arXiv subjects

Duc An Phan

Publications and source records attributed to Duc An Phan.

4 recordsLinked to original sources

arXiv2026

A Fujita-type threshold for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces

In this paper, we consider the semilinear damped wave equation with Hartree-type nonlinearity $\mathcal{I}_γ\left(|u|^{p_1}\right)|u|^{p_2}$, where $0<γ 0$. This formulation is particularly suited to the nonlinear analysis. We then establish the existence of a unique global mild solution for sufficiently small initial data whenever $$ p_1+p_2\geq 1+\frac{4+2γ}{n+2β}, $$ under the remaining admissibility conditions stated in the existence theorem. In particular, the critical case is covered whenever the critical line satisfies these conditions. Conversely, for initial data satisfying an explicit positive lower bound, the test-function method rules out global weak solutions when $$ 2<p_1+p_2<1+\frac{4+2γ}{n+2β}. $$ Thus, whenever the critical line is admissible and the subcritical interval is nonempty, $1+(4+2γ)/(n+2β)$ gives a Fujita-type threshold for the sum $p_1+p_2$.

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

The application of decay character on the global behavior of damped wave equation with Riesz potential-type power nonlinearity

In this paper, our first objective is to investigate the decay rates and the global (in time) existence of solutions to the semilinear damped wave equation with the Riesz potential-type power nonlinearity $\mathcal{I}_γ\left(|u|^p\right)$, where $γ\in[0,n)$, in terms of the decay character of the initial data. This approach enables us to establish global existence results for several classes of initial data. Our second objective is to show, via a blow-up argument, that the conditions imposed on the nonlinearity in the global existence theorem are sharp for initial data belonging to the pseudo-measure space $\mathcal{Y}^q$. As a consequence, we derive the new critical exponent $$ p_{\mathrm{crit}}(n,q,γ):=1+\frac{2+γ}{n-q} $$ for $1\leq n\leq 4$ and $0\leq γ<q<n/2$. Furthermore, we establish a sharp lifespan estimate for solutions that blow up in finite time.

math.AP↗