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Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation. (English) Zbl 0614.35043
Die Untersuchung der semilinearen Wärmeleitungsgleichung \[ \partial_ t\psi (t,x)=\Delta \psi (t,x)+| \psi (t,x)|^{\gamma -1} \psi (t,x) \] für \((t,x)\in (0,\infty)\times\mathbb R^ n\), \(\gamma >1\), auf Lösungen \(\psi (t,x)=t^{-1/(\gamma -1)} u(| x| / \sqrt{t})\), \(u: [0,\infty)\to \mathbb R\), \(u\in C^ 2\), führt auf die gewöhnliche Differentialgleichung \[ u''(x)+((n- 1)/x+x/2)u'(x)+(k/2)u(x)+| u(x)|^{\gamma -1} u(x)=0 \tag{*} \] mit \(u'(0)=0\), \(k=2/(\gamma -1)\) und \(x>0\).
Verf. setzt in der vorliegenden Arbeit die Untersuchungen der Gleichung (*) aus A. Haraux und Verf. [Indiana Univ. Math. J. 31, 167–189 (1982; Zbl 0465.35049)] auf Lösungsverhalten für \(x\to \infty\), Nullstellen der Lösungen und verschiedene Anfangsbedingungen bei \(x=0\) fort.
Reviewer: L. Jantscher

MSC:
35K55 Nonlinear parabolic equations
35K58 Semilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35C06 Self-similar solutions to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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