arXiv subjects

Wanderley Nunes do Nascimento

Publications and source records attributed to Wanderley Nunes do Nascimento.

5 recordsLinked to original sources

arXiv2026

Blow-up for a semilinear Tricomi equation in the oscillatory regime at the critical Strauss-type exponent

We study finite-time blow-up for a semilinear shifted Tricomi equation with decreasing propagation speed and an oscillatory scale-invariant mass. We focus on the Strauss-type critical regime and prove that every weak solution with finite speed of propagation, arising from nonnegative nontrivial energy data satisfying a suitable localization condition, blows up in finite time. For sufficiently small initial data, we also obtain the corresponding critical exponential upper bound for the lifespan. The proof relies on a positive self-similar solution of the homogeneous adjoint equation represented by the Gauss hypergeometric function. Combined with a previously established lower bound for the nonlinear term and new estimates adapted to the critical case, this construction reduces the PDE problem to a nonlinear differential inequality. An ODE comparison argument then yields both finite-time blow-up and the lifespan estimate.

math.AP↗

arXiv2026

Strauss and Glassey exponents for semilinear wave equations with a non-effective and not scattering producing damping

In this paper, we study the Cauchy problems for a semilinear damped wave equation with a time-dependent coefficient for the damping term belonging to the class of non-effective damping terms with critical decay rates involving iterated logarithmic factors. As nonlinearities we consider both $|u|^p$ and $|u_t|^p$. Assuming nonnegative and compactly supported initial data, we establish the blow-up for weak solutions in the sub-Strauss range $1 < p \leq p_{\mathrm{Str}}(n)$ for the power of the nonlinear term $|u|^p$. The proof in the sub-critical case relies on an iteration frame for the space average of the solution, obtained by employing a time-dependent multiplier related to the coefficient of the damping term. On the other hand, in the limit case $p=p_{\mathrm{Str}}(n)$, we work with solutions of the homogeneous equation with separated variables, and we investigate the properties of a fundamental system of solutions for the corresponding time-dependent ODE, in order to derive an iteration frame for a suitable weighted space average of the solution. Finally, for the derivative type nonlinearity $|u_t|^p$ we prove the blow-up of weak solutions in the sub-Glassey range $1 < p \leq \frac{n+1}{n-1}$ by using a comparison argument for a suitable time-dependent function associated with the corresponding local in time solution.

math.AP↗

arXiv2020

The move from Fujita to Kato type exponent for a class of semilinear evolution equations with time-dependent damping

In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leq p\leq 2\leq q\leq \infty$, for the solutions to the $σ$-evolution equation, $σ>1$, with scale-invariant time-dependent damping and power nonlinearity~$|u|^p$, \[ u_{tt}+(-Δ)^σu + \fracμ{1+t} u_t= |u|^{p}, \] where $μ>0$, $p>1$. The critical exponent $p=p_c$ for the global (in time) existence of small data solutions to the Cauchy problem is related to the long time behavior of solutions, which changes accordingly $μ\in (0, 1)$ or $μ>1$. Under the assumption of small initial data in $L^1\cap L^2$, we find the critical exponent \[ p_c=1+ \max \left\{\frac{2σ}{[n-σ+σμ]_+}, \frac{2σ}{n} \right\} =\begin{cases} 1+ \frac{2σ}{[n-σ+σμ]_+}, \quad μ\in (0, 1)\\ 1+ \frac{2σ}{n}, \quad μ>1. \end{cases} \] For $μ>1$ it is well known as Fujita type exponent, whereas for $μ\in (0, 1)$ one can read it as a shift of Kato exponent.

math.AP↗

arXiv2017

A classification for wave models with time-dependent mass and speed of propagation

In this paper, we study the long time behavior of energy solutions for a class of wave equation with time-dependent mass and speed of pro\-pagation. We introduce a classification of the potential term, which clarifies whether the solution behaves like the solution to the wave equation or Klein-Gordon equation. Moreover, $L^q-L^2, q\in [1, 2]$ estimates for scale-invariant models are derived and applied to obtain global in time small data energy solutions for the semilinear Klein-Gordon equation in anti de Sitter spacetime.

math.AP↗

arXiv2015

Wave equations with mass and dissipation

In this paper we consider a wave model with non-effective mass and dissipation terms and provide asymptotic descriptions of its representation of solutions. In particular we conclude sharp estimates for a corresponding energy and estimates of dispersive type.

math.AP↗