Article Contents
Article Contents

# Singular parabolic problems with possibly changing sign data

• 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.
Mathematics Subject Classification: 35K55, 35K67.

 Citation:

•  [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.