Initial trace of positive solutions of a class of degenerate heat equation with absorption

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May 2013, 33(5): 2033-2063. doi: 10.3934/dcds.2013.33.2033

Initial trace of positive solutions of a class of degenerate heat equation with absorption

Tai Nguyen Phuoc ^1, and Laurent Véron ^2,

1.	Department of Mathematics, Technion, 32000 Haifa, Israel
2.	Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Université François Rabelais, Tours, France

Received April 2012 Revised September 2012 Published December 2012

Abstract
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We study the initial value problem with unbounded nonnegative functions or measures for the equation $ ∂_t u-Δ_p u+f(u)=0$ in $\mathbb{R}^ × (0,\infty)$ where $p>1$, $Δ_p u = div(|∇ u|^{p-2} ∇ u )$ and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data $kΔ_0$ and we study their limit when $k → ∞$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)= ∫_0^s f(σ)dσ$. We also give new results dealing with uniqueness and non uniqueness for the initial value problem with unbounded initial data. If $p>2$, we prove that, for a large class of nonlinearities $f$, any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case $f(u)=u^α ln^β (u+1)$, where $α>0$ and $β ≥ 0$.

Keywords: Borel measure., semigroup, P-Laplace operator, initial trace, singularities.

Mathematics Subject Classification: Primary: 35K92, 35K61; Secondary: 35A0.

Citation: Tai Nguyen Phuoc, Laurent Véron. Initial trace of positive solutions of a class of degenerate heat equation with absorption. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2033-2063. doi: 10.3934/dcds.2013.33.2033

References:

[1]	G. I. Barenblatt, On self-similar motions of compressible fluids in porous media, Prikl. Mat. Mech., 16 (1952), 679-698 (Russian)
[2]	M. F. Bidaut-Véron, E. Chasseigne and L. Véron, Initial trace of solution of some quasilinear parabolic equations with absorption, J. Funct. Anal., 193 (2002), 140-205. doi: 10.1006/jfan.2002.3912.
[3]	X. Chen, Y. Qi and M. Wang, Singular solution of the parabolic p-Laplacian with absorption, Trans. Amer. Math. Soc., 359 (2007), 5653-5668. doi: 10.1090/S0002-9947-07-04336-X.
[4]	M. G. Crandall and T. A. Liggett, Generation of seigroups of nonlinear transformations in general Banach spaces, Amer. J. Math., 93 (1971), 265-298.
[5]	E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, Series Universitext, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[6]	E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution p-Laplacian equation. Initial traces and cauchy problem when $1 < p < 2$, Arch. Rat. Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111.
[7]	A. Friedman and L. Véron, Singular solutions of some quasilinear elliptic equations, Arch. Rat. Mech. Anal., 96 (1986), 359-387. doi: 10.1007/BF00251804.
[8]	M. Guedda and L. Véron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.
[9]	M. Herrero and J. L. Vazquez, Asymptotic behaviour of the solution of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse (5)ème serie, 3 (1981), 113-127.
[10]	S. Kamin and J. L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-352. doi: 10.4171/RMI/77.
[11]	S. Kamin and J. L. Vazquez, Singular solutions of of some nonlinear parabolic equations, J. Analyse Math., 59 (1992), 51-74. doi: 10.1007/BF02790217.
[12]	J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
[13]	F. Li, Regularity for entropy solutions of a class of parabolic equations with irregular data, Comment. Math. Univ. Carolin., 48 (2007), 69-82.
[14]	M. Marcus and L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Studies, 2 (2002), 395-436.
[15]	T. Nguyen Phuoc and L. Véron, Local and global properties of solutions of heat equation with superlinear absorption, Adv. Differential Equations, 16 (2011), 487-522.
[16]	S. Segura de Leon and J. Toledo, Regularity for entropy solutions of parabolic p-Laplacian type equations, Publicacions Matemàtiques, 43 (1999), 665-683. doi: 10.5565/PUBLMAT_43299_08.
[17]	J. L. Vazquez, An a priori* interior estimate for the solutions of a nonlinear problem representing weak diffusion*, Nonlinear Anal., 5 (1981), 95-103. doi: 10.1016/0362-546X(81)90074-2.
[18]	J. L. Vazquez and L. Véron, Isolated singularities of some semilinear elliptic equations, J. Differential Equations, 60 (1985), 301-321. doi: 10.1016/0022-0396(85)90127-5.
[19]	J. L. Vazquez and L. Véron, Different kinds of singular solutions of nonlinear parabolic equations, Nonlinear Problems in Applied Mathematics, 240-249, SIAM, Philadelphia, PA, (1996).
[20]	L. Véron, Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces, Math. Comp., 39 (1982), 325-337. doi: 10.2307/2007318.
[21]	L. Véron, "Singularities of Solutions of Second Other Quasilinear Equations," Pitman Research Notes in Math. Series 353, Adison Wesley, Longman, 1996.

