arXiv subjects

The Anh Cung

Publications and source records attributed to The Anh Cung.

5 recordsLinked to original sources

arXiv2026

Critical thresholds for the damped wave system with mixed product-type nonlinearities

In this paper, we study the Cauchy problem for a coupled system of damped wave equations with mixed product-type nonlinearities. We have succeeded in determining the critical curve, formulated in terms of the power exponents of nonlinearities, for the global (in time) existence of small data mild solutions and the nonexistence of global mild solutions. Moreover, the global existence result is established by means of Schauder's fixed point theorem and Banach's contraction principle, whereas the nonexistence analysis relies on a new iteration scheme adapted to the product-type structure of nonlinearities. To the best of our knowledge, this work provides the first critical threshold results for such systems.

math.AP↗

arXiv2026

Nonexistence for effectively damped waves with time-dependent mass

In this paper, we study the semilinear wave equations $$ u_{tt}-Δu+b(t)u_t+m^2(t)u=|u|^p, \quad t \geq 0, \quad x\in\mathbb{R}^n $$ with effective time-dependent damping and a time-dependent mass dominated by the damping. D'Abbicco, Girardi and Reissig established global small-data existence in supercritical ranges and identified the scale $$ p_{β,η}(n)=1+\frac{2η}{n+2ηβ} $$ for initial data in $(L^η(\mathbb R^n)\cap H^1(\mathbb R^n))\times(L^η(\mathbb R^n)\cap L^2(\mathbb R^n))$ with $1\leqη<2$, where $β$ is the lower mass index associated with the damping-mass pair. To support the expected sharpness of this scale, they also established an analogous subcritical nonexistence result for the corresponding diffusion equation with nonnegative initial data in $L^η(\mathbb{R}^n)$, leaving the wave-equation counterpart with effective damping and time-dependent mass open. We address this problem for $η=1$ under an intrinsic accumulated-mass balance and a Liouville nonoscillation condition. By constructing a positive slow adjoint mode, we prove nonexistence of global weak solutions for $$ 1<p<p_{β,1}(n)=1+\frac{2}{n+2β}, $$ and also treat the critical case $p=p_{β,1}(n)$ under an Osgood divergence condition. Conditional lifespan upper bounds and explicit admissible coefficient families are also given.

math.AP↗

arXiv2026

New definitions of decay indicators and critical exponent for fractional semi-linear structurally damped evolution equations on the Heisenberg group

In this paper, we introduce the lower and upper decay indicators and the associated decay character on the Heisenberg group. These notions are used to derive decay estimates for the linear fractional diffusion equation and to characterize the decay of initial data in several function spaces. We then study the Cauchy problem $$ \partial_t^2u+\left(-Δ{\mathrm H}\right)^{δ_1}u+\left(-Δ_{\mathrm H}\right)^{δ_2}\partial_tu=0, \quad δ_1\in\left[0,\frac{δ_2}{2}\right], $$ and establish decay estimates for solutions and their derivatives in homogeneous fractional Sobolev spaces in terms of the decay characters of the initial data. These results recover known estimates and extend them to new classes of data. We also investigate the corresponding semilinear problem with nonlinearity $|u|^p$. Global existence and decay are proved for $$ p>1+\frac{2ωδ_1}{Q-2ωδ_2}, \quad ω=\frac{Q}{Q+\min{r_{\mathrm H}(u_0),r_{\mathrm H}(u_1)-2δ_2}+2δ_2}, $$ together with the corresponding critical case. Finally, by constructing test functions adapted to the nonlocal fractional sub-Laplacians, we establish blow-up results and identify the critical exponent $$ p=1+\frac{2mδ_1}{Q+mγ-2mδ_2} $$ for initial data in $\dot H_m^{-γ}(\mathbf H_n)$, where $m\in(1,2]$ and $γ\in\left[0,Q-\frac{Q}{m}\right)$.

math.AP↗

arXiv2026

On a system of semilinear damped $σ$-evolution equations with different damping types in the critical case

In this paper, we study the non-symmetric system of semilinear damped $σ$-evolution equations with different damping types, where two power exponents of nonlinearities belong to the critical curve, by using moduli of continuity in nonlinear terms. Our goal is to determine the sharp conditions on these moduli of continuity that guarantee the global (in time) existence of Sobolev solutions or, conversely, lead to finite-time blow-up. Furthermore, by employing the analysis introduced in the proof of our blow-up result, we provide a positive answer to an open problem for the symmetric models posed in the literature.

math.AP↗

arXiv2026

Blow-up and sharp lifespan estimates to the weakly coupled system of structurally damped wave equations with critical nonlinearities

In this paper, we would like to study the weakly coupled system of semilinear structurally damped wave equations with moduli of continuity in nonlinear terms whose powers belong to the critical curve in the $p-q$ plane. Our main purpose is to find a sharp condition for these moduli of continuity by investigating the global (in time) existence of small data Sobolev solutions and the blow-up result for solutions in finite time as well. Furthermore, when the blow-up phenomenon occurs, we are going to achieve the sharp lifespan estimates for the local (in time) Sobolev solution.

math.AP↗