Let be a bounded subset of R N , a ∈ C 1 ( ) with a > 0 in and A be the operator defined by Au := ∇ · (a∇u) with the generalized Wentzell boundary conditionIf ∂ is in C 2 , β and γ are nonnegative functions in C 1 (∂ ), with β > 0, and := {x ∈ ∂ : a(x) > 0} = ∅, then we prove the existence of a (C o ) contraction semigroup generated by A, the closure of A, on a suitable L p space, 1 ≤ p < ∞ and on C( ). Moreover, this semigroup is analytic if 1 < p < ∞.

We prove a very general form of the Angle Concavity Theorem, which says that if ((T(t)) defines a one parameter semigroup acting over various L^p spaces (over a fixed measure space), which is analytic in a sector of opening angle \theta_p, then the maximal choice for \theta_p is aconcave function of 1-1/p. This and related results are applied to get improved estimates on the optimal L^p angle of ellipticity for a parabolic equation of the form {\partial u}{\partial t}=Au, where A is a uniformly elliptic second order partial differential operator with Wentzell or dynamic boundary condition. Similar results are obtained for the higher order equation {\partial u}{\partial t}=(-1)^{m+1}A^m u\ud for all positive integers m

SUMMARYMotivated by a neurobiological problem, we discuss a class of diffusion problems on a network. The celebrated Rall lumped soma model for the spread of electrical potential in a dendritical tree prescribes that the common cable equation must be coupled with particular dynamic conditions in some nodes (the cell bodies, or somata). We discuss the extension of this model to the case of a whole network of neurons, where the ramification nodes can be either active (with excitatory time-dependent boundary conditions) or passive (where no dynamics take place, i.e. only Kirchhoff laws are imposed). While well-posedness of the system has already been obtained in previous works, using abstract tools based on variational methods and semigroup theory we are able to prove several qualitative properties, including asymptotic behaviour, regularity of solutions, and monotonicity of the semigroups in dependence on the physical coefficients.

We deal with the problem of analyticity for the semigroup generated by the second order differential operator Au ≔ αu″ + βu′ (or by some restrictions of it) in the spaces Lp(0, 1), with or without weight, and in W1,p(0, 1), 1 < p < ∞. Here α and β are assumed real‐valued and continuous in [0, 1], with α(x) > 0 in (0, 1), and the domain of A is determined by the generalized Neumann boundary conditions and by Wentzell boundary conditions.

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