In this paper we study existence and regularity results for solution to a nonlinear and singular parabolic problem. The model is ∂u∂t-div((a(x,t)+|u|q)∇u)=fuγinQ,u(x,t)=0onΓ,u(x,0)=u0(x)inΩ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{lll} \dfrac{\partial u}{\partial t}-\text{ div }((a(x,t)+|u|^{q})\nabla u)=\frac{f}{u^{\gamma }} &{} \text{ in } &{} Q,\\ u(x,t)=0 &{} \text{ on } &{} \Gamma ,\\ u(x,0)=u_{0}(x) &{} \text{ in } &{} \varOmega , \end{array} \right. \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega $$\end{document} is a bounded open subset of RN,\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^{N},$$\end{document}N≥2,\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 2,$$\end{document}Q is the cylinder Ω×(0,T),\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega \times (0,T),$$\end{document}T>0,\documentclass[12pt]{minimal}
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\begin{document}$$T>0,$$\end{document}Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} the lateral surface ∂Ω×(0,T),\documentclass[12pt]{minimal}
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\begin{document}$$\partial \varOmega \times (0,T),$$\end{document}q>0,\documentclass[12pt]{minimal}
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\begin{document}$$q>0,$$\end{document}γ>0,\documentclass[12pt]{minimal}
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\begin{document}$$\gamma >0,$$\end{document} and f is non-negative function belonging to some Lebesgue space Lm(Q),\documentclass[12pt]{minimal}
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\begin{document}$$L^{m}(Q),$$\end{document}m≥1\documentclass[12pt]{minimal}
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\begin{document}$$m\ge 1$$\end{document} and u0∈L∞(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$u_{0}\in L^{\infty }(\varOmega )$$\end{document} such that ∀ω⊂⊂Ω,∃Dω>0:u0≥Dωinω.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \forall \; \omega \subset \subset \varOmega ,\; \exists \; D_{\omega }>0: \; u_{0}\ge D_{\omega }\;\; \text{ in }\; \omega . \end{aligned}$$\end{document}