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References:

[1]	G. I. Barenblatt, On self-similar motions of compressible fluids in porous media, Prikl. Mat. Mech., 16 (1952), 679-698 (Russian)
[2]	M. F. Bidaut-Véron, E. Chasseigne and L. Véron, Initial trace of solution of some quasilinear parabolic equations with absorption, J. Funct. Anal., 193 (2002), 140-205. doi: 10.1006/jfan.2002.3912.
[3]	X. Chen, Y. Qi and M. Wang, Singular solution of the parabolic p-Laplacian with absorption, Trans. Amer. Math. Soc., 359 (2007), 5653-5668. doi: 10.1090/S0002-9947-07-04336-X.
[4]	M. G. Crandall and T. A. Liggett, Generation of seigroups of nonlinear transformations in general Banach spaces, Amer. J. Math., 93 (1971), 265-298.
[5]	E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, Series Universitext, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[6]	E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution p-Laplacian equation. Initial traces and cauchy problem when $1 < p < 2$, Arch. Rat. Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111.
[7]	A. Friedman and L. Véron, Singular solutions of some quasilinear elliptic equations, Arch. Rat. Mech. Anal., 96 (1986), 359-387. doi: 10.1007/BF00251804.
[8]	M. Guedda and L. Véron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.
[9]	M. Herrero and J. L. Vazquez, Asymptotic behaviour of the solution of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse (5)ème serie, 3 (1981), 113-127.
[10]	S. Kamin and J. L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-352. doi: 10.4171/RMI/77.
[11]	S. Kamin and J. L. Vazquez, Singular solutions of of some nonlinear parabolic equations, J. Analyse Math., 59 (1992), 51-74. doi: 10.1007/BF02790217.
[12]	J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
[13]	F. Li, Regularity for entropy solutions of a class of parabolic equations with irregular data, Comment. Math. Univ. Carolin., 48 (2007), 69-82.
[14]	M. Marcus and L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Studies, 2 (2002), 395-436.
[15]	T. Nguyen Phuoc and L. Véron, Local and global properties of solutions of heat equation with superlinear absorption, Adv. Differential Equations, 16 (2011), 487-522.
[16]	S. Segura de Leon and J. Toledo, Regularity for entropy solutions of parabolic p-Laplacian type equations, Publicacions Matemàtiques, 43 (1999), 665-683. doi: 10.5565/PUBLMAT_43299_08.
[17]	J. L. Vazquez, An a priori* interior estimate for the solutions of a nonlinear problem representing weak diffusion*, Nonlinear Anal., 5 (1981), 95-103. doi: 10.1016/0362-546X(81)90074-2.
[18]	J. L. Vazquez and L. Véron, Isolated singularities of some semilinear elliptic equations, J. Differential Equations, 60 (1985), 301-321. doi: 10.1016/0022-0396(85)90127-5.
[19]	J. L. Vazquez and L. Véron, Different kinds of singular solutions of nonlinear parabolic equations, Nonlinear Problems in Applied Mathematics, 240-249, SIAM, Philadelphia, PA, (1996).
[20]	L. Véron, Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces, Math. Comp., 39 (1982), 325-337. doi: 10.2307/2007318.
[21]	L. Véron, "Singularities of Solutions of Second Other Quasilinear Equations," Pitman Research Notes in Math. Series 353, Adison Wesley, Longman, 1996.

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