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 nonlinearities.
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.
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 p…
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）
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.
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.
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.
We consider a beam-string-beam transmission problem, where two structurally damped or undamped beams are coupled with a frictionally damped string by transmission conditions. We show that for this type of structure, the dissipation produced by the frictional part is strong enough to produce exponential decay of the solution no matter how small is its size: for the exponential stability in the damped-damped-damped situation we use energy method and in the undamped-damped-undamped situation we use a frequency domain method from the semigroups theory, which combines a contradiction argument with …
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…
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 …
The Boussinesq system for buoyancy driven fluids couples the momentum equation forced by the buoyancy with the convection-diffusion equation for the temperature. One fundamental issue on the Boussinesq system is the stability problem on perturbations near the hydrostatic balance. This problem can be extremely difficult when the system lacks full dissipation. This paper solves the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion. We establish the stability for the nonlinear system and derive precise large-time behavior for…
We study the stability and dynamics of traveling-front solutions of a modified Kuramoto--Sivashinsky equation arising in the modeling of nanoscale ripple patterns that form when a nominally flat solid surface is bombarded with a broad ion beam at an oblique angle of incidence. Structurally, the linearized operators associated with these fronts have unstable essential spectrum---corresponding to instability of the spatially asymptotic states---and stable point spectrum---corresponding to stability of the transition profile of the front. We show that these waves are linearly orbitally asymptotic…
