equations 23 (2005), 67-81 | MR | Zbl

[8] A. Friedman, B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447 | MR | Zbl

[9] M. Fila, H. Matano, P. Poláčik, Immediate regularization after blow-up, SIAM J. Math. Anal. 37 (2005), 752-776 | MR | Zbl

[10] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124 | MR | Zbl

[11] V. Galaktionov, J.L. Vazquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1-67 | MR | Zbl

[12] Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319 | MR | Zbl

[13] Y. Giga, R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40 | MR | Zbl

[14] Y. Giga, R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845-884 | MR | Zbl

[15] Y. Giga, S. Matsui, S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), 483-514 | MR | Zbl

[16] Y. Giga, S. Matsui, S. Sasayama, On blow-up rate for sign-changing solutions in a convex domain, Math. Methods Appl. Sci. 27 (2004), 1771-1782 | MR | Zbl

[17] S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305-330 | MR | Zbl

[18] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural'Ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. vol. 23, American Mathematical Society, Providence, RI (1967) | MR

[19] H.A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form ${\mathrm{𝑃𝑢}}_t=-\mathrm{𝐴𝑢}+ℱ\left(u\right)$, Arch. Ration. Mech. Anal. 51 (1973), 371-386 | MR

[20] H. Matano, F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), 1494-1541 | MR | Zbl

[21] N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its application, Indiana Univ. Math. J. 54 (2005), 1047-1059 | MR | Zbl

[22] N. Mizoguchi, Multiple blowup of solutions for a semilinear heat equation, Math. Ann. 331 (2005), 461-473 | MR | Zbl

[23] N. Mizoguchi, Various behaviors of solutions for a semilinear heat equation after blowup, J. Funct. Anal. 220 (2005), 214-227 | MR | Zbl

[24] W.-M. Ni, P. Sacks, J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), 97-120 | MR | Zbl

[25] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 485-502 | MR | Zbl