本文研究具有两个变系数的非线性阻尼梁方程 Cauchy 问题解的渐近剖面，方程为 ∂_t^2 u + b(t)∂_t u − a(t)∂_x^2 u + ∂_x^4 u = ∂_x(N(∂_x u))。在作者先前的文献 [17] 中，已根据对系数 a(t) 和 b(t) 的假设，对线性化问题（N≡0）解的渐近剖面进行分类，并证明了有效阻尼情形下的渐近行为。本文给出关于系数和非线性项的条件，以使当 t→∞ 时，该解的渐近表现与热方程 b(t)∂_t u − a(t)∂_x^2 u=0 的解一致。
英文摘要：
In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: \[ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u
= \partial_x \left( N(\partial_x u) \right). \] In the authors' previous article [17], the asymptotic profile of solutions for linearized problem ($N \equiv 0$) was classified depending on the assumptions for the coefficients $a(t)$ and $b(t)$ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $b(t) \partial_t u - a(t) \partial_x^2 u=0$ asymptotically as $t \to \infty$.
We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $|u|^p+|u_t|^q$ in an exterior domain, where $p,q>1$. Assuming the smallness of initial data in exponentially weighted spaces and some conditions on powers of nonlinearity, we prove global (in time) existence of small data energy solution with suitable higher regularity by using a weighted energy method.
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, a strongly damped semilinear wave equation with a general nonlinearity is considered. With the help of a newly constructed auxiliary functional and the concavity argument, a general finite time blow-up criterion is established for this problem. Furthermore, the lifespan of the weak solution is estimated from both above and below. This partially extends some results obtained in recent literatures and sheds some light on the similar effect of power type nonlinearity and logarithmic nonlinearity on finite time blow-up of solutions to such problems.
We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.
We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.
In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity $|u|^p$ for any $p>1$, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.
This paper is devoted to the lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity $$u_{tt}+Δ^2u-Δu-ωΔu_t+α(t)u_t=|u|^{p-2}u\ln|u|.$$ Finite time blow-up criteria for solutions at both lower and high initial energy levels are established, and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.
In this research, we would like to study the global (in time) existence of small data solutions to the following damped $σ$-evolution equations with nonlocal (in space) nonlinearity: \begin{equation*} \partial_{t}^{2}u+(-Δ)^σu+\partial_{t}u+(-Δ)^σ\partial_{t}u=I_α(|u|^{p}), \ \ t>0, \ \ x\in \mathbb{R}^{n}, \end{equation*} where $σ\geq1$, $p>1$ and $I_α$ is the Riesz potential of power nonlinearity $|u|^{p}$ for any $α\in (0,n)$. More precisely, by using the $(L^{m}\cap L^{2})-L^{2}$ and $L^{2}-L^{2}$ linear estimates, where $m\in[1,2]$, we show the new influence of the parameter $α$ on the ad…
In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions …
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schrödinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. This improves known results in Sobolev spaces in some sense. As …
In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-…
We consider the damped wave equation αu_tt + u_t = u_xx - V'(u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x,t) = h(x-st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V. We show that, if the initial data are sufficiently close to the profile of a front for large |x|, the solution of the damped wave equation converges uniformly on R to a travelling front as t goes to plus infinity. The proof of this global stability result is inspired by a rec…
We study the Cauchy problem for the fractional Schrödinger equation $$ i\partial_tu = (m^2-Δ)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < α< 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = λ(\frac{ψ(\cdot)}{|\cdot|^γ} * |u|^2)u$$ with $λ= \pm1, 0 <γ< n$, and $0 \le ψ\in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $α$, $γ$, $λ$, $ψ$. We also remark on finite time blowup of solutions when $λ= -1$.
We consider the Cauchy problem for the fourth order nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R}…
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.
The paper investigates the existence of global attractors and their upper semicontinuity for a structural damped wave equation on $\mathbb{R}^{N}: u_{tt}-Δu+(-Δ)^αu_{t}+u_{t}+u+g(u)=f(x)$, where $α\in (1/2, 1)$ is called a dissipative index. We propose a new method based on the harmonic analysis technique and the commutator estimate to exploit the dissipative effect of the structural damping $(-Δ)^αu_{t}$ and to overcome the essential difficulty: "both the unbounded domain $\mathbb{R}^N$ and the supercritical nonlinearity cause that the Sobolev embedding loses its compactness"; Meanwhile we sh…
We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\R ^d)$ with $s> \max \le…
We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah--Struwe solutions, which satisfy the Strichartz estimates and are coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, uniform, and cocycle attractors and the relations between them. We also prove that these non-autonomous attractors converge upper-semicontinuously to the global attractor for the limit autonomous problem if the time-dependent nonlinearity tends to tim…
We consider the Cauchy problem for the $L^{2}$-critical damped nonlinear Schrödinger equation. We prove existence and stability of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.
We consider a Cauchy problem of energy-critical fractional Schrödinger equation with Hartree nonlinearity below the energy space. Using a method of randomization of functions on $\mathbb{R}^d$ associated with the Wiener decomposition, introduced by Á. Bényi, T. Oh, and O. Pocovnicu \cite{beohpo1,beohpo2}, we prove that the Cauchy problem is almost surely locally well-posed. Our result includes Hartree Schrödinger equation ($α= 2$).
We consider the Cauchy problem for the damped wave equations with variable coefficients a(x) having power type nonlinearity |u|^p. We discuss the global existence of solutions for small initial data and investigate the relation between the range of a(x) and the order p.
