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Tuan Anh Dao

Publications and source records attributed to Tuan Anh Dao.

45 records · Page 3Linked to original sources
arXiv2019

Critical exponent for a weakly coupled system of semi-linear $σ$-evolution equations with frictional damping

We are interested in studying the Cauchy problem for a weakly coupled system of semi-linear $σ$-evolution equations with frictional damping. The main purpose of this paper is two-fold. We would like to not only prove the global (in time) existence of small data energy solutions but also indicate the blow-up result for Sobolev solutions when $σ$ is assumed to be any fractional number.

math.AP↗
arXiv2019

A blow-up result for semi-linear structurally damped $σ$-evolution equations

We would like to prove a blow-up result for semi-linear structurally damped $σ$-evolution equations, where $σ\ge 1$ and $δ\in [0,σ)$ are assumed to be any fractional numbers. To deal with the fractional Laplacian operators $(-Δ)^σ$ and $(-Δ)^δ$ as well-known non-local operators, in general, it seems difficult to apply the standard test function method directly. For this reason, in this paper we shall construct new test functions to overcome this difficulty.

math.AP↗
arXiv2019

Some remarks on the asymptotic profile of solutions to structurally damped $σ$-evolution equations

In this paper, we are interested in analyzing the asymptotic profiles of solutions to the Cauchy problem for linear structurally damped $σ$-evolution equations in $L^2$-sense. Depending on the parameters $σ$ and $δ$ we would like to not only indicate approximation formula of solutions but also recognize the optimality of their decay rates as well in the distinct cases of parabolic like damping and $σ$-evolution like damping. Moreover, such results are also discussed when we mix these two kinds of damping terms in a $σ$-evolution equation to investigate how each of them affects the asymptotic profile of solutions.

math.AP↗
arXiv2019

Study of semi-linear $σ$-evolution equations with frictional and visco-elastic damping

In this article, we study semi-linear $σ$-evolution equations with double damping including frictional and visco-elastic damping for any $σ\ge 1$. We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the $L^p-L^q$ framework, with $1\le p\le q\le \infty$, to the corresponding linear equations. By assuming additional $L^{m}$ regularity on the initial data, with $m\in [1,2)$, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of the global obtained solutions as well to semi-linear equations. Moreover, we also determine the so-called critical exponent when $σ$ is integers.

math.AP↗
arXiv2018

A note on a weakly coupled system of semi-linear visco-elastic damped $σ$-evolution models with different power nonlinearities and different $σ$ values

In this article, we prove the global (in time) existence of small data solutions from energy spaces basing on $L^q$ spaces, with $q \in (1,\infty)$, to the Cauchy problems for a weakly coupled system of semi-linear visco-elastic damped $σ$-evolution models. Here we consider different power nonlinearities and different $σ$ values in the comparison between two single equations. To do this, we use $(L^m \cap L^q)- L^q$ and $L^q- L^q$ estimates, i.e., by mixing additional $L^m$ regularity for the data on the basis of $L^q- L^q$ estimates for solutions, with $m \in [1,q)$, to the corresponding linear Cauchy problems. In addition, allowing loss of decay and the flexible choice of parameters $σ$, $m$ and $q$ bring some benefits to relax the restrictions to the admissible exponents $p$.

math.AP↗
arXiv2018

An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $σ$-evolution models

We study the following Cauchy problems for semi-linear structurally damped $σ$-evolution models: \begin{equation*} u_{tt}+ (-Δ)^σu+ μ(-Δ)^δu_t = f(u,u_t),\, u(0,x)= u_0(x),\, u_t(0,x)=u_1(x) \end{equation*} with $σ\ge 1$, $μ>0$ and $δ\in (0,\fracσ{2})$. Here the function $f(u,u_t)$ stands for the power nonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. We are interested in investigating $L^{1}$ estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faà di Bruno's formula. By assuming additional $L^{m}$ regularity on the initial data, we use $(L^{m}\cap L^{q})- L^{q}$ and $L^{q}- L^{q}$ estimates with $q\in (1,\infty)$ and $m\in [1,q)$, to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on $L^q$ spaces.

math.AP↗
arXiv2018

$L^1$ estimates for oscillating integrals and their applications to semi-linear models with $σ$-evolution like structural damping

The present paper is a continuation of our recent paper \cite{DaoReissig}. We will consider the following Cauchy problems for semi-linear structurally damped $σ$-evolution models: \begin{equation*} u_{tt}+ (-Δ)^σu+ μ(-Δ)^δu_t = f(u,u_t),\, u(0,x)= u_0(x),\, u_t(0,x)=u_1(x) \end{equation*} with $σ\ge 1$, $μ>0$ and $δ\in (\fracσ{2},σ]$. Our aim is to study two main models including $σ$-evolution models with structural damping $δ\in (\fracσ{2},σ)$ and those with visco-elastic damping $δ=σ$. Here the function $f(u,u_t)$ stands for power nonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. We are interested in investigating the global (in time) existence of small data solutions to the above semi-linear models from suitable spaces basing on $L^q$ space by assuming additional $L^{m}$ regularity on the initial data, with $q\in (1,\infty)$ and $m\in [1,q)$.

math.AP↗
arXiv2018

Global existence for weakly coupled systems of semi-linear structurally damped $σ$-evolution models with different power nonlinearities

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped $σ$-evolution models with different power nonlinearities. By assuming additional $L^m$ regularity on the initial data, with $m \in [1,2)$, we use $(L^m \cap L^2)- L^2$ and $L^2- L^2$ estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces.

math.AP↗