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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
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