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Local and global solvability of a class of semilinear parabolic equations. (English) Zbl 0622.35033

The authors study the problem \[ u_ t=\Delta u+q(x)u^ n,\quad u(x,0)=u_ 0(x),\quad x\in R^ N. \] Here \(n>1\) and the functions \(u_ 0\) and q are continuous and nonnegative on \(R^ N\). Necessary conditions, expressed in terms of \(u_ 0\) and q, for the existence of a local resp. global solution of the above problem are formulated. Also some rather involved sufficient conditions are stated.
Reviewer: O.Vejvoda

MSC:

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:

[1] P. Baras and M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, in “Analyse non Linéaire, Annales de l”I. H. P.”; P. Baras and M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, in “Analyse non Linéaire, Annales de l”I. H. P.” · Zbl 0599.35073
[2] Kalashnikov, A. S., On a heat conduction equation for a medium with non-uniformly distributed non-linear heat sources or absorbers, Bull. Univ. Moscou, Math., Mech., 3, 20-24 (1983) · Zbl 0523.35060
[3] Levine, H. A.; Sacks, P. E., Some existence and nonexistence theorems for solutions of degenerate parabolic equations, Differential Equations, 52, 135-161 (1984) · Zbl 0487.34003
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