In the present note, we consider a semilinear damped wave equation on a compact Lie group with a non-autonomous nonlinearity φ(t)|u|^p. We are interested in describing how the nonnegative time-dependent factor φ affects the global in time prolongability of a local solution. In particular, the summability of the function φ provides a criterion to distinguish between the blow-up in finite time and global existence of small data. Finally, we derive sharp lifespan estimates for local in time solutions when φ∉L^1([0,+∞))) and satisfies a certain scaling condition, that we named uniform upper scaling condition.

. We investigate semilinear wave-type equations that can be recast as wave equations with derivatives perturbed by zero-order terms. This framework covers several well-studied cases, including the scale-invariant wave equation. In this setting, we refine existing blow-up results for radial initial data with suitable decay, and identify conditions on the zero-order terms that govern the interplay between derivative perturbations, initial data size, and the nonlinearity exponent.

Abstract Communicating Science is a voluntary, inclusive, general-purpose course aimed at equipping students with some basic non-disciplinary skills. This paper evaluates the five-year trajectory of the course (2020–2024) through a robust methodological framework. Using the Science Communication Training Effectiveness (SCTE) scale, it assesses key outcomes, including learning expectations, self-confidence, and skill acquisition, across a diverse cohort of undergraduate and graduate students, researchers, and professionals. Findings from this longitudinal study provide insights into the course’s effectiveness, the influence of motivation and work position on outcomes, and the representativeness of specific disciplinary subgroups, Physics among those, with implications for the broader design and delivery of science communication training programs.

A system of two linear wave equations with cross friction is a hyperbolic analogue of a system of two linear heat equations with cross diffusion. Such wave equation systems often have illposed initial value problems, but they can have some desirable asymptotic properties on certain closed subspaces of the underlying Hilbert space. Our main result concerns asymptotic equipartition of operator-weighted energy for some nonautonomous cross friction systems.

Some nonlinear evolutive equations of Mathematical Physics present infinitely many solutions described in many paper by different methods. For example Korteweg De Vries equation and Kadomtsev -Petviashvili equation are completely integrable despite the presence of quasilinear terms. In the present paper we perturb these kind of equations by positive nonlinear terms having polynomial growth. Assuming that the quasilinear term in the original equation has divergence form, we may apply test function method and establish a range of exponents for the perturbation so that a non-existence result of global weak solutions holds. Concerning initial data condition, an important difference with other equations studied by similar methods (wave, Tricomi and so on) appears. Indeed, we present a class of quasilinear equations for which the sign assumption on the initial data can be omitted and non-existence results still hold. Our basic examples are the perturbation of Boiti Leon Manna Pempinelli equation and Yu Toda Sasa Fukuyama equation. Finally we suggest open problems for other equations and other kind of perturbations.

Scientific communication relies on two main different structures, the IMRAD (Introduction Methods Results Analysis Discussion) and the 5W+H (Who Where Why What When How). The former is quite rigidly applied to all peer-to-peer communication, while the latter appears more loosely adaptable to dissemination and communication to the general public. We analyzed the evolution of the content of Calculus books widely adopted in Italian scientific academic courses over the past sixty years and the syllabus of a Communication of Science short course offered by the University of Bari in the last three years. We conclude that the 5Ws can be commuted to some extend with no harm to final result, while the same does not hold for the H that should always remain as the final question. Notwithstanding this flexibility, it is compulsory to answer to all 5Ws and H questions for effective communication.

On the one hand, science is increasingly entering the governance of modern society. On the other hand, society seems to be lagging behind scientific achievements, increasingly often unable to understand its language. The language gap is based on the well-known intimidating appeal of STEM programs and the unreasonable timidity of scientific degree courses to include communication skills as mandatory. An open course was experimented to provide students with the necessary tools to decode and encode the essential elements of science communication, regardless of disciplinary content. The course, which was designed following the 5W+H "golden rule" of non-academic science communication, has been offered for three consecutive years to nearly one hundred students. The students' perspective is analysed in terms of their perception that they have achieved their own learning expectations. The main result is that it is possible to involve undergraduates, graduates and professionals in rethinking their approach to communicating scientific content to the general public, with rewarding results for all players, when focusing on the greatest common divisor instead of the least common multiple among the disciplinary languages and communication templates.

