arXiv subjects

Firas Kaabi

Publications and source records attributed to Firas Kaabi.

5 recordsLinked to original sources

arXiv2026

Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping

We study the maximal existence time $T^{*}(\varrho)$ of the solution of the semilinear wave equation $u_{tt}-Δu=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ in a bounded domain, with Dirichlet boundary condition and initial data $(\varrho f,\varrho g)$, where $1 2p/(p+1)$, whereas the energy method gives a lower bound of order $\varrho^{1-p}$ only. We prove lower bounds with the same exponents as the upper ones, so that $T^{*}(\varrho)\asymp\varrho^{-\vartheta(p,q)}$ with $\vartheta(p,q)=\min\{(p-1)/2,\,(p-q)/q\}$. The proof rests on a hyperbolic rescaling which converts the large amplitude into a dilation of the domain and a coefficient $\varrho^{(q(p+1)-2p)/2}$ in front of the damping term, on a local existence theory in uniformly local energy norms whose existence time does not depend on that coefficient, and on a quantitative use of the dissipation when the coefficient is large. The threshold $2p/(p+1)$ is the value of $q$ at which the damping term is invariant under the rescaling. No attempt is made to optimize the constants: sharpness is meant throughout at the level of the exponent.

math.AP↗

arXiv2026

Small-Data Lifespan for a One-Dimensional Wave Equation with Mixed Characteristic-Time Derivative Source

We study the lifespan of classical solutions to \[ v_{tt}-v_{xx}=|v_t+v_x|^m|v_t|^n, \qquad x\in\R,\quad t>0, \] where $m>1$ and $n>1$. For compactly supported data $(ηϕ,ηψ)$, with $ϕ\in C_0^2(\R)$ and $ψ\in C_0^1(\R)$, we prove the two-sided estimate \[ cη^{-(m+n-1)} \leq T(η) \leq Cη^{-(m+n-1)} \] under the single one-sided assumption $ψ-ϕ'\geq0$ on $\R$, the data being nontrivial. The lower bound is obtained from the characteristic integral system and holds without any sign restriction; the upper bound follows from a scalar superlinear inequality along a selected characteristic. A short argument shows that the sign assumption already forces $ψ(x_0)+ϕ'(x_0)>0$ at some point, so that no separate activation hypothesis is needed. We also show that the compatible cancellation condition $ψ+ϕ'\equiv0$ produces the global free wave $v(x,t)=ηϕ(x-t)$, and that this cancellation regime meets the sign assumption only for trivial data. The model therefore separates a cancellation regime from a finite-time amplification regime with an exactly determined lifespan scale.

math.AP↗

arXiv2026

Two-sided estimates of the blow-up time for a semilinear wave equation with fractional structural damping

We consider the initial--boundary value problem for the semilinear wave equation with fractional structural damping $$ u_{tt}+(-Δ)^θu_{t}-Δu=|u|^{p-1}u $$ in a bounded domain, where the exponent $θ\in[0,1]$ interpolates between external frictional damping ($θ=0$) and internal Kelvin--Voigt viscoelastic damping ($θ=1$), and therefore parametrises the frequency dependence of the dissipation mechanism. For initial data with energy below the depth of the potential well and negative Nehari functional we prove finite-time blow-up together with an explicit upper bound for the blow-up time. The bound comes from a single concavity functional in which the fractional dissipation cancels identically; it therefore has the same form for every $θ\in[0,1]$, and it admits a variant that is uniform in $θ$. Conversely, for $1<p\le\frac{n+2θ}{n-2}$ when $n\ge3$, we establish an explicit lower bound for the blow-up time. Mechanically, the two bounds delimit a guaranteed interval of existence and a guaranteed failure time for the model. The admissible range of exponents in the lower bound widens linearly with $θ$, which quantifies how the strength of the internal damping enlarges the class of nonlinear loads for which such a guarantee can be computed; we do not claim monotonicity in $θ$ of the numerical value of the bound. Both theorems are proved for a class of energy solutions specified by a short list of requirements, so that they are independent of any particular local existence theorem, and they carry over unchanged to the elasticity and plate operators used in structural models. The two endpoint cases recover known results for frictional and strong damping.

math.AP↗

arXiv2026

Lifespan Lower Estimates for a Strongly Damped Semilinear Wave Equation

We consider a strongly damped semilinear wave equation with initial data prescribed as $(\varrhoϕ,\varrho h)$, where the profiles are fixed and only the amplitude $\varrho>0$ is allowed to vary. The question addressed here is how this rescaling affects a guaranteed lower bound for the maximal existence time. We show that the solution exists at least on a time interval of length comparable to $\varrho^{-(p-2)}$. The proof is based on the growth of a quadratic phase-space norm: after the source term is estimated by the relevant Sobolev embedding, the problem reduces to a scalar differential inequality. The constants produced in the argument are independent of $\varrho$, so the dependence on the initial amplitude remains explicit throughout.

math.AP↗

arXiv2026

Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$).

math.AP↗