American Institute of Mathematical Sciences

Advanced
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

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 & 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)  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), 140-205. 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), 5653-5668. 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), 265-298.  Google Scholar

[5]

E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, Series Universitext, New York, 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), 225-290. 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), 359-387. 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), 159-189. 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), 113-127.  Google Scholar

[10]

S. Kamin and J. L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-352. 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), 51-74. doi: 10.1007/BF02790217.  Google Scholar

[12]

J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  Google Scholar

[13]

F. Li, Regularity for entropy solutions of a class of parabolic equations with irregular data, Comment. Math. Univ. Carolin., 48 (2007), 69-82.  Google Scholar

[14]

M. Marcus and L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Studies, 2 (2002), 395-436.  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), 487-522.  Google Scholar

[16]

S. Segura de Leon and J. Toledo, Regularity for entropy solutions of parabolic p-Laplacian type equations, Publicacions Matemàtiques, 43 (1999), 665-683. 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), 95-103. 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), 301-321. 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, 240-249, SIAM, Philadelphia, PA, (1996).  Google Scholar

[20]

L. Véron, Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces, Math. Comp., 39 (1982), 325-337. doi: 10.2307/2007318.  Google Scholar

[21]

L. Véron, "Singularities of Solutions of Second Other Quasilinear Equations," Pitman Research Notes in Math. Series 353, Adison Wesley, Longman, 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), 679-698 (Russian)  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), 140-205. 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), 5653-5668. 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), 265-298.  Google Scholar

[5]

E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, Series Universitext, New York, 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), 225-290. 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), 359-387. 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), 159-189. 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), 113-127.  Google Scholar

[10]

S. Kamin and J. L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-352. 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), 51-74. doi: 10.1007/BF02790217.  Google Scholar

[12]

J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  Google Scholar

[13]

F. Li, Regularity for entropy solutions of a class of parabolic equations with irregular data, Comment. Math. Univ. Carolin., 48 (2007), 69-82.  Google Scholar

[14]

M. Marcus and L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Studies, 2 (2002), 395-436.  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), 487-522.  Google Scholar

[16]

S. Segura de Leon and J. Toledo, Regularity for entropy solutions of parabolic p-Laplacian type equations, Publicacions Matemàtiques, 43 (1999), 665-683. 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), 95-103. 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), 301-321. 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, 240-249, SIAM, Philadelphia, PA, (1996).  Google Scholar

[20]

L. Véron, Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces, Math. Comp., 39 (1982), 325-337. doi: 10.2307/2007318.  Google Scholar

[21]

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

[1]

Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301
[2]

Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024
[3]

Manas Kar, Jenn-Nan Wang. Size estimates for the weighted p-Laplace equation with one measurement. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2011-2024. doi: 10.3934/dcdsb.2020188
[4]

Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177
[5]

Yangrong Li, Jinyan Yin. Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1939-1957. doi: 10.3934/dcdss.2016079
[6]

Joachim Naumann. On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 837-852. doi: 10.3934/dcdss.2017042
[7]

Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783
[8]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715
[9]

Shingo Takeuchi. Partial flat core properties associated to the $p$-laplace operator. Conference Publications, 2007, 2007 (Special) : 965-973. doi: 10.3934/proc.2007.2007.965
[10]

Runlin Zhang. Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021183
[11]

George Baravdish, Yuanji Cheng, Olof Svensson, Freddie Åström. Generalizations of $ p $-Laplace operator for image enhancement: Part 2. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3477-3500. doi: 10.3934/cpaa.2020152
[12]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
[13]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016
[14]

Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. doi: 10.3934/dcds.2020367
[15]

Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic & Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036
[16]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[17]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[18]

Patrizia Pucci, Raffaella Servadei. Nonexistence for $p$--Laplace equations with singular weights. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1421-1438. doi: 10.3934/cpaa.2010.9.1421
[19]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43
[20]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy