arXiv subjects

Wanderley Nunes do Nascimento

Publications and source records attributed to Wanderley Nunes do Nascimento.

6 recordsLinked to original sources

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↗

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

An Exact Mass Threshold for Global Existence and Blow-up in Semilinear Anti-de Sitter Equations

We study the influence of the mass parameter on global existence and finite-time blow-up for the semilinear anti-de Sitter Klein--Gordon equation \[ ϕ_{tt}-e^{2t}Δϕ+nϕ_t+m^2ϕ=|ϕ|^p \qquad\text{in }[0,\infty)\times\R^d. \] After removing the damping term, we establish $L^2$ energy estimates for the resulting Klein--Gordon equation with constant coefficient in the potential. They imply global existence for arbitrary sufficiently small energy data, without spatial localization, whenever $ m^2>-\frac{d(2n+d)}{4} \quad\text{and}\quad p>1. $ In the remaining mass parameter interval above the structural threshold, an exact change to conformal time reduces the problem to a scale-invariant effective damped wave equation. Exponentially weighted energy estimates then extend global existence to $ -d(n+d) 1. $ In the complementary regime $m^2\le-d(n+d)$, we prove finite-time blow-up for every $p>1$ and for weak solutions with finite speed of propagation arising from nonnegative, nontrivial initial data. We further obtain lifespan estimates that are polynomial at the threshold and logarithmic below it. Thus, $m^2=-d(n+d)$ is the exact structural mass parameter threshold separating the global-existence and blow-up regimes.

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.

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