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July  2013, 12(4): 1769-1782. doi: 10.3934/cpaa.2013.12.1769

Propagation of singularities of nonlinear heat flow in fissured media

Andrey Shishkov 1, and  Laurent Véron 2,

1. 

Institute of Applied Mathematics and Mechanics, 83114 Donetsk
2. 

Laboratoire de Mathématiques et Physique Théorique CNRS UMR 6083, Université François-Rabelais, 37200 Tours, France

Received  June 2011 Revised  June 2012 Published  November 2012

Let $\Gamma=\{\gamma(\tau)\in R^N\times [0,T], \gamma(0)=(0,0)\}$ be $C^{0,1}$ -- space-time curve and continuos function $h(x,t)>0$ in $ R^N\times [0,T]\setminus \Gamma (h(x,t)=0$ on $\Gamma$). We investigate the behaviour as $k\to \infty$ of the fundamental solutions $u_k$ of equation $u_t-\Delta u+h(x,t)u^p=0$, $p>1$, satisfying singular initial condition $u_k(x,0)=k\delta_0$. The main problem is whether the limit $u_\infty$ is still a solution of the above equation with isolated point singularity at $(0,0)$, or singularity set of $u_\infty$ contains some part or all $\Gamma$.
Keywords: propagation of singularity., very singular solution, strong degenerate absorption, Semilinear parabolic equation.
Mathematics Subject Classification: Primary: 35K60; Secondary: 35K5.
Citation: Andrey Shishkov, Laurent Véron. Propagation of singularities of nonlinear heat flow in fissured media. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1769-1782. doi: 10.3934/cpaa.2013.12.1769
References:
[1]

H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial coinditions, J. Math. Pures Appl., 62 (1983), 73-97.

[2]

H. Brezis, L. A. Peletier and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal., 95 (1986), 185-206.

[3]

A. Gmira and L. Véron, Asymptotic behaviour of the solution of a semilinear parabolic equation, Monat. für Math., 94 (1982), 299-311.

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., (1964), xiv+347 pp.

[5]

O. Ladyshenskaya, V. A. Solonnikov and N. Ural'Ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Math. Monographs, Amer. Math. Soc., Providence, R.I., xi+648 pp., Vol. 23, 1967.

[6]

M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations, Comm. Part. Diff. Equ., 24 (1999), 1445-1499.

[7]

M. Marcus and L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Studies, 2 (2002), 395-436.

[8]

A. Shishkov and L. Véron, The balance between diffusion and absorption in semilinear parabolic equations, Rend. Lincei, Mat. Appl., 18 (2007), 59-96.

[9]

A. Shishkov and L. Véron, Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption, Calc. Var. & Part. Diff. Equ., 33 (2008), 343-375.

[10]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations," Pitman Research Notes in Math, Addison Wesley Longman, 353, 1996.

show all references

References:
[1]

H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial coinditions, J. Math. Pures Appl., 62 (1983), 73-97.

[2]

H. Brezis, L. A. Peletier and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Anal., 95 (1986), 185-206.

[3]

A. Gmira and L. Véron, Asymptotic behaviour of the solution of a semilinear parabolic equation, Monat. für Math., 94 (1982), 299-311.

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., (1964), xiv+347 pp.

[5]

O. Ladyshenskaya, V. A. Solonnikov and N. Ural'Ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Math. Monographs, Amer. Math. Soc., Providence, R.I., xi+648 pp., Vol. 23, 1967.

[6]

M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations, Comm. Part. Diff. Equ., 24 (1999), 1445-1499.

[7]

M. Marcus and L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Studies, 2 (2002), 395-436.

[8]

A. Shishkov and L. Véron, The balance between diffusion and absorption in semilinear parabolic equations, Rend. Lincei, Mat. Appl., 18 (2007), 59-96.

[9]

A. Shishkov and L. Véron, Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption, Calc. Var. & Part. Diff. Equ., 33 (2008), 343-375.

[10]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations," Pitman Research Notes in Math, Addison Wesley Longman, 353, 1996.

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