# American Institute of Mathematical Sciences

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 and 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), 1355-1371. doi: 10.1016/j.na.2010.10.008. [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), 2771-2795. doi: 10.1016/j.jde.2010.05.009. [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), 4006-4042. doi: 10.1016/j.jde.2009.01.016. [4] D. Arcoya and S. Segura de Léon, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM: Control, Optimization and the Calculus of Variations, 16 (2010), 327-336. doi: 10.1051/cocv:2008072. [5] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426. doi: 10.1051/cocv:2008031. [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), 97-126. doi: 10.1016/j.anihpc.2005.02.006. [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), 155-188. doi: 10.1007/s00028-007-0362-3. [8] D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior, Boll. Unione Mat. Ital., 9 (2009), 349-370. [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), 1875-1895. doi: 10.3934/cpaa.2012.11.1875. [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), 913-948. [11] O. A. Ladyzenskaja, V. A. Solonnikov and N .N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Math. Monographs, Providence, 1968. [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), 135-158. [13] P. J. Martínez-Aparicio and F. Petitta, Parabolic equations with nonlinear singularities, Nonlinear Analysis, 74 (2011), 114-131. doi: 10.1016/j.na.2010.08.023. [14] J. Simon, Compact sets in the space $L^p(0, T, B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

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##### 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), 1355-1371. doi: 10.1016/j.na.2010.10.008. [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), 2771-2795. doi: 10.1016/j.jde.2010.05.009. [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), 4006-4042. doi: 10.1016/j.jde.2009.01.016. [4] D. Arcoya and S. Segura de Léon, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM: Control, Optimization and the Calculus of Variations, 16 (2010), 327-336. doi: 10.1051/cocv:2008072. [5] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426. doi: 10.1051/cocv:2008031. [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), 97-126. doi: 10.1016/j.anihpc.2005.02.006. [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), 155-188. doi: 10.1007/s00028-007-0362-3. [8] D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior, Boll. Unione Mat. Ital., 9 (2009), 349-370. [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), 1875-1895. doi: 10.3934/cpaa.2012.11.1875. [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), 913-948. [11] O. A. Ladyzenskaja, V. A. Solonnikov and N .N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Math. Monographs, Providence, 1968. [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), 135-158. [13] P. J. Martínez-Aparicio and F. Petitta, Parabolic equations with nonlinear singularities, Nonlinear Analysis, 74 (2011), 114-131. doi: 10.1016/j.na.2010.08.023. [14] J. Simon, Compact sets in the space $L^p(0, T, B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
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