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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…

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and…

In this paper, we would like to study the critical exponent for semi-linear damped wave equations with power nonlinearity and the initial data belonging to Sobolev spaces of negative order $\dot{H}_m^{-\gamma}$. Precisely, we obtain a new…

We prove that the 3D stable Muskat problem is globally well-posed in the critical Sobolev space $\dot H^2 \cap \dot W^{1,\infty}$ provided that the semi-norm $\Vert f_0 \Vert_{\dot H^{2}}$ is small enough. Consequently, this allows the…

Given a compact Riemannian Manifold (M,g) of dimension n > 2, a point x_0 in M and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. The Hardy-Sobolev embedding yields the existence of A,B > 0 such that…

We give a sufficient criterion, which we call stability, for a coarse Lipschitz map $f$ from a complete manifold $X$ with Ricci curvature bounded below to a proper Hadamard space $Y$ to be within bounded distance of a harmonic map. We prove…

We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted…

We consider a class of defocusing energy-supercritical nonlinear Schr\"odinger equations in four space dimensions. Following a concentration-compactness approach, we show that for $1<s_c<3/2$, any solution that remains bounded in the…

In this paper, we first establish a kind of weighted space-time $L^2$ estimate, which belongs to Keel-Smith-Sogge type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the…

In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in dimensions two. Building on such a regularity, we then establish the existence of a global weak solution to the…

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform…

In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( \Omega ,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this…

In this paper, we consider a wave equation with strong damping and logarithmic nonlinearity. This paper aims to study the local and global existence, uniqueness and the uniform energy decay rate of a weak solution under some sufficient…

This paper establishes the global existence of solutions for a class of wave-Klein-Gordon coupled systems with specific nonlinearities in 3+1-dimensional Minkowski spacetime. The study demonstrates that imposing certain constraints on the…

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $L^2$-based…

Let $X, Y \subset \mathbb{R}^n$ be Lipschitz domains, and suppose there is a homeomorphism $\varphi \colon \overline{X} \to \overline{Y}$. We consider the class of Sobolev mappings $f \in W^{1,n} (X, \mathbb{R}^n)$ with a strictly positive…

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend…

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions…

We study global existence and decay estimates for quasilinear wave equations with dissipative terms in the Sobolev space $H^L \times H^{L-1}$, where $L \geq [d/2]+3$. The linear dissipative terms depend on space variable coefficient, and…

We construct a gauge theoretic change of variables for the wave map from $R \times R^n$ into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global…