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| Doubly nonlinear reaction-diffusion equations allow to model reactions of non-Newtonian fluids inside a porous medium like catalytic pellets. In the talk, the prototypical doubly nonlinear reaction-diffusion equation
\begin{equation*}
\frac{\partial u^{m-1}}{\partial t} - \Delta_p u = u^{q-1}
\end{equation*}
is considered. After a survey of extinction and blow up, positivity of solutions is discussed for $L^1$-data resp. measures as initial values. While the weak comparison principle guarantees non-negative solutions to $L^1$-data $u_0^{m-1} \ge 0$ and blow up occurs for $1 < m < p < q < q_c := p(1+\frac{m-1}{n})$ (where $q_c$ is Fujita's critical exponent of blow-up), we were able to show that in the same situation there exist self-similar solutions $u$ which have a positive multiple of the Dirac measure at the origin as renormalized initial value and do not blow up in finite time, but instantly become sign-changing for $t>0$. |
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