Harmonic solutions to a class of differential-algebraic equations with separated variables

We study the set of T-periodic solutions of a class of T-periodically perturbed Differential-Algebraic Equations with separated variables. Under suitable hypotheses, these equations are equivalent to separated variables ODEs on a manifold. By combining known results on Differential-Algebraic Equations, with an argument about ODEs on manifolds, we obtain a global continuation result for the T-periodic solutions to the considered equations. As an application of our method, a multiplicity result is provided

On a class of differential-algebraic equations with infinite delay

We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed Differential-Algebraic Equations, allowing the perturbation to contain a distributed and possibly infinite delay. Under suitable assumptions, the perturbed equations are equivalent to Retarded Functional (Ordinary) Differential Equations on a manifold. Our study is based on known results about the latter class of equations.Comment: 13 pages. Revision: Incorporate changes suggested by readers. Corrected a few typos across the paper, definition of BU added, revised the (previously incorrect) definition of solution of RFDAE, made slight changes in the Introduction. Replacement of Dec. 6, 2012: introduced further changes suggested by referee, bundled addendum/erratum containing a corrected version of Lemma 5.5 and Corollary 5.

On a non-homogeneous and non-linear heat equation

We consider the Cauchy-problem for a parabolic equation of the following type: \begin{equation*} \frac{\partial u}{\partial t}= \Delta u+ f(u,|x|), \end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial condition $u(x,0)=\phi$, and we allow $\phi$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;\phi)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, Ground States with slow decay lie on the threshold between blowing up initial data and the basin of attraction of the null solution. Our results extend previous ones allowing Matukuma-type potential and more generic dependence on $u$. Then, we further explore such a threshold in the subcritical case too. We find two families of initial data $\zeta(x)$ and $\psi(x)$ which are respectively above and below the threshold, and have arbitrarily small distance in $L^{\infty}$ norm, whose existence is new even for $f(u,r)=u^{q-1}$. Quite surprisingly both $\zeta(x)$ and $\psi(x)$ have fast decay (i.e. $\sim |x|^{2-n}$), while the expected critical asymptotic behavior is slow decay (i.e. $\sim |x|^{2/q-2}$).Comment: 2 figure

Periodic solutions of semi-explicit differential-algebraic equations with time-dependent constraints

In this paper we investigate the properties of the set of T-periodic solutions of semi-explicit parametrized Differential-Algebraic Equations with non-autonomous constraints of a particular type. We provide simple, degree theoretic conditions for the existence of branches of T-periodic solutions of the considered equations. Our approach is based on topological arguments about differential equations on implicitly defined manifolds, combined with elementary facts of matrix analysis