Messaouda Ben Attia, Elmehdi Zaouche, Mahmoud Bousselsal
By choosing convenient test functions and using the method of doubling variables, we prove the uniqueness of the solution to a nonlinear evolution dam problem in an arbitrary heterogeneous porous medium of IR^n with an impermeable horizontal bottom.
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We establish uniqueness of the solution of the unsteady state dam problem in the heterogeneous and rectangular case assuming the dam wet at the bottom and dry near to the top.
This paper is concerned with an uniqueness of solution of the weak formulation of an evolution dam problem related to a compressible fluid flow through a two-dimensional, rectangular and heterogeneous porous medium. Note that our problem associated with the equation a(x_1)(u_{x_2}+χ)_{x_2}-(u+χ)_t=0. Our technique is based on the idea that we transform the weak form of this equation into a similar situation to the proof of the uniqueness in the incompressible case (see [12]). It is also difficult to adapt the proof obtained in [12] by using some properties of the solutions as in [12, Sect. 2].
Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements
本文研究时间分数阶阻尼波动方程的反系数问题，通过Dirichlet-to-Neumann映射唯一确定分数阶阻尼系数和零阶势，利用卷积型积分恒等式和高频波解渐近分析证明唯一性。
In this paper we prove the existence and uniqueness of strong solution of the incompressible Navier-Stokes equations with damping $α(e^{β|u|^2}-1)u$.
The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, …
Quantitative Uniqueness and Rough Damping on $\mathbb T^2$
受Burq和Gérard猜想启发，研究$\text{T}^2$上固定宽度环带谱函数的定量唯一性，获一致可观测性估计，推出粗糙阻尼波动方程指数衰减，还建立凸多边形边界伸缩谱的类似不确定性原理，阻尼仅需$L^\boldsymbol{\text{∞}}$正则性。
We consider a second-order hyperbolic equation on an open bounded domain $Ω$ in $\mathbb{R}^n$ for $n\geq2$, with $C^2$-boundary $Γ=\paΩ=\bar{Γ_0\cupΓ_1}$, $Γ_0\capΓ_1=\emptyset$, subject to non-homogeneous Neumann boundary conditions on the entire boundary $Γ$. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit sub-portion $Γ_1$ of the boundary $Γ$, and over a computable time interval $T>0$. Under sharp conditions on the complementary pa…
The present paper is devoted to the study of existence, uniqueness and stability of transition fronts of nonlocal dispersal evolution equations in time heterogeneous media of bistable type under the unbalanced condition. We first study space nonincreasing transition fronts and prove various important qualitative properties, including uniform steepness, stability, uniform stability and exponential decaying estimates. Then, we show that any transition front, after certain space shift, coincides with a space nonincreasing transition front (if it exists), which implies the uniqueness, up to space …
In this paper we analyse a one-dimensional debonding model for a thin film peeled from a substrate when viscosity is taken into account. It is described by the weakly damped wave equation whose domain, the debonded region, grows according to a Griffith's criterion. Firstly we prove that the equation admits a unique solution when the evolution of the debonding front is assigned. Finally we provide an existence and uniqueness result for the coupled problem given by the wave equation together with Griffith's criterion.
In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coefficients.
In this article, we consider the inverse problems of determining the damping coefficient appearing in the wave equation. We prove the unique determination of the coefficient from the data coming from a single coincident source-receiver pair. Since our problem is under-determined, so some extra assumption on the coefficient is required to prove the uniqueness.
We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that the uniqueness holds at the origin provided that the initial condition is appropriately chosen. We show that the choice of the initial condition leading to uniqueness is related to a fine version of unique continuation property for elliptic operators. We also establish a Lipschitz directional stability estimate at the origin, which is obtained by a linearization process.
Optimal backward uniqueness and polynomial stability of second order equations with unbounded damping
针对无界阻尼二阶方程，给出排除有限时间熄灭的最优条件，通过控制估计与伸缩方法导出显式衰减率，并应用于奇异阻尼、重力水波和欧拉-伯努利板等系统。
Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments
本文研究了带logistic源的抛物型-抛物型趋化模型在异质有界域中的渐进行为，证明了在特定参数条件下系统存在唯一的正整体解，并且该解全局渐近稳定。
Generalization of 3D-NSE Global Weak Solution with damping
本文推广了带阻尼项的3D不可压缩Navier-Stokes方程，证明了其全局弱解的存在性、唯一性及在L^2中的连续性，其中阻尼函数为递增凸可微且零点为零。
In this paper, we study the $L^p$-asymptotic stability with $p\in (1,\infty)$ of the one-dimensional nonlinear damped wave equation with a localized damping and Dirichlet boundary conditions in a bounded domain $(0,1)$. We start by addressing the well-posedness problem. We prove the existence and the uniqueness of weak solutions for $p\in [2,\infty)$ and the existence and the uniqueness of strong solutions for all $p\in [1,\infty)$. The proofs rely on the well-posedness already proved in the $L^\infty$ framework by [4] combined with a density argument. Then we prove that the energy of strong s…
Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities
本文研究了一维非均质两相流模型，其中毛管压力函数在空间上可以是不连续的。首先给出一个模型，导出一个由非线性透射条件空间耦合的退化非线性抛物型方程组。我们近似的解决方案，我们的问题感谢一个单调的有限体积计划。证明了当离散步数趋于0时，基本离散解收敛到一个弱解。我们还表明，在初始数据的假设下，一个统一的估计通量，然后使用的唯一性证明。密度参数允许我们放松对初始数据的假设，并将存在唯一性框架扩展到作为近似极限获得的解族。最后给出了一个数值例子来说明模型的性能。
We study the sharp constant for the embedding of $W^{1,p}_0(Ω)$ into $L^q(Ω)$, in the case $2<p<q$. We prove that for smooth connected sets, when $q>p$ and $q$ is sufficiently close to $p$, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace--type equations by L. Damascelli an…
Existence and Uniqueness for Double-Phase Poisson Equations with Variable Growth
本文研究变指数双相泊松方程，通过模函数空间建立梯度模的一致凸性，在自然最小假设下证明了弱解的存在性与唯一性。
On the wave equation with logarithmic damping: wellposedness, blow-up and numerical analysis
研究具有对数型非线性阻尼和多项式源项的波动方程，证明了问题的适定性和唯一性，在初始能量为负且源项指数临界时建立了爆破结果，并进行了数值验证。
We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The leading idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form. The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with a diffusion equation, accounting for the presence of the pathological zone in the heart tissue.
