In this paper, we study the Stokes system in the half-space $\mathbb{R}^n_+$, with $n {\geqslant} 2$. We consider data and give solutions which live in weighted Sobolev spaces, for a whole scale of weights. We start to study the kernels of the biharmonic and Stokes operators. After the central case of the generalized solutions, we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument.

We consider the structure-interaction model, introduced by J.-L. Lions, describing the interaction of an elastic body and an incompressible fluid. Recently, many works have addressed well posedness for this model. In this paper, we prove local existence of strong solutions under the initial condition $(u_0,w_0,w_1)\in H^{1}\times H^{3/2+k}\times H^{1/2+k}$ for every $k>0$ sufficiently small and where $u_0$, $w_0$, and $w_1$ are the initial velocity of the fluid, initial displacement of the body, and the initial velocity of the body respectively. We also propose new alternative matching stress boundary conditions for this model.

In this paper, we show that critical points of the one-dimensional Ambrosio-Tortorelli functional with fidelity term converge to those of the corresponding Mumford-Shah functional, a famous model for image segmentation. Equi-partition and convergence of the energy-density are also derived.

We consider non-negative solutions to the Dirichlet problem of a semilinear heat equation with localized reaction in $\Omega$: $u_t = \Delta u +f(u(x_0(t),t))$, where $\Omega$ is a smooth bounded domain, $x_0(t)$ is a locally Hölder continuous function from $[0,\infty)$ into $\Omega$ and $f$ satisfies $f(0)=f'(0)=0$ and some blow-up condition. We show that, if $x_0(t)$ remains in some compact subset of $\Omega$ as $t\to\infty$, then all global solutions are bounded in $\Omega\times(0,\infty)$ and, if $x_0(t)$ approaches the boundary of $\Omega$ as $t\to \infty$, then some unbounded global solution (infinite time blow-up solution) exists. These results are parts of our main results on the classification of all solutions.

In this paper we study the dynamics in the semiclassical limit of some classes of solutions for an energy-subcritical focusing nonlinear Schrödinger equation in the presence of an electric potential. We prove the existence of a symmetric configuration of two solitons rotating around a common pole. The existence of such solutions derives mainly from the balance between the attracting force of the two solitons and the centrifugal force due to the rotation and not from the trapping effect of the potential.

We give a new example of nonuniqueness of weak solutions of the initial-boundary-value problem for the heat flow of director fields in an infinitely long cylinder in $\mathbb R^{3}$. The example confirms the connection between nonuniqueness of axially symmetric solutions for the harmonic map heat flow and the occurrence of point singularities in the solutions. The result is compared with earlier nonuniqueness results. Traveling wave solutions are used as barrier functions.

In 1966, Fujita started the close investigation of the blow-up phenomena arising in some semilinear parabolic problems and found the so-called Fujita exponent. Later, Galaktionov and Levine researched nonlinear parabolic equations involving $p$-Laplace operators with nonlinear boundary conditions. In the present paper we extend their results to a non-Newtonian filtration system coupled via nonlinear boundary conditions. In terms of a characteristic algebraic system introduced for the problem, we obtain a clear and simpler presentation of both the critical Fujita exponent and the blow-up rate for this complicated coupled system containing six nonlinear exponent parameters. The proof for establishing the critical exponents is based on careful constructions of the comparison functions, especially for the blow-up case, by using the Barlenbratt-type solutions. The analysis of the blow-up rate relies on the appropriate scale transformation of the independent and the dependent variables with some differential inequalities satisfied by the $L^\infty$-norm of blowing up