In the present paper we consider the non-existence of weak solutions related to a class of differential equations connected with the Airy operator. More precisely, we deal with a positive semilinear perturbation of important PDEs involved in fluido-dynamic: Airy, KdV and BLMP equation.

An active area of recent research is the study of global existence and blow up for nonlinear wave equations where time depending mass or damping are involved. The interaction between linear and nonlinear terms is a crucial point in determination of global evolution dynamics. When the nonlinear term depends on the derivatives of the solution, the situation is even more delicate. Indeed, even in the constant coefficients case, the null conditions strongly relate the symbol of the linear operator with the form of admissible nonlinear terms which leads to global existence. Some peculiar operators with time-dependent coefficients lead to a wave operator in which the time derivative becomes a covariant time derivative. In this paper we give a blow up result for a class of quasilinear wave equations in which the nonlinear term is a combination of powers of first and second order time derivatives and a time-dependent factor. Then we apply this result to scale invariant damped wave equations with nonlinearity involving the covariant time derivatives.

In this paper we give the notion of equivalent damped wave equations. As an application we study global in time existence for the solution of special scale invariant damped wave equation with small data. To gain such results, without radial assumption, we deal with Klainerman vector fields. In particular we can treat some potential behind the forcing term.

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.

In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.

In this paper we consider a quasilinear Cauchy problem for the scale invariant damped wave equation $$\displaystyle v_{tt}-\Delta v+\frac {\mu }{(1+t)}v_t+\frac {\mu }{2} \left (\frac {\mu }{2}-1 \right )\frac {v}{(1+t)^2} =\left | \frac {\mu }{2(1+t)}v+v_t\right |{ }^p, \\ $$ with μ ≥ 0, v = v(t, x) and $$x\in \mathbb {R}^n$$ . The particular structure of the nonlinear term, guarantees a blow up result and a lifespan estimate, assuming radial initial data having slow decay. In particular the range of admissible exponents p depends on μ, n and the rate of the initial data decay.

In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation $$\mathcal {T}_\ell u=|\partial _t u|^p$$ , where $$\mathcal {T_\ell }=\partial _t^2-t^{2\ell }\Delta $$ . Smooth solutions blow up in finite time for positive Cauchy data when the exponent p of the nonlinear term is below $$\frac{\mathcal {Q}}{\mathcal {Q} -2}$$ , where $$\mathcal {Q}=(\ell +1)n+1$$ is the quasi-homogeneous dimension of the generalized Tricomi operator $$\mathcal {T}_\ell $$ . Furthermore, we get also an upper bound estimate for the lifespan.

In this paper we study local and global in time existence for a class of nonlinear evolution equations having order eventually greater than 2 and not integer. The linear operator has an homogeneous damping term; the nonlinearity is of polynomial type without derivatives: u(tt) + (-Delta)(2 theta)u + 2 mu(-Delta)(theta)u(t) + vertical bar u vertical bar(p-1) u = 0, t >= 0, x is an element of R-n, with mu > 0, theta > 0. Since we are treating an absorbing nonlinear term, large data solutions can be considered.

In this paper, the city of Matera is described from a mathematical point of view. Previous papers on this subject have concentrated on seeing Matera as a fractal city. Here, this analysis is also dealt with as an extension of Euclidean dimensions. The idea is to create a double presentation narrative which is useful for the promotion of cultural heritage and also for the popularizing of mathematics. Those who like geometrical vision will discover some aspects of one of the most ancient cities in Italy. Those who like travelling will have new words to describe the wonders of this country. We reach this objective by using a combinatoric puzzle and suitable story telling.

In this chapter, we prove the large data almost global existence of the 4-dimensional weakly hyperbolic equation: $$\displaystyle u_{tt}-(t_0-t)^2\varDelta u=-(t_0-t)^4|u|u\,. $$

In the present paper we recall the main points of the Fourier Transform developments, in particular the historical origin of the inversion formula. Hence we construct explicit examples of functions in different zones of the range of the Fourier transform in L-1. These can be used as exercises in a basic course of signal processing or harmonic analysis.