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Global existence and blowup for sign-changing solutions of the nonlinear heat equation. (English) Zbl 1176.35010

Given \(0<\alpha<2/N\), it is proved that a function \(\psi\) exists with the following properties: The solution of the equation \(u_t+\Delta u=|u|^\alpha u\) in \(\mathbb{R}^N\) with the initial condition \(\psi\) is global while the solution with the initial condition \(\lambda\psi\) blows up in finite time if \(\lambda >0\) is either sufficiently small or sufficiently large.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
35K15 Initial value problems for second-order parabolic equations
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