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Tuan Anh Dao

Publications and source records attributed to Tuan Anh Dao.

At least 19 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.

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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.

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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.

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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.

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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.

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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.

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arXiv2026

Non-existence results for a system of wave inequalities on locally finite graphs

Let $V$ be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system $$ \begin{cases} u_{t t}-Δu \geq h_1|v|^p & \text { in } V \times(0, \infty), v_{t t}-Δv \geq h_2|u|^q & \text { in } V \times(0, \infty), u=u_0;\;v=v_0 & \text { in } V \times\{0\}, u_t=u_1;\;v_t=v_1 & \text { in } V \times\{0\}, \end{cases}$$ where $p,q>1$, $h_1, h_2$ are positive potentials. Under some volume growth condition of a ball, we prove that the system has no non-trivial non-negative solutions. In particular, our result is a natural extension of that in [\textit{D.~D.~Monticelli, F.~Punzo, and J.~Somaglia. Nonexistence results for the semilinear wave equation on graphs. arXiv.2506.08697, 2025.}] from a single inequality to a system.

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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.

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arXiv2025

Sharp behavior of semilinear damped wave equations driven by mixed local-nonlocal operators

This paper investigates the Cauchy problem for the semilinear damped wave equation $u_{tt}+\mathcal{L}_{a,b}u+u_t=|u|^p$ with the mixed local-nonlocal operator $\mathcal{L}_{a,b}:=-aΔ+b(-Δ)^σ$, where $a,b\in\mathbb{R}_+$ and $σ\in(0,1)\cup (1,+\infty)$. We determine the critical exponent for this problem being $p_{\mathrm{crit}}=1+\frac{2\min\{1,σ\}}{n}$, which sharply separates global in-time existence and finite-time blow-up of solutions. Furthermore, for the super-critical case $p>p_{\mathrm{crit}}$, we establish the asymptotic profiles of global in-time solutions, showing the anomalous diffusion when $σ\in(0,1)$ and the classical diffusion when $σ\in(1,+\infty)$, together with the sharp decay estimates. For solutions blowing up in finite time when $1<p\leqslant p_{\mathrm{crit}}$, we derive the sharp estimates for upper and lower bounds of lifespan. Our results reveal the crucial influence of mixed operators on the qualitative properties of solutions, fundamentally governing their critical phenomena, large-time behavior and blow-up dynamics, via $\max\{1,σ\}$ or $\min\{1,σ\}$.

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arXiv2025

Blow up results and lifespan estimates for nonlinear damped wave equations on weighted graphs

In this article, we are interested in studying the Cauchy problems for nonlinear damped wave equations and their systems on a weighted graph. Our main purpose is two-fold, namely, under certain conditions for volume growth of a ball and the initial data we would like to not only prove nonexistence of global (in time) weak solutions but also indicate lifespan estimates for local (in time) weak solutions when a blow-up phenomenon in finite time occurs. Throughout the present paper, we will partially give a positive answer for the optimality of our results by an application to the $n$-dimensional integer lattice graph $\Z^n$ to recover the well-known results in the Euclidean setting.

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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

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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.

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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.

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arXiv2024

A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics

We present a novel high-order nodal artificial viscosity approach designed for solving Magnetohydrodynamics (MHD) equations. Unlike conventional methods, our approach eliminates the need for ad hoc parameters. The viscosity is mesh-dependent, yet explicit definition of the mesh size is unnecessary. Our method employs a multimesh strategy: the viscosity coefficient is constructed from a linear polynomial space constructed on the fine mesh, corresponding to the nodal values of the finite element approximation space. The residual of MHD is utilized to introduce high-order viscosity in a localized fashion near shocks and discontinuities. This approach is designed to precisely capture and resolve shocks. Then, high-order Runge-Kutta methods are employed to discretize the temporal domain. Through a comprehensive set of challenging test problems, we validate the robustness and high-order accuracy of our proposed approach for solving MHD equations.

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arXiv2024

Viscous regularization of the MHD equations

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection.

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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.

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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.

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