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Singular behavior in nonlinear parabolic equations. (English) Zbl 0573.35046
At the beginning the authors study the question of singular solutions for elliptic boundary value problems $$\Delta u+f(u)=0$$ on $$\Omega$$, $$u=0$$ on $$\partial \Omega$$, $$u>0$$ on $$\Omega$$. Singular solution is the $$C^ 2(\Omega \setminus \{0\})$$ solution of the problem for which, $$\overline{\lim}_{x\to 0}u(x)=+\infty.$$ In theorem 1 the conditions on f (general) are formulated under which the problem has (i) infinitely many singular solutions, (ii) no singular solution. Theorem 2 contains the result for the special case $$f=\lambda | x|^{\ell} u^ p.$$
Further, the boundary value problems with the conditions $$u(x,t)=0$$, (x,t)$$\in \partial \Omega$$, $$t>0$$ and $$u(x,0)=u^ 0(x)$$, $$x\in \Omega$$ for either the equation (1) $$u_ t=\Delta u+u^ p$$, or the equation (2) $$u_ t=\Delta (u^ m)$$ are considered. In theorem 3, which concerns the equation (1), the conditions on p are given guaranteeing the existence of $$u^ 0\in L_ p(\Omega)$$ such that the problem has at least two solutions from $$C([0,T];L_ p(\Omega))$$. The asymptotic behaviour of the non uniformly bounded solutions of this problems is established in theorem 4.
Finally for the problem of ”fast diffusion” (equation (2)) there are given the conditions under which for each $$t_ 0>0$$ there is a solution with $$u_ 0\in L_ q(\Omega)$$ and $$u(.,t_ 0)\not\in L^{\infty}(\Omega)$$.
Reviewer: O.John

##### MSC:
 35K55 Nonlinear parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J60 Nonlinear elliptic equations 35R25 Ill-posed problems for PDEs
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