Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation
- Universidad Catolica, Santiago (Chile)
- Universidad de Chile, Santiago (Chile)
We will consider in this paper a semilinear elliptic equation {triangle}u + f(u) = 0 in {Omega}, (1.5) where the function f is locally Lipschitz in (0,{infinity}) and continuous in (0,{infinity}). We study symmetry properties of nonnegative solutions of this equation in two different situations: first we assume {Omega} = IR{sup N}, and second we consider {Omega} {ne} IR{sup N} and we provide (1.5) with overdetermined boundary conditions. Next we describe our results in the first case, that is, when {Omega} = IR{sup N}. We will consider the following hypothesis on the nonlinear function f (F) f(0) {le} 0, f continuous in (0,+{infinity}), locally Lipschitz in (0,+{infinity}) and there exists {alpha} > 0 so that f is strictly decreasing in [0,{alpha}]. We note that the support of a solution of (1.5) is not known a priori and so we have in fact a free boundary involved. Our goal is to determine the shape of this support and the symmetry properties of the solution.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 441144
- Journal Information:
- Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 3-4 Vol. 21; ISSN 0360-5302; ISSN CPDIDZ
- Country of Publication:
- United States
- Language:
- English
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