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Parabolic equations with singular and supercritical reaction terms. (English) Zbl 1374.35193
The authors study existence of positive solutions to the problem $u_t-\operatorname{div}(M(x)\nabla u)={\lambda\over u^\gamma}+\mu u^p$ in a bounded set $$\Omega\subset R^n,\;n\geq 3$$, with bounded nonnegative initial value and homogeneous Dirichlet boundary conditions. $$M$$ is supposed to be bounded and strongly elliptic matrix, the coefficients $$\lambda,\gamma$$ and $$p$$ are positive, $$\mu\geq 0$$. Influence of the singular term $${\lambda\over u^p}$$ on the local and global existence of solutions is investigated. It is shown that for $$p\in (0,1)$$ no restrictions on the length of the time interval and of the size of the data are needed, and the distributional solutions exist in the class $$u\in L^2(0,T;W_0^{1,2}(\Omega))$$ for $$\gamma\in (0,1]$$, $$u^{\gamma+1\over2}\in L^2(0,T;W_0^{1,2}(\Omega))$$, $$u^{-\gamma} \in L^1(0,T;L^1_{loc}(\Omega))$$ if $$\gamma>1$$. On the other hand, $$p\geq1$$ requires smallness assumptions on the time interval or on the data to get solutions in the same class as above.

##### MSC:
 35K10 Second-order parabolic equations 35K58 Semilinear parabolic equations 35K67 Singular parabolic equations