We revisit the problem of recovering wave speeds and density across a curved interface from reflected wave amplitudes. Such amplitudes have been exploited for decades in (exploration) seismology in this context. However, the analysis in seismology has been based on linearization and mostly flat interfaces. Here, we present a nonlinear analysis allowing curved interfaces, establish uniqueness and provide a reconstruction, while making the notion of amplitude precise through a procedure rooted in microlocal analysis.
We study a nonlinear wave equation on finite connected weighted graphs. Using Rothe's and energy methods, we prove the existence and uniqueness of solution under certain assumption. For linear wave equation on graphs, Lin and Xie \cite{Lin-Xie} obtained the existence and uniqueness of solution. The main novelty of this paper is that the wave equation we considered has the nonlinear damping term $|u_t|^{p-1}\cdot u_t$ ($p>1$).
Nonlinear biphasic mixture model: existence and uniqueness results
本文研究体外实体瘤内孔隙弹性流体动力学的双相混合模型，将水力电阻率设为各向异性且非均匀，导出弱形式并构建等价不动点问题，结合Galerkin方法与不动点定理证明了强耦合非线性系统的存在性与唯一性。
The inverse source problem where an unknown source is to be identified from the knowledge of its radiated wave is studied. The focus is placed on the effect that multi-frequency data has on establishing uniqueness. In particular, it is shown that data obtained from finitely many frequencies is not sufficient. On the other hand, if the frequency varies within a set with an accumulation point, then the source is determined uniquely, even in the presence of highly heterogeneous media. In addition, an algorithm for the reconstruction of the source using multi-frequency data is proposed. The algori…
Scalar conservation laws with discontinuous flux: existence and uniqueness
针对具有间断通量的标量守恒律柯西问题，研究了两类间断通量情形下的适定性，构造稳定解半群并建立适定求解框架，证明了解的存在性与唯一性。
We consider a simplified model of a two-phase flow through a heterogeneous porous medium, in which the convection is neglected. This leads to a nonlinear degenerate parabolic problem in a domain shared in an arbitrary finite number of homogeneous porous media. We introduce a new way to connect capillary pressures on the interfaces between the homogeneous domains, which leads to a general notion of solution. We then compare this notion of solution with an existing one, showing that it allows to deal with a larger class of problems. We prove the existence of such a solution in a general case, th…
We consider an initial-boundary value problem for the Maxwell's system in a bounded domain with a linear inhomogeneous anisotropic instantaneous material law subject to a nonlinear Silver-Muller-type boundary feedback mechanism incorporating both an instantaneous damping and a time-localized delay effect. By proving the maximal monotonicity property of the underlying nonlinear generator, we establish the global well-posedness in an appropriate Hilbert space. Further, under suitable assumptions and geometric conditions, we show the system is exponentially stable.
We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $Ω$ for any $t\ge T_\star,$ for some $T_\star>0)$ for solutions of the nonlinear Schrödinger problem ${\rm i} u_t+Δu+a|u|^{-(1-m)}u=f(t,x),$ on a bounded domain $Ω$ of $\mathbb{R}^N,$ $N\le 3,$ $a\in\mathbb{C}$ with $\Im(a)>0$ (the damping case) and under the crucial assumptions $0<m<1$ and the dominating condition $2\sqrt m\,\Im(a)\ge(1-m)|\Re(a)|.$ We use an energy method as well as several a priori estimates to prove the main conclusion. The presence of the non-Lipschitz nonlinear term in the equation introduces a lack…
Mathematical analysis of fully coupled approach to creep damage
我们证明了一个经典的蠕变损伤问题解的存在唯一性。我们制定了一个充分条件的问题有一个独特的光滑的解决方案，当地的时间。该条件是根据给定数据的平滑度来陈述的，例如实体几何形状、边界条件、施加的载荷和初始条件。反例与任意小的寿命的结构也给出了，机械的解释强加的光滑条件。所提出的理论给出了一个严格的框架，应变局部化分析。在此框架内的损伤梯度的应变局部化过程的影响的特点和损伤局部化的措施提出。
$L^p$ asymptotic stability of 1D damped wave equation with nonlinear distributed damping
本文在$L^p$框架下研究带局域非线性阻尼的一维波动方程，证明了解的存在唯一性，给出几乎最优的能量衰减估计，结合多种方法完成证明。
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