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Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations

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  • The aim of this short note is to study self-similiar radially symmetric solutions of the scalar doubly nonlinear reaction-diffusion equation \begin{equation*} \frac{\partial u^{m-1}}{\partial t} - \Delta_p u = \lambda u^{q-1} \end{equation*} on $\mathbb{R}^n$, where the parameters $1 < m, p,q < \infty$ and $0 < \lambda < \infty$ are fixed. Particularly, for $m < p < q < q_c := p(1+\frac{m-1}{n})$ (where $q_c$ is Fujita's critical exponent of blow-up) we show that there exist self-similar and radially symmetric solutions $u$, which do not blow up in finite time, but instantly become sign-changing for $t>0$ inside some subdomain.
    Mathematics Subject Classification: Primary: 35B05, 35K92; Secondary: 35K65.

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