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Liouville type theorems and complete blow-up for indefinite superlinear parabolic equations. (English) Zbl 1093.35037
Chipot, Michel (ed.) et al., Nonlinear elliptic and parabolic problems. A special tribute to the work of Herbert Amann, Zürich, Switzerland, June 28–30, 2004. Basel: Birkhäuser (ISBN 3-7643-7266-4/hbk). Progress in Nonlinear Differential Equations and their Applications 64, 391-402 (2005).
Let \(h\in C(\mathbb{R})\) and \(f\in C^1(\mathbb{R}^+,\mathbb{R}^+)\) be nondecreasing functions such that \(h(0)=0\), \(h\) is strictly increasing for \(x_1>0\), \(\lim_{x_1\to\infty} h(x_1)=+\infty\), \(f(0)=f'(0)=0\), and \(f(u)>0\) for \(u>0\). The paper shows that the only nonnegative bounded classical solution of the parabolic equation, \[ u_t-\Delta u=h(x_1)f(u),\;x=(x_1,x_2,x_2,\dots,x_N)\in\mathbb{R}^N,\quad t\in\mathbb{R}, \] is the trivial solution. The complete blow-up for a related initial-boundary value problem in a bounded domain is given.
For the entire collection see [Zbl 1077.00009].

35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs