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Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs

  • * Corresponding author

    * Corresponding author 

In memoriam Karl-Peter Hadeler 1936-2017

The second author is supported by MINECO grant MTM2014-52402-C3-1-P.
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  • We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction-diffusion-equations on the edges of a finite metric graph $ G$ under continuity and Kirchhoff flow transition conditions at the vertices.

    $(1) \ \ \ \ \ \ \ \ \ \ \begin{cases} u∈ \mathcal{C}(G×[0,∞))\cap \mathcal{C}^{2,1}_{K}(G×(0,∞)),\\\partial_t u_j=\partial_j^2u_{j}+f(u_j) & \text{on the edges }k_j,\\ \displaystyle(K)\ \ \ \ \sum\limits_{j=1}^N d_{ij} c_{ij}\partial_ju_{j}(v_i,t)=0 &\text{at the vertices } v_i.\end{cases} $

    Mathematics Subject Classification: Primary: 35K57, 35B35, 35B41, 35R02, 35J25.

    Citation:

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  • Figure 1.  Cutting at $p$ with $\partial u(p) = 0 $. The original graph $\Gamma$ is drawn on the left, while the resulting graph $\tilde{\Gamma}$ is drawn on the right

    Figure 2.  Yanagida's exceptional graphs

    Figure 3.  More "exceptional" graphs by Theorem 5.2

    Figure 4.  Proof of Lemma 6.2. The indicated signs are those of the $\Delta_{ij}$. The two thin arrows indicate the nodes $v_m$ and $v_{1}$ fulfilling the assertion

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