The nonexistence of stable stationary nonconstant solutions of reaction-diffusion-equations partial derivative(t)u(j) = partial derivative(j)(a(j)(x(j))partial derivative(j)u(j)) + f(j)(u(j)) on the edges of a finite (topological) graph is investigated under continuity and consistent Kirchhoff flow conditions at all vertices of the graph. In particular, it is shown that in the balanced autonomous case f(u) = u - u(3), no such stable stationary solution can exist on any finite graph. Finally, the balanced autonomous case is discussed on the two-sided unbounded path with equal edge lengths.
The nonexistence of stable stationary nonconstant solutions of reaction–diffusion–equations on the edges of a finite (topological) graph is investigated under continuity and consistent Kirchhoff flow conditionsat all vertices of the graph. In particular, it is shown that in the balanced autonomous case f(u) = u - u^3, no such stable stationary solution can exist if the graph is analytic. Under more general, eventually inconsistent Kirchhoff conditions, there are no stable stationary nonconstant solutions on finitetrees if the derivatives of the nonlinearities are uniformly bounded from above.
Finally, the above balanced autonomous case is discussed on the two–sided unbounded path with equal edge lengths.
We present some general bounds for the algebraic and geometric multiplicity of eigenvalues of second order elliptic operators on finite networks under continuity and weighted Kirchhoff flow conditions at the vertices. In particular the algebraic multiplicity of an eigenvalue is shown to be strictly bounded from above by the number of vertices if there are no eigenfunctions vanishing in all nodes, and to be bounded from above by the number of edges if there are such eigenfunctions.
We consider the continuous Laplacian on an infinite uniformly locally finite network under natural transition conditions as continuity at the ramification nodes and the classical Kirchhoff flow condition at all vertices in a L8-setting. The characterization of eigenvalues of infinite multiplicity for trees with finitely many boundary vertices (von Below and Lubary, Results Math 47:199–225, 2005, 8.6) is generalized to the case of infinitely many boundary vertices. Moreover, it is shown that on a tree, any eigenspace of infinite dimension contains a subspace isomorphic to l8(N) . As for the zero eigenvalue, it is shown that a locally finite tree either is a Liouville space or has infinitely many linearly independent bounded harmonic functions if the edge lengths do not shrink to zero anywhere. This alternative is shown to be false on graphs containing circuits.
Cabre, X.; Valencia, M.; Consul, M. N.; Lubary, J.; Mande, J.; Gonzalez, M.; Aguareles, M.; Sola-morales, J.; Lucia, M.; Haro, J.; Serra, J.; Ros, X.; Charro, F.; De La Torre, A.; Mas, A. Projecte R+D+I competitiu