We consider the Cauchy problem for semi-linear heat equations with exponential nonlinearity. The main purpose of this paper is to prove the existence of solutions lying on the borderline between global existence and blow-up infinite time. The existence has been shown for semi-linear heat equations with power type nonlinearity. We explain the main strategy to prove the existence. By using the definition of exponential function, we approximate the solution to exponential type equation by that of power type equation. Then we can use directly the knowledge for power type equation.
We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the case when $1\leq n\leq3.$ Moreover, we derive a blow-up result under some positive data in any dimensional space.
We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. Our work assumes $(-Δ_W)^θ\partial_tu$ with $θ\in[\frac{1}{2},1)$ and where $Δ_W$ is the Wentzell-Laplacian. Hence, the associated linear operator admits a compact resolvent. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth on each potential, as well as mixed dissipative/anti-d…
In this paper, we deal with the initial value fractional damped wave equation on $G$, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on $G$. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, that is, damping and mass with the same power type nonlinearity o…
In this paper, we establish a standard $L^p$-theory of solutions to one dimensional nonlinear Schrödinger equations with the power like nonlinearity. More precisely, we extend the following three well-known results in the $L^2$ space into $L^p$ setting: 1. Large data local well-posedness for subcritical nonlinearities, 2. Small data global well-posedness for critical nonlinearities, 3. Large data global well-posedness if the subcritical nonlinearity is given by $|u|^{α-1}u$.
We study the global existence of solutions to semilinear damped wave equations in the scattering case with derivative power-type nonlinearity on (1+3) dimensional nontrapping asymptotically Euclidean manifolds. The main idea is to exploit local energy estimate, together with local existence to convert the parameter $μ$ to small one.
We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.
We consider the asymptotic behavior of the soltion to the wave equation with time-dependent damping and analytic nonlinearity. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Lojasiewicz-Simon type inequality.
In this paper, we are interested in the Cauchy problem for the viscoelastic damped wave equation with memory of type I. By applying WKB analysis and Fourier analysis, we explain the memory's influence on dissipative structures and asymptotic profiles of solutions to the model with weighted $L^1$ initial data. Furthermore, concerning standard energy and the solution itself, we establish singular limit relations between the Moore-Gibson-Thompson equation with memory and the viscoelastic damped wave equation with memory.
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 line…
We consider the strongly damped Klein Gordon equation for defocusing nonlinearity and we study the asymptotic behaviour of the energy for periodic solutions. We prove first the exponential decay to zero for zero mean solutions. Then, we characterize the limit of the energy, when the time tends to infinity, for solutions with small enough initial data and we finally prove that such limit is not necessary zero.
This paper is concerned about the lifespan estimate to the Cauchy problem of semilinear damped wave equations with the Fujita critical exponent in high dimensions$(n\geq 4)$. We establish the sharp upper bound of the lifespan in the following form \begin{equation}\nonumber\\ \begin{aligned} T(\varepsilon)\leq \exp(C\varepsilon^{-\frac 2n}), \end{aligned} \end{equation} by using the heat kernel as the test function.
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 $σ$.
In this paper we consider the Cauchy problem for linear dissipative generalized Klein-Gordon equations with nonlinear memory in the right hand side. Our goal is to study the effect of this nonlinearity on both the decay estimates of global solutions as well as the admissible range of the exponent p.
We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.
We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of the limit problem as $p>2$ varies.
We prove the global existence of small data solution in all space dimension for weakly coupled systems of semi-linear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, nonlinearity terms $ f(t,u) $ and $ g(t,v) $ satisfying some properties of the parabolic equation. We study the problem in several classes of regularity.
We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $n$ dimensional nontrapping asymptotically Euclidean manifolds, when $n=3, 4$. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.
The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.
We consider the Cauchy problem in the whole space for strongly damped Klein-Gordon equations. We derive asymptotic profles of solutions with weighted initial data by a simple method introduced by R. Ikehata. The obtained results show that the wave effect will be weak because of the mass term, especially in the low dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases.
We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.
We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations \[ i\partial_t u + Δu + i a u= \pm |u|^αu, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $α>0$. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up $H^1$ solutions to the focusing problem in the mass-critical and mass-supercritical cases.
This note is to conclude $L^1-L^1$ estimates for solutions to the following Cauchy problem for visco-elastic damped $σ$-evolution models: \begin{equation} \begin{cases} u_{tt}+ (-Δ)^σu+ (-Δ)^σu_t = 0, &\quad x\in \mathbb{R}^n,\, t \ge 0, \\ u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x), &\quad x\in \mathbb{R}^n, \label{pt1.1} \end{cases} \end{equation} where $σ> 1$, in all space dimensions $n\ge 1$.
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.
We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impact on the critical exponent of the corresponding nonlinear Cauchy problem with small initial data. The main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical Strauss exponent in 3D to the critical 5D Strauss exponent for the wave equation without damping coefficients.
In this paper we obtain higher order asymptotic profilles of solutions to the Cauchy problem of the linear damped wave equation in $\textbf{R}^n$ \begin{equation*} u_{tt}-Δu+u_t=0, \qquad u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x), \end{equation*} where $n\in\textbf{N}$ and $u_0$, $u_1\in L^2(\textbf{R}^n)$. Established hyperbolic part of asymptotic expansion seems to be new in the sense that the order of the expansion of the hyperbolic part depends on the spatial dimension.
