September  2014, 19(7): 2047-2064. doi: 10.3934/dcdsb.2014.19.2047

Singular parabolic problems with possibly changing sign data

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via Scarpa 16, 00161 Roma, Italy

2. 

Dip. Metodi e Modelli Matematici per le Scienze Applicate, Univ. Roma 1, Via Antonio Scarpa 16, 00161 Roma

Received  April 2013 Revised  September 2013 Published  August 2014

We show the existence of bounded solutions $u\in L^2(0,T;H^1_0(\Omega))$ for a class of parabolic equations having a lower order term $b(x,t,u,\nabla u)$ growing quadratically in the $\nabla u$-variable and singular in the $u$-variable on the set $\{u=0\}$.
    We refer to the model problem $$\left\{ \begin{array}{ll} u_t - \Delta u = b(x,t) \frac{|\nabla u|^2}{|u|^k} + f(x,t) &     in \Omega \times (0,T)\\ u(x,t) = 0 &     on \partial\Omega\times(0,T)\\ u(x,0) = u_0 (x)   &
    in \Omega \end{array}\right. $$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N, N \geq 2, 0 < T < + \infty$ and $0 < k < 1$. The data $f(x,t), u_0(x)$ can change their sign, so that the possible solution $u$ can vanish inside $Q_T=\Omega\times(0,T)$ even in a set of positive measure. Therefore, we have to carefully define the meaning of solution. Also $b(x,t)$ can have a quite general sign.
Citation: Ida De Bonis, Daniela Giachetti. Singular parabolic problems with possibly changing sign data. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2047-2064. doi: 10.3934/dcdsb.2014.19.2047
References:
[1]

B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary,, Nonlinear Analysis, 74 (2011), 1355.  doi: 10.1016/j.na.2010.10.008.  Google Scholar

[2]

D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms,, J. Differential Equations, 249 (2010), 2771.  doi: 10.1016/j.jde.2010.05.009.  Google Scholar

[3]

D. Arcoya, J. Carmona, T. Leonori, P. J. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations,, J. Differential Equations, 246 (2009), 4006.  doi: 10.1016/j.jde.2009.01.016.  Google Scholar

[4]

D. Arcoya and S. Segura de Léon, Uniqueness of solutions for some elliptic equations with a quadratic gradient term,, ESAIM: Control, 16 (2010), 327.  doi: 10.1051/cocv:2008072.  Google Scholar

[5]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM Control Optim. Calc. Var., 14 (2008), 411.  doi: 10.1051/cocv:2008031.  Google Scholar

[6]

A. Dall'Aglio, D. Giachetti, C. Leone and S. Segura de León, Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term,, Ann. I. H. Poincaré, 23 (2006), 97.  doi: 10.1016/j.anihpc.2005.02.006.  Google Scholar

[7]

D. Giachetti and G. Maroscia, Existence results for a class of porous medium type equations with quadratic gradient term,, Journal of Evolution Equations, 8 (2008), 155.  doi: 10.1007/s00028-007-0362-3.  Google Scholar

[8]

D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior,, Boll. Unione Mat. Ital., 9 (2009), 349.   Google Scholar

[9]

D. Giachetti, F. Petitta and S. Segura De Léon, Elliptic equations having a singular quadratic gradient term and a changing sign datum,, Communications on Pure and Applied Analysis, 11 (2012), 1875.  doi: 10.3934/cpaa.2012.11.1875.  Google Scholar

[10]

D. Giachetti, S. Segura De Léon and F. Petitta, A priori estimates for elliptic problems with a strongly singular gradient term and a general datum,, Differential Integral Equations, 26 (2013), 913.   Google Scholar

[11]

O. A. Ladyzenskaja, V. A. Solonnikov and N .N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translations of Math. Monographs, (1968).   Google Scholar

[12]

R. Landes and V. Mustonen, On Parabolic initial-boundary value problems with critical growth for the gradient,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 135.   Google Scholar

[13]

P. J. Martínez-Aparicio and F. Petitta, Parabolic equations with nonlinear singularities,, Nonlinear Analysis, 74 (2011), 114.  doi: 10.1016/j.na.2010.08.023.  Google Scholar

[14]

