Blow-up of solutions to a degenerate parabolic equation not in divergence form. (English) Zbl 1028.35081

Winkler, Michael

J. Differ. Equations 192, No. 2, 445-474 (2003).

The author studies the so-called blow-up phenomenon for solutions to the Dirichlet problem \[ \begin{gathered} u_t = u^p(\Delta u + u),\quad (x,t) \in \Omega\times (0,T),\\ \left. u\right|_{\partial\Omega} = 0,\quad \left. u\right|_{t = 0} = u_0, \end{gathered} \] where \(0 < p < 2\), \(\Omega \subset \mathbb{R}^n\) is a smooth bounded domain and \(u_0\in C^0(\overline {\Omega})\) is a positive in \(\Omega\) and vanishes at \(\partial\Omega\). It is assumed that the principal eigenvalue \(\lambda_1(\Omega)\) of the Laplacian in \(\Omega\) is less than one. The author proves that the set of points at which \(u\) blows up has positive measure and the blow-up rate is exactly \((T - t)^{-1/p}\). The \(\omega\)-limit set of \((T - t)^{1/p}u(t)\) consists of continuous functions which solve \(\Delta w + w = (1/p)w^{1-p}\) in the case of \(n =1\) or \(p < 1\). In one space dimension it is shown that \((T - t)^{1/p}u(t) \to w\) as \(t\to T\), where \(w\) coincides with an element of a one-parameter family of functions inside each component of its positivity set. The rest of the article is devoted to studying the size of the components of \(\{w > 0\}\).

Reviewer: Vladimir N.Grebenev (Novosibirsk)

MSC:

35K65	Degenerate parabolic equations
35K55	Nonlinear parabolic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

Dirichlet problem; blow-up rate; \(\omega\)-limit set; blow-up set

References:

[1]	Angenent, S., The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390, 79-96 (1988) · Zbl 0644.35050
[2]	Angenent, S., On the formation of singularities in the curve shortening flow, J. Differential Geom., 33, 3, 601-633 (1991) · Zbl 0731.53002
[3]	Bebernes, J.; Galaktionov, V. A., On classification of blow-up patterns for a quasilinear heat equation, Differential Integral Equations, 9, 4, 655-670 (1996) · Zbl 0851.35057
[4]	Bernis, F.; Friedman, A., Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83, 1, 179-206 (1990) · Zbl 0702.35143
[5]	Friedman, A., Variational Principles and Free-Boundary Problems (1982), Wiley: Wiley New York · Zbl 0564.49002
[6]	Friedman, A.; McLeod, B., Blow-up of solutions of nonlinear degenerate parabolic equations, Arch. Rational Mech. Anal., 96, 55-80 (1987) · Zbl 0619.35060
[7]	Gage, M. E., On the size of the blow-up set for a quasilinear parabolic equation, Contemp. Math., 127, 41-58 (1992) · Zbl 0770.35029
[8]	Galaktionov, V. A., On a blow-up set for the quasilinear heat equation \(u_t=(u^{σ\) · Zbl 0802.35065
[9]	Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin · Zbl 0691.35001
[10]	Hamilton, R. S., The Ricci flow on surfaces, Contemp. Math., 71, 237-262 (1988) · Zbl 0663.53031
[11]	B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, Springer, Berlin, 1985.; B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, Springer, Berlin, 1985. · Zbl 0593.35002
[12]	Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and Quasi-linear Equations of Parabolic Type (1968), AMS: AMS Providence, RI · Zbl 0174.15403
[13]	Low, B. C., Resistive diffusion of force-free magnetic fields in a passive medium II, A nonlinear analysis of the one-dimensional case, Astrophys. J., 184, 917-929 (1973)
[14]	Luckhaus, S.; Dal Passo, R., A degenerate diffusion problem not in divergence form, J. Differential Equations, 69, 1-14 (1987) · Zbl 0688.35045
[15]	Porzio, M. M.; Vespri, V., Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103, 1, 146-178 (1993) · Zbl 0796.35089
[16]	Samarskii, A. A.; Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P., Blow-up in quasilinear parabolic equations (1995), De Gruyter Expositions in Mathematics: De Gruyter Expositions in Mathematics Berlin · Zbl 1020.35001
[17]	B.J. Schmitt, M. Wiegner, Remarks to the blow-up rate of a degenerate parabolic equation, Lecture Notes in Pure and Applied Sciences, Dekker, New York, Proceedings of the Sixth International Conference on Evolution Equations, Bad Herrenalb, 1998.; B.J. Schmitt, M. Wiegner, Remarks to the blow-up rate of a degenerate parabolic equation, Lecture Notes in Pure and Applied Sciences, Dekker, New York, Proceedings of the Sixth International Conference on Evolution Equations, Bad Herrenalb, 1998. · Zbl 1009.35049
[18]	Tsutsumi, T.; Ishiwata, M., Regional blow-up of solutions to the initial boundary value problem for \(u_t=u^{δ\) · Zbl 0883.35068
[19]	Wiegner, M., Blow-up for solutions of some degenerate parabolic equations, Differential Integral Equations, 7, 5-6, 1641-1647 (1994) · Zbl 0797.35100
[20]	Wiegner, M., A degenerate diffusion equation with a nonlinear source term, Nonlinear Anal. TMA, 28, 1977-1995 (1997) · Zbl 0874.35061
[21]	M. Wiegner, Remarks to the blow-up profile of a degenerate parabolic equation, in: H. Amann, et al. (Eds.), Progress in Partial Differential Equations, Papers from the Third European Conference on Elliptic and Parabolic Problems, Pont-a-Mousson, France, June 1997, Vol. 2, Longman, Harlow, Pitman Res. Notes Math. Ser. 384, Longman Sci. Tech., Harlow, 1998, 201-212.; M. Wiegner, Remarks to the blow-up profile of a degenerate parabolic equation, in: H. Amann, et al. (Eds.), Progress in Partial Differential Equations, Papers from the Third European Conference on Elliptic and Parabolic Problems, Pont-a-Mousson, France, June 1997, Vol. 2, Longman, Harlow, Pitman Res. Notes Math. Ser. 384, Longman Sci. Tech., Harlow, 1998, 201-212. · Zbl 0919.35072
[22]	M. Winkler, Some results on degenerate parabolic equations not in divergence form, Ph.D. Thesis,; M. Winkler, Some results on degenerate parabolic equations not in divergence form, Ph.D. Thesis,
[23]	Winkler, M., On the Cauchy problem for a degenerate parabolic equation, Z. Anal. Anwendungen, 20, 3, 677-690 (2001) · Zbl 0987.35089
[24]	T.U. Zimmer, Theorie und Numerik einer degenerierten parabolischen Gleichung, Ph.D. Thesis, Bayreuth, 1997.; T.U. Zimmer, Theorie und Numerik einer degenerierten parabolischen Gleichung, Ph.D. Thesis, Bayreuth, 1997.