Fujita modified exponent for scale invariant damped semilinear wave equations

The aim of this paper is to prove a blow-up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition, we give an upper bound estimate for the lifespan of the solution. It depends not only on the exponent of the nonlinear term and not only on the damping coefficient but also on the size of the decay rate of the initial data.

Polynomial Roth Theorems on Sets of Fractional Dimensions

Let $E\subset{\mathbb{R}}$ be a closed set of Hausdorff dimension $\alpha \in (0, 1)$. Let $P: {\mathbb{R}}\to{\mathbb{R}}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Łaba and Pramanik [ 11] to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot et al. [ 7].

THE CHURCH OF SANT'ANDREA IN BERGAMO: AN INTEGRATED SURVEY FOR KNOWLEDGE AND CONSERVATION

<p><strong>Abstract.</strong> The church dedicated to Sant'Andrea (St. Andrew) in <q>Porta Dipinta</q> street in Bergamo city is a treasure that keep inside it a rich heritage of great historical and cultural value, both from the architectural and from the artistic point of view. Lacking of the façade (left unfinished), it is often neglected, despite being on the main road leading to the old town from <q>Sant'Agostino</q> Gate. The approach to an historical building like this requires a multi-disciplinary integration, in order to join the technical competence of engineering sciences to the sensitivity of human and fine arts sciences. For a better understanding of the structural performances of the building, historical research, measurement survey, material and decay condition study have to complement each other.</p>

Radiation conditions and integral representations for Clifford algebra-valued null solutions of the iterated perturbed Dirac operator

In 1912 Arnold Sommerfeld introduced a special decay condition at infinity to address uniqueness issues for certain boundary value problems involving the Helmholtz operator in an exterior domain. Examples of such boundary value problems arise in optical diffraction theory and radio wave propagation. This decay condition, which has become known as Sommerfeld's radiation condition, has been subsequently adapted to various other operators of interest in mathematics, engineering, and physics. Examples include the Silver-Muller radiation condition for the Maxwell system, and radiation conditions for certain perturbed Dirac operators. In this dissertation, we continue this line of research by considering iterated perturbed Dirac operators. Among other things, suitable radiation conditions are identified which allow us to prove integral representation formulas for Clifford algebra-valued null-solutions of iterated perturbed Dirac operators.

Traveling Wave Solutions to the Burgers-αβ Equations

The Burgers-αβ equation, which was first introduced by Holm and Staley [4], is considered in the special case where ${\nu=0}$ and ${b=3}$. Traveling wave solutions are classified to the Burgers-αβ equation containing four parameters ${b,\alpha,\nu}$, and β, which is a nonintegrable nonlinear partial differential equation that coincides with the usual Burgers equation and viscous b-family of peakon equation, respectively, for two specific choices of the parameter ${\beta=0}$ and ${\beta=1}$. Under the decay condition, it is shown that there are smooth, peaked and cusped traveling wave solutions of the Burgers-αβ equation with ${\nu=0}$ and ${b=3}$ depending on the parameter β. Moreover, all traveling wave solutions without the decay condition are parametrized by the integration constant ${k_{1}\in\mathbb{R}}$. In an appropriate limit ${\beta=1}$, the previously known traveling wave solutions of the Degasperis–Procesi equation are recovered.

A multiplicity result for the scalar field equation

We prove the existence of N - 1 distinct pairs of nontrivial solutions of the scalar field equation in ℝN under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more solutions than the existing results in the literature when N ≥ 6. When the ground state is the only positive solution, we also obtain the stronger result that at least N - 1 of the first N minimax levels are critical, i.e., we locate our solutions on particular energy levels with variational characterizations. Finally we prove a symmetry breaking result when the potential is radial. To overcome the difficulties arising from the lack of compactness we use the concentration compactness principle of Lions, expressed as a suitable profile decomposition for critical sequences.

Supersymmetric rigidity of asymptotically locally hyperbolic manifolds

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.