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Singular parabolic problems with possibly changing sign data

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  • 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.


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