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We provide some new integral estimates for solutions to Hamilton-Jacobi equations and we discuss several consequences, ranging from $L^p$-rates of convergence for the vanishing viscosity approximation to regularizing effects for the Cauchy…

We study the local properties of positive solutions of the equation $-\Delta u+ m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $\Omega\setminus\{0\}$ of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly…

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption $$ \partial_{t}u-\Delta u^m+u^q=0 \quad \quad \hbox{in} \ (0,\infty)\times\real^N\, $$ with…

We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-\Delta u= |\nabla u|^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known…

This paper is devoted to study a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under suitable conditions, we prove that any weak solutions with negative initial energy…

We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a…

We exhibit a sufficient condition in terms of decay at infinity of the initial data for the finite time blowup of strong solutions to the Camassa--Holm equation: a wave breaking will occur as soon as the initial data decay faster at…

We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $\ren$ for times up to the extinction time $T>0$.…

We study a system of semilinear wave equations satisfying the weak null condition, which can be regarded as a simplified model for the Einstein vacuum equations. The main objective is to establish precise pointwise decay estimates, as both…

In this paper, a two-component variant of the Degasperis-Procesi equation on the real line is discussed. Applying Kato's theory, we first prove the local well-posedness for the equation under consideration in $H^s\times H^{s-1}$, for $s\geq…

We show that, in the periodic homogenization of uniformly elliptic Hamilton-Jacobi equations in any dimension, the effective Hamiltonian does not necessarily inherit the quasiconvexity property (in the momentum variables) of the original…

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi Equation with Neumann boundary condition and initial data a continious function. Then, we study the large time behavior of the solutions.

We study the degenerate and singular parabolic equation with a forcing term \[ |x|^{\sigma_1}u_t = \Delta u + |x|^{\sigma_2}|u|^p + t^\varrho \mathbf{w}(x), \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, \] where $N\ge 2$,…

We study the large time behavior of solutions to the porous medium equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u^m, \quad \hbox{in} \ \real^N\times(0,\infty), $$ where $m>1$ and $N\geq3$.…

We investigate the local existence, finite time blow-up and global existence of sign-changing solutions to the inhomogeneous parabolic system with space-time forcing terms $$ u_t-\Delta u =|v|^{p}+t^\sigma w_1(x),\,\, v_t-\Delta v…

We prove H\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x|^{2+\beta}}u=0,\,\,(\beta\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01}…

We consider positive solutions for the fractional heat equation with critical exponent \begin{equation*} \begin{cases} u_t = -(-\Delta)^{s}u + u^{\frac{n+2s}{n-2s}}\text{ in } \Omega\times (0, \infty), u = 0\text{ on }…

We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order…

Let $s \in (0, 1]$ and $N > 2s$. It is known that positive solutions to the (fractional) fast diffusion equation $\partial_t u + (-\Delta)^s (u^\frac{N-2s}{N+2s}) = 0$ on $(0, \infty) \times \mathbb R^N$ with regular enough initial datum…

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on…