arXiv subjects

Dinh Van Duong

Publications and source records attributed to Dinh Van Duong.

11 recordsLinked to original sources

arXiv2026

Global existence of solutions for nonlinear damped wave equations in an exterior domain with nonlinearities of derivative type

In this paper, we are interested in considering the semi-linear damped wave equation with the power nonlinearity of derivative type in $2$D exterior domains to indicate the global (in time) existence of small data solutions, which has never appeared in previous studies. Our proof is based on constructing a judicious time-weighted function space and demonstrating nonlinear estimates compatible with fractional powers of the Dirichlet Laplacian linked to the semigroup decay structure. Moreover, the large time behavior of the time derivative of the obtained global solutions and its sharp estimate are also discussed in this paper.

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↗

arXiv2026

Critical exponent for semilinear damped wave equations with weighted nonlinear terms and data from Sobolev spaces of negative order

In this paper, we would like to study the critical exponent for semilinear damped wave equations with the nonlinearity terms of Coulomb-type singularities $|x|^{-α} |u(t,x)|^p$ and the initial data belonging to Sobolev spaces of negative order $\dot{H}^{-β}$. Precisely, we obtain a critical exponent $$p_{\rm c}(α,β,n): = 1 + \frac{4-2α}{n+2β} $$ for $1 \leq n \leq 4$ and $ 0 \leq α, β< n/2,$ by proving the global (in time) existence of small data solutions when $p \geq p_{\rm c}(α,β,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(α,β,n)$. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.

math.AP↗

arXiv2026

Existence of global solutions to semilinear damped wave equations with nonlinearities of derivative type

In this paper, we would like to consider the semi-linear damped wave equation with the power nonlinearity of derivative type $|u_t|^p$. The main contribution of this work is to improve the results for global (in time) solution existence in a comparison with the pioneering paper \cite{Matsumura1976} of Matsumura, who first established that the solutions exist globally for $p > 1$ ($n = 1$) and $p \ge 2$ ($n \ge 2$). More precisely, we have extended such a result for any $p > 1$ ($n = 1,2$) and $p > 3/2$ ($n = 3$). Our approach relies on constructing appropriately weighted solution spaces linked to the delicate application of several tools from Harmonic Analysis and Banach fixed-point theorem.

math.AP↗

arXiv2025

On the asymptotic profile of solutions to semilinear damped wave equations with critical nonlinearities

In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} μ(|u|)$, where $μ$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020)) and Girardi (Nonlinear Differ. Equ. Appl. 32 (2025)), the authors obtained a sharp critical condition on $μ$ in low space dimensions $n=1,2,3$, which determines the threshold between global (in time) existence of small data solutions and blow-up of solutions in finite time. Our new results are to prove that this condition remains valid in dimension $n=4$, together with the asymptotic profiles of global solutions. From this, we see that the behavior of the solution at $t \to \infty$ is identified by the Gauss kernel. Finally, a sharp lifespan estimate for local solutions is also derived in the case when blow-up occurs.

math.AP↗

arXiv2025

Critical curve for the weakly coupled system of damped wave equations with mixed nonlinearities

In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear damped wave equations with mixed nonlinear terms. Our main objective is to draw conclusions about the critical curve of this problem using tools from Harmonic Analysis. Precisely, we obtain a new critical curve $pq = 1+ \frac{2}{n}$ for $n =1,2$ by proving global (in time) existence of small data Sobolev solutions when $pq > 1 +\frac{2}{n}$ and blow-up of weak solutions in finite time even for small data when $pq < 1+ \frac{2}{n}$ for $n \geq 1$. From this, we infer the impact of the nonlinearities of time derivative-type on the critical curve associated with the system.

math.AP↗

arXiv2025

Critical curve for a weakly coupled system of semi-linear $σ$-evolution equations with different damping types

In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi linear $sigma$ evolution equations with different damping mechanisms for any $σ>1$, parabolic like damping and $σ$ evolution like damping. Motivated strongly by the well known Nakao's problem, the main goal of this work is to determine the critical curve between the power exponents $p$ and $q$ of nonlinear terms by not only establishing the global well posedness property of small data solutions but also indicating blow up in finite time solutions. We want to point out that the application of a modified test function associated with a judicious choice of test functions really plays an essential role to show a blow up result for solutions and upper bound estimates for lifespan of solutions, where $σ$ is assumed to be any fractional number. To end this paper, lower bound estimates for lifespan of solutions are also shown to verify their sharp results in some spatial dimensions

math.AP↗

arXiv2025

Sharp higher-order $L^2$-asymptotic expansion of solutions to σ-evolution equations with different damping types

In this paper, our main goal is to achieve the high-order asymptotic expansion of solutions to $σ$-evolution equations with different damping types in the $L^2$ framework. Throughout this, we observe the influence of parabolic like models corresponding to $σ_1 \in [0, σ/2)$ and $σ$-evolution like models corresponding to $σ_2 \in (σ/2, σ]$ on the asymptotic behavior of solutions.

math.AP↗

arXiv2025

Effect of additional regularity for the initial data on semi-linear $σ$-evolution equations with different damping types

In this paper, we would like to study the critical exponent for semi-linear $σ$-evolution equations with different damping types under the influence of additional regularity for the initial data. On the one hand, we establish the existence of global (in time) solutions for small initial data and the blow-up in finite time solutions in the supercritical case and the subcritical case, respectively. The very interesting phenomenon is that the critical case belonging to the global solution range or the blow-up solution range depends heavily on the assumption of additional regularity for the initial data. Furthermore, we are going to provide lifespan estimates for solutions when the blow-up phenomenon occurs.

math.AP↗

arXiv2023

Applications of $L^p-L^q$ estimates for solutions to semi-linear $σ$-evolution equations with general double damping

In this paper, we would like to study the linear Cauchy problems for semi-linear $σ$-evolution models with mixing a parabolic like damping term corresponding to $σ_1 \in [0,σ/2)$ and a $σ$-evolution like damping corresponding to $σ_2 \in (σ/2,σ]$. The main goals are on the one hand to conclude some estimates for solutions and their derivatives in $L^q$ setting, with any $q\in [1,\infty]$, by developing the theory of modified Bessel functions effectively to control oscillating integrals appearing the solution representation formula in a competition between these two kinds of damping. On the other hand, we are going to prove the global (in time) existence of small data Sobolev solutions in the treatment of the corresponding semi-linear equations by applying $(L^{m}\cap L^{q})- L^{q}$ and $L^{q}- L^{q}$ estimates, with $q\in (1,\infty)$ and $m\in [1,q)$, from the linear models. Finally, some further generalizations will be discussed in the end of this paper.

math.AP↗

arXiv2023

On asymptotic properties of solutions to $σ$-evolution equations with general double damping

In this paper, we would like to consider the Cauchy problem for semi-linear $σ$-evolution equations with double structural damping for any $σ\ge 1$. The main purpose of the present work is to not only study the asymptotic profiles of solutions to the corresponding linear equations but also describe large-time behaviors of globally obtained solutions to the semi-linear equations. We want to emphasize that the new contribution is to find out the sharp interplay of ``parabolic like models" corresponding to $σ_1 \in [0,σ/2)$ and ``$σ$-evolution like models" corresponding to $σ_2 \in (σ/2,σ]$, which together appear in an equation. In this connection, we understand clearly how each damping term influences the asymptotic properties of solutions.

math.AP↗