Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state
The paper is devoted to the investigation of the nonlinear heat equation \[ u_t = \Delta u + |u|^{\alpha} u, \qquad t>0, \quad x \in \mathbb{R}^N, \tag{1} \] with the singular initial condition of the form \[ \lim_{t \to 0+} u(t) = \mu V \tag{2} \] (the limit in \(S'\)), where \(V(x) = \beta^{1/\alpha} |x|^{-2/\alpha}\) with \(\beta = (2/\alpha) (N-2 - 2/\alpha)\) is a stationary solution (in the sense of tempered distributions) of (1). The case \(\mu = 1\) was studied in [\textit{V. A. Galaktionov} and \textit{J. L. Vazquez}, Commun. Pure Appl. Math. 50, 1-67 (1997; Zbl 0874.35057)]. It was proved there that for \(N > 2\) and \(\frac{2}{N-2} < \alpha < \alpha^*\) (with suitably defined \(\alpha^* > \frac{4}{N-2}\)) problem (1), (2) has a \(C^{\infty}\) positive solution \(u(t)\) for \(t>0\). The authors extend this result to \(\mu \in (1, 1+ \varepsilon)\) with some \(\varepsilon > 0\). It is proved as well that \[ \lim_{t \to \infty} u(t) = 0 \] in \(L^q(R^N)\) for \(q > \frac{N\alpha}{2}\). Moreover the nonuniqueness of the solution is proved in the following cases: \[ \frac{2}{N-2} < \alpha < \frac{4}{N-2}, \qquad \mu \in (0, 1+\varepsilon), \] \[ \alpha = \frac{4}{N-2}, \qquad \mu \in [1,1+ \varepsilon), \] \[ \frac{4}{N-2} < \alpha < \alpha^*, \qquad \mu \in (1- \varepsilon, 1 + \varepsilon) \] with suitably defined \(\varepsilon > 0\).
existence, uniqueness and asymptotic behavior
scientific article; zbMATH DE number 1912449
scientific article