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We consider a class of abstract second order evolution equations with a restoring force that is strictly superlinear at infinity with respect to the position, and a dissipation mechanism that is strictly superlinear at infinity with respect…
We consider a degenerate abstract wave equation with a time-dependent propagation speed. We investigate the influence of a strong dissipation, namely a friction term that depends on a power of the elastic operator. We discover a threshold…
This paper develops a characterisation of when solutions of forced second order linear differential equations converge to the zero solution of the asymptotically stable and unforced second order equation, or when the solution is bounded,…
We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value…
We consider an abstract second order evolution equation with damping. The "elastic" term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We…
For general second order evolution equations, we prove an optimal condition on the degree of unboundedness of the damping, that rules out finite-time extinction. We show that control estimates give energy decay rates that explicitly depend…
We prove sharp $L^p$ estimates for a singular transport equation by building what we call a \emph{cascading solution}; the equation studies the combined effect of multiplying by a bounded function and application of the Hilbert transform.…
We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover…
We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates…
We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing…
We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form $\alpha/x$, $\alpha>0$. We establish the exponential stability of the semigroup for all…
We derive explicit pointwise bounds for the spatial derivative $\left| \frac{\partial V}{\partial x} \right|$ of solutions to linear parabolic PDEs with Neumann boundary conditions. The bound is fully explicit in the sense that it depends…
We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive…
We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real…
The global existence of smooth solutions to the vacuum free boundary problem with physical singularity of compressible Euler equations with damping and gravity is proved in space dimensions $n=1, 2, 3$, for the initial data being small…
We consider a class of viscous fluids with a general monotone dependence of the viscous stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to the associated initial boundary value problem inspired by…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a…
Local behaviors near boundary are analyzed for solutions of the Stokes and Navier-Stoke equations in the half space with localized non-smooth boundary data. We construct solutions of Stokes equations whose velocity field is not bounded near…
We investigate the large-time dynamics of solutions of multi-dimensional reaction-diffusion equations with ignition type nonlinearities. We consider solutions which are in some sense locally persistent at large time and initial data which…