Elisa Sovrano

This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator \[\label{P}-\mathrm{div}\left(\nabla u/\sqrt{1+|\nabla u|^2}\right)=\lambda a(x)f(u) \quad \text{in }\Omega,\qquad u=0 \quad\text{on } \partial\Omega,\] with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a regular boundary \(\partial \Omega,\) \(\lambda>0\) represents a diffusivity parameter, \(a\) is a continuous weight which may change sign in \(\Omega,\) and \(f\colon[0,L]\to\mathbb{R},\) with \(L>0\) a given constant, is a continuous function satisfying \(f(0)=f(L)=0\) and \(f(s)>0\) for every \(s\in ]0,L[.\) Depending on the behavior of \(f\) at zero, three qualitatively different bifurcation diagrams appear by varying \(\lambda\). Typically, the solutions we find are regular as long as \(\lambda\) is small, while as a consequence of the saturation of the flux they may develop singularities when \(\lambda\) becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, \(f(s) = s(L-s)\) and \(a\equiv 1\), having no similarity with the case of linear diffusion based on the Fick-Fourier's law.

This paper analyzes the superlinear indefinite prescribed mean curvature problem \[-\mathrm{div}\left({\nabla u}/{\sqrt{1+|\nabla u|^2}}\right)=\lambda a(x)h(u) \quad \text{in }\Omega,\qquad u=0 \quad\text{on } \partial\Omega,\] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a regular boundary \(\partial \Omega\), \(h\in C^0(\mathbb{R})\) satisfies \(h(s) \sim s^{p}\), as \(s\to0^+,\) \(p>1\) being an exponent with \(p< \frac{N+2}{N-2}\) if \(N\geq 3\), \(\lambda> 0\) represents a parameter, and \(a\in C^0(\overline \Omega)\) is a sign-changing function. The main result establishes the existence of positive regular solutions when \(\lambda\) is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for \(\lambda\) small is further discussed assuming that \(h\) satisfies \(h(s) \sim s^{q}\) as \(s\to +\infty,\) \(q>0\) being such that \(q< \frac{1}{N-1}\) if \(N\geq 2\); thus, in dimension \(N\ge 2\), the function \(h\) is not superlinear at \(+\infty,\) although its potential \(H(s) = \int_0^sh(t)\, \mathrm{d}t\) is. Imposing such different degrees of homogeneity of \(h\) at \(0\) and at \(+\infty\) is dictated by the specific features of the mean curvature operator.

We discuss the appearance of chaos in time-periodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension~\(1\) reversible vector fields and discuss the ways a time-dependent periodic forcing term of pulse form may be added to them to yield topological chaotic behaviour. Chaos here means that the resulting dynamics is semiconjugate to a shift in a finite alphabet. The results rely on the classification of reversible vector fields and on the theory of topological horseshoes. This work is part of a project of studying periodic forcing of symmetric vector fields.