J. Simon, Compact sets in the space $L^p(0, T, B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary,, Nonlinear Analysis, 74 (2011), 1355.  doi: 10.1016/j.na.2010.10.008.  Google Scholar

[2]

D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms,, J. Differential Equations, 249 (2010), 2771.  doi: 10.1016/j.jde.2010.05.009.  Google Scholar

[3]

D. Arcoya, J. Carmona, T. Leonori, P. J. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations,, J. Differential Equations, 246 (2009), 4006.  doi: 10.1016/j.jde.2009.01.016.  Google Scholar

[4]

D. Arcoya and S. Segura de Léon, Uniqueness of solutions for some elliptic equations with a quadratic gradient term,, ESAIM: Control, 16 (2010), 327.  doi: 10.1051/cocv:2008072.  Google Scholar

[5]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM Control Optim. Calc. Var., 14 (2008), 411.  doi: 10.1051/cocv:2008031.  Google Scholar

[6]

A. Dall'Aglio, D. Giachetti, C. Leone and S. Segura de León, Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term,, Ann. I. H. Poincaré, 23 (2006), 97.  doi: 10.1016/j.anihpc.2005.02.006.  Google Scholar

[7]

D. Giachetti and G. Maroscia, Existence results for a class of porous medium type equations with quadratic gradient term,, Journal of Evolution Equations, 8 (2008), 155.  doi: 10.1007/s00028-007-0362-3.  Google Scholar

[8]

D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior,, Boll. Unione Mat. Ital., 9 (2009), 349.   Google Scholar

[9]

D. Giachetti, F. Petitta and S. Segura De Léon, Elliptic equations having a singular quadratic gradient term and a changing sign datum,, Communications on Pure and Applied Analysis, 11 (2012), 1875.  doi: 10.3934/cpaa.2012.11.1875.  Google Scholar

[10]

D. Giachetti, S. Segura De Léon and F. Petitta, A priori estimates for elliptic problems with a strongly singular gradient term and a general datum,, Differential Integral Equations, 26 (2013), 913.   Google Scholar

[11]

O. A. Ladyzenskaja, V. A. Solonnikov and N .N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translations of Math. Monographs, (1968).   Google Scholar

[12]

R. Landes and V. Mustonen, On Parabolic initial-boundary value problems with critical growth for the gradient,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 135.   Google Scholar

[13]

P. J. Martínez-Aparicio and F. Petitta, Parabolic equations with nonlinear singularities,, Nonlinear Analysis, 74 (2011), 114.  doi: 10.1016/j.na.2010.08.023.  Google Scholar

[14]

J. Simon, Compact sets in the space $L^p(0, T, B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

[1]

Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe, Atsushi Yagi. An $L^p$-approach to singular linear parabolic equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 989-1008. doi: 10.3934/dcds.2008.22.989

[2]

Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure & Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163

[3]

Rosaria Di Nardo. Nonlinear parabolic equations with a lower order term and $L^1$ data. Communications on Pure & Applied Analysis, 2010, 9 (4) : 929-942. doi: 10.3934/cpaa.2010.9.929

[4]

N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119

[5]

Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure & Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587

[6]

Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012

[7]

Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975

[8]

Byungsoo Kang, Hyunseok Kim. W1, p-estimates for elliptic equations with lower order terms. Communications on Pure & Applied Analysis, 2017, 16 (3) : 799-822. doi: 10.3934/cpaa.2017038

[9]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89

[10]

S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305

[11]

Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure & Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923

[12]

Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541

[13]

C. García Vázquez, Francisco Ortegón Gallego. On certain nonlinear parabolic equations with singular diffusion and data in $L^1$. Communications on Pure & Applied Analysis, 2005, 4 (3) : 589-612. doi: 10.3934/cpaa.2005.4.589

[14]

Simona Fornaro, Maria Sosio, Vincenzo Vespri. Harnack type inequalities for some doubly nonlinear singular parabolic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5909-5926. doi: 10.3934/dcds.2015.35.5909

[15]

Rui Huang, Yifu Wang, Yuanyuan Ke. Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1005-1014. doi: 10.3934/dcdsb.2005.5.1005

[16]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621

[17]

Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507

[18]

Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583

[19]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005

[20]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]