Counterexamples for Uniqueness in Strongly Coupled Parabolic Systems.
Counter-Examples to Boundary Estimates for Two-Dimensional Elliptic Systems.
Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary
In this paper we consider a smooth and bounded domain of dimension with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains...
Coupled string-beam equations as a model of suspension bridges
We consider nonlinearly coupled string-beam equations modelling time-periodic oscillations in suspension bridges. We prove the existence of a unique solution under suitable assumptions on certain parameters of the bridge.
Crank-Nicolson-Galerkin approximation of the periodic solutions of weakly nonlinear parabolic equations
C?-Regularity for the Porous Medium Equation.
Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition
Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties by using assumptions...
Critical boundary constants and Pohozaev identity
Critical convective-type equations on a half-line.
Critical exponent and minimization problem in
Critical global asymptotics in higher-order semilinear parabolic equations.
Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data
Critical Neumann problem with competing Hardy potentials.
Critical nonlinear elliptic equations with singularities and cylindrical symmetry
Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in RN with cylindrical symmetry and with singularities on a whole subspace of RN. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.
Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws
Critical point theory applied to a class of the systems of the superquadratic wave equations.
Critical points for reaction-diffusion system with one and two unilateral conditions
We show the location of so called critical points, i.e., couples of diffusion coefficients for which a non-trivial solution of a linear reaction-diffusion system of activator-inhibitor type on an interval with Neumann boundary conditions and with additional non-linear unilateral condition at one or two points on the boundary and/or in the interior exists. Simultaneously, we show the profile of such solutions.
Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case
Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.
Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case
Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.
Critical points of solutions of elliptic equations in two variables