This work is devoted to the analysis of a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2-D domain. The damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero as time goes to infinity. Using discrete multiplier techniques, we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size)...

In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial...

We prove unique continuation for solutions of the inequality $\left|\Delta u\right(x\left)\right|\le V\left(x\right)\left|\nabla u\right(x\left)\right|$, $x\in \Omega$ a connected set contained in ${\mathbf{R}}^n$ and $V$ is in the Morrey spaces $F^{\alpha ,p}$, with $p\ge (n-2)/2(1-\alpha )$ and $\alpha \<1$. These spaces include $L^q$ for $q\ge (3n-2)/2$ (see [H], [BKRS]). If $p=(n-2)/2(1-\alpha )$, the extra assumption of $V$ being small enough is needed.

Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of $W$ concentrating at the origin.

In this paper, we consider one-dimensional wave equation with real-valued square-summable potential. We establish the long-time asymptotics of solutions by, first, studying the stationary problem and, second, using the spectral representation for the evolution equation. In particular, we prove that part of the wave travels ballistically if q ∈ L2(ℝ+) and this result is sharp.

We investigate a generalized class of fractional hemivariational inequalities involving the time-fractional aspect. The existence result is established by employing the Rothe method in conjunction with the surjectivity of multivalued pseudomonotone operators and the properties of the Clarke generalized gradient. We are also exploring a numerical approach to address the problem, utilizing both spatially semi-discrete and fully discrete finite elements, along with a discrete approximation of the fractional...

L’equazione $u_{tt}={\sum }_{ij=1}^n{(a_{ij}(x,t)u_{x_j})}_{x_i}$ in condizioni di debole iperbolicità $\left({\sum }_{ij=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\ge 0\right)$, è ben posta negli spazi di Gevrey ${\gamma }_{loc}^{(s)}$ con , purché $a_{ij}$ sia di Gevrey in $x$ di ordine $s$ e risulti $$\left[\sum _{ij=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\right]{}^{1/\sigma }\in BV([0,T]:{𝐋}_{loc}^{\mathrm{\infty }})$$

We prove that the Cauchy problem for a class of hyperbolic equations with non-Lipschitz coefficients is well-posed in ${𝒞}^{\infty }$ and in Gevrey spaces. Some counter examples are given showing the sharpness of these results.