Souplet, Philippe; Weissler, Fred B. Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state. (English) Zbl 1029.35106 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 2, 213-235 (2003). 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 [V. A. Galaktionov and 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 \(\varepsilon > 0\). Reviewer: H.Marcinkowska (Wrocław) Cited in 23 Documents MSC: 35K15 Initial value problems for second-order parabolic equations 35K55 Nonlinear parabolic equations Keywords:profile equation; existence, uniqueness and asymptotic behavior Citations:Zbl 0874.35057 PDFBibTeX XMLCite \textit{P. Souplet} and \textit{F. B. Weissler}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 2, 213--235 (2003; Zbl 1029.35106) Full Text: DOI Numdam EuDML References: [1] Cazenave, T.; Weissler, F. B., Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z., 228, 83-120 (1998) · Zbl 0916.35109 [2] Dohmen, C.; Hirose, M., Structure of positive radial solutions to the Haraux-Weissler equation, Nonlinear Anal. TMA, 33, 51-69 (1998) · Zbl 0934.34028 [3] Galaktionov, V. A.; Vazquez, J. L., Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math., 50, 1-67 (1997) · Zbl 0874.35057 [4] Haraux, A.; Weissler, F. B., Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31, 167-189 (1982) · Zbl 0465.35049 [5] Joseph, D. D.; Lundgren, T. S., Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49, 241-269 (1973) · Zbl 0266.34021 [6] Peletier, L. A.; Terman, D.; Weissler, F. B., On the equation \(Δu+12x·∇u+f(u)\), Arch. Rat. Mech. Anal., 94, 83-99 (1986) · Zbl 0615.35034 [7] Vazquez, J. L., Domain of existence and blowup for the exponential reaction-diffusion equation, Indiana Univ. Math. J., 48, 677-709 (1999) · Zbl 0928.35080 [8] Weissler, F. B., Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation, Arch. Rat. Mech. Anal., 91, 231-245 (1986) · Zbl 0614.35043 [9] Weissler, F. B., \(L^p\)-energy and blow-up for a semilinear heat equation, Proc. Symp. Pure Math. Part II, 45, 545-551 (1986) [10] Yanagida, E., Uniqueness of rapidly decaying solutions of the Haraux-Weissler equation, J. Differential Equations, 127, 561-570 (1996) · Zbl 0856.34058 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.