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

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

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 $β ≥ 0$.
Citation: Tai Nguyen Phuoc, Laurent Véron. Initial trace of positive solutions of a class of degenerate heat equation with absorption. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  doi: 10.1006/jfan.2002.3912.  Google Scholar

[3]

X. Chen, Y. Qi and M. Wang, Singular solution of the parabolic p-Laplacian with absorption,, Trans. Amer. Math. Soc., 359 (2007), 5653.  doi: 10.1090/S0002-9947-07-04336-X.  Google Scholar

[4]

M. G. Crandall and T. A. Liggett, Generation of seigroups of nonlinear transformations in general Banach spaces,, Amer. J. Math., 93 (1971), 265.   Google Scholar

[5]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[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.  doi: 10.1007/BF00400111.  Google Scholar

[7]

A. Friedman and L. Véron, Singular solutions of some quasilinear elliptic equations,, Arch. Rat. Mech. Anal., 96 (1986), 359.  doi: 10.1007/BF00251804.  Google Scholar

[8]

M. Guedda and L. Véron, Local and global properties of solutions of quasilinear elliptic equations,, J. Differential Equations, 76 (1988), 159.  doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[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.   Google Scholar

[10]

S. Kamin and J. L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation,, Rev. Mat. Iberoamericana, 4 (1988), 339.  doi: 10.4171/RMI/77.  Google Scholar

[11]

S. Kamin and J. L. Vazquez, Singular solutions of of some nonlinear parabolic equations,, J. Analyse Math., 59 (1992), 51.  doi: 10.1007/BF02790217.  Google Scholar

[12]

J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.   Google Scholar

[13]

F. Li, Regularity for entropy solutions of a class of parabolic equations with irregular data,, Comment. Math. Univ. Carolin., 48 (2007), 69.   Google Scholar

[14]

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

[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.   Google Scholar

[16]

S. Segura de Leon and J. Toledo, Regularity for entropy solutions of parabolic p-Laplacian type equations,, Publicacions Matemàtiques, 43 (1999), 665.  doi: 10.5565/PUBLMAT_43299_08.  Google Scholar

[17]

J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion,, Nonlinear Anal., 5 (1981), 95.  doi: 10.1016/0362-546X(81)90074-2.  Google Scholar

[18]

J. L. Vazquez and L. Véron, Isolated singularities of some semilinear elliptic equations,, J. Differential Equations, 60 (1985), 301.  doi: 10.1016/0022-0396(85)90127-5.  Google Scholar

[19]

J. L. Vazquez and L. Véron, Different kinds of singular solutions of nonlinear parabolic equations,, Nonlinear Problems in Applied Mathematics, (1996), 240.   Google Scholar

[20]

L. Véron, Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces,, Math. Comp., 39 (1982), 325.  doi: 10.2307/2007318.  Google Scholar

[21]

L. Véron, "Singularities of Solutions of Second Other Quasilinear Equations,", Pitman Research Notes in Math. Series 353, 353 (1996).   Google Scholar

show all references

References:
[1]

G. I. Barenblatt, On self-similar motions of compressible fluids in porous media,, Prikl. Mat. Mech., 16 (1952), 679.   Google Scholar

[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.  doi: 10.1006/jfan.2002.3912.  Google Scholar

[3]

X. Chen, Y. Qi and M. Wang, Singular solution of the parabolic p-Laplacian with absorption,, Trans. Amer. Math. Soc., 359 (2007), 5653.  doi: 10.1090/S0002-9947-07-04336-X.  Google Scholar

[4]

M. G. Crandall and T. A. Liggett, Generation of seigroups of nonlinear transformations in general Banach spaces,, Amer. J. Math., 93 (1971), 265.   Google Scholar

[5]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[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.  doi: 10.1007/BF00400111.  Google Scholar

[7]

A. Friedman and L. Véron, Singular solutions of some quasilinear elliptic equations,, Arch. Rat. Mech. Anal., 96 (1986), 359.  doi: 10.1007/BF00251804.  Google Scholar

[8]

M. Guedda and L. Véron, Local and global properties of solutions of quasilinear elliptic equations,, J. Differential Equations, 76 (1988), 159.  doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[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.   Google Scholar

[10]

S. Kamin and J. L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation,, Rev. Mat. Iberoamericana, 4 (1988), 339.  doi: 10.4171/RMI/77.  Google Scholar

[11]

S. Kamin and J. L. Vazquez, Singular solutions of of some nonlinear parabolic equations,, J. Analyse Math., 59 (1992), 51.  doi: 10.1007/BF02790217.  Google Scholar

[12]

J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.   Google Scholar

[13]

F. Li, Regularity for entropy solutions of a class of parabolic equations with irregular data,, Comment. Math. Univ. Carolin., 48 (2007), 69.   Google Scholar

[14]

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

[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.   Google Scholar

[16]

S. Segura de Leon and J. Toledo, Regularity for entropy solutions of parabolic p-Laplacian type equations,, Publicacions Matemàtiques, 43 (1999), 665.  doi: 10.5565/PUBLMAT_43299_08.  Google Scholar

[17]

J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion,, Nonlinear Anal., 5 (1981), 95.  doi: 10.1016/0362-546X(81)90074-2.  Google Scholar

[18]

J. L. Vazquez and L. Véron, Isolated singularities of some semilinear elliptic equations,, J. Differential Equations, 60 (1985), 301.  doi: 10.1016/0022-0396(85)90127-5.  Google Scholar

[19]

J. L. Vazquez and L. Véron, Different kinds of singular solutions of nonlinear parabolic equations,, Nonlinear Problems in Applied Mathematics, (1996), 240.   Google Scholar

[20]

L. Véron, Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces,, Math. Comp., 39 (1982), 325.  doi: 10.2307/2007318.  Google Scholar

[21]

L. Véron, "Singularities of Solutions of Second Other Quasilinear Equations,", Pitman Research Notes in Math. Series 353, 353 (1996).   Google Scholar

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