Mario Michele Coclite

On a Model for the Evolution of Morphogens in a Growing Tissue

We analyze a recently proposed model [I. Averbukh et al., Development, 141 (2014), pp. 2150--2156] for the regulation of growth and patterning in developing tissues by diffusing morphogens. We show that solutions of the underlying coupled systems of nonlinear PDEs exist, are unique, and are stable in a suitable sense. The key tool in the analysis is the transformation of the underlying system to a porous medium equation. Numerical experiments illustrating the model are also presented.

An existence result for variational problems and applications to some nonlinear hyperbolic and elliptic equations involving critical Sobolev exponents

SummaryWe prove the existence of nontrivial solutions for nonlinear equations of the type Lu=g(x, u) + ¦u¦¯p−2u, ¯ p > 2, where L is a continuous self- adjoint linear operator in a Hilbert space H and u↦g(x, u) is a lower order perturbation of ¦u¦¯p−1. We assume that ¯p is the critical exponent in the sense that the embedding H ↪Lp, If is compact for 1⩽p<¯p and is continuous (not necessarily compact) for p=¯p. From this result we deduce, for example, that utt -Δu- λu=¦u¦2/Nu, uε L2(SN×S1) has at least one pair (−u, u) of solutions nonconstant with respect to t, provided that λ is sufficiently close to some eigenvalue of ∂tt−Δ.

Perturbazioni singolari per un problema quasi-ellittico a coefficienti costanti nel semispazio

SummaryA singular perturbation analysis is performed for a case in which both the reduced and the perturbed problems are semi-elliptic boundary problems with constant coefficients in a half space. Our treatment is based on the results presented in the first part of this paper, in which a theorem by Ostrowski is employed to analyse the asymptotic behavior of the roots of the symbol of P(D)+ɛQ(D) for ɛ→0+.