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.
This work is devoted to improve the time decay estimates for the solution and some of its derivatives of the linear structurally damped $σ$-evolution equations. The Pitt inequality is the main tool provided that the initial data lies in some weighted spaces.
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.
Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped wave equations. Main goal is to find the threshold, which classifies the global (in time) existence of small data solutions or the nonexistence of global solutions under the growth condition of the nonlinearities.
Main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: $$u_{tt}- Δu+ μ(-Δ)^{σ/2} u_t= |u_t|^p,\quad u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x),$$ with $μ>0$, $n\geq1$, $σ\in (0,2]$ and $p>1$. In particular, we are going to prove the non-existence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.
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.
Blow-up and sharp lifespan estimates to the weakly coupled system of structurally damped wave equations with critical nonlinearities
School of Mathematics and Computer Sciences, Hanoi National University of Education（河内国家教育大学数学与计算机科学学院）；Faculty of Mathematics and Informatics, Hanoi University of Science and Technology（河内科技大学数学与信息学院）；Department of Mathematics, Hanoi National University of Education（河内国家教育大学数学系）
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $ν(-Δ)^2 u_t$, where $ν>0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain global solutions with sharp decay properties in time. Our main purpose of this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical n…
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 fun…
We investigate the internal controllability of the wave equation with structural damping on the one dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.
We consider the transmission problem for a coupled system of undamped and structurally damped plate equations in two sufficiently smooth and bounded subdomains. It is shown that, independently of the size of the damped part, the damping is strong enough to produce uniform exponential decay of the energy of the coupled system.
We consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions. We show that exponential stability holds in the damped-damped situation and polynomial stability (but no exponential stability) holds in the damped-undamped case. Additionally, we show that the solutions first defined by the weak formulation, in fact have higher Sobolev space regularity.
We study an abstract damped wave equation. We prove that the solution of the damped wave equation becomes closer to the solution of a heat type equation as time tend to infinity. As an application of our approach, we also study the asymptotic behavior of the damped wave equation in Euclidean space under the geometric control condition.
In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.
A condition which guaranties the exponential decay of the solutions of the initial-boundary value problem for the damped wave equation is proved. A method for the effective computability of the coefficient of exponential decay is also presented.
In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations $\partial_{t}^{2} u - Δu + ν(-Δ)^σ \partial_{t} u=0$, where $ν>0$ and $0< σ\le1$. Our result is the approximation formula of the solution by a constant multiple of a special function as $t \to \infty$, which states that the asymptotic profiles of the solutions are classified into $5$ patterns depending on the values $ν$ and $σ$.
The large deflections of panels in subsonic flow are considered. Specifically, a fully clamped von Karman plate accounting for both rotational inertia in plate filaments and structural damping of square root type is considered. The panel is taken to be embedded in the boundary of a linear, subsonic potential flow on the positive halfspace in $\mathbb R^3$. Solutions are constructed via a semigroup approach despite the lack of natural dissipativity associated to the generator of the linear dynamics. The flow-plate dynamics are then reduced---via an explicit Neumann-to-Dirichlet (downwash-to-pre…
This article is to study the nonexistence of global solutions to semi-linear structurally damped wave equation with nonlinear memory in $\R^n$ for any space dimensions $n\ge 1$ and for the initial arbitrarily small data being subject to the positivity assumption. We intend to apply the method of a modified test function to establish blow-up results and to overcome some difficulties as well caused by the well-known fractional Laplacian $(-Δ)^{σ/2}$ in structural damping terms.
In this short notice, we prove the non-existence of global solutions to the semilinear damped wave equation on the half-space, and we determine the critical exponent for any space dimension.
In this article, we indicate that under suitable assumptions of a modulus of continuity we obtain either the global (in time) existence of small data Sobolev solutions or the blow-up result of local (in time) Sobolev solutions to semi-linear damped $σ$-evolution equations with a modulus of continuity term in nonlinearity.
In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $\cap$ W 2,$\infty$ and the damped wave or Schrödinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x $\in$ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.
The article is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier-Stokes-Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an $L^p-L^q$ setting for small time or for small data. Through a change of variables and a fixed point argument, the proof of the main resu…
The motivation of the present study is to discuss the global (in time) existence of small data solutions to the following semi-linear structurally damped $σ$-evolution models: \begin{equation*} \partial_{tt}u+(-Δ)^σu+(-Δ)^{σ/2}\partial_{t}u=\left|u\right| ^{p}, \ σ\geq 1, \ \ p>1, \end{equation*} where the Cauchy data $(u(0,x), \partial_{t}u(0,x))$ will be chosen from energy space on the base of $L^{q}$ with different additional $L^{m}$ regularity, namely \begin{equation*} u(0,x)\in H^{σ,q}(\mathbb{R}^{n})\cap L^{m_{1}}(\mathbb{R}^{n}) , \ \ \partial_{t}u(0,x)\in L^{q}(\mathbb{R}^{n})\cap L^{m…
In this paper we consider second order evolution equations with unbounded dynamic feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.
We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier--Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier--Stokes system with damping our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2D Navier--Stokes sys…
We consider the two-dimensional stochastic damped nonlinear wave equation (SdNLW) with the cubic nonlinearity, forced by a space-time white noise. In particular, we investigate the limiting behavior of solutions to SdNLW with regularized noises and establish triviality results in the spirit of the work by Hairer, Ryser, and Weber (2012). More precisely, without renormalization of the nonlinearity, we establish the following two limiting behaviors; (i) in the strong noise regime, we show that solutions to SdNLW with regularized noises tend to 0 as the regularization is removed and (ii) in the w…
In this paper we consider the initial-boundary value problem for the heat, damped wave, complex-Ginzburg-Landau and Schr"odinger equations with the power type nonlinearity $|u|^p$ with $p in (1,2]$ in a two-dimensional exterior domain. Remark that $2=1+2/N$ is well-known as the Fujita exponent. If $p>2$, then there exists a small global-in-time solution of the damped wave equation for some initial data small enough (see Ikehata'05), and if $p<2$, then global-in-time solutions cannot exist for any positive initial data (see Ogawa-Takeda'09 and Lai-Yin'17). The result is that for given initial d…
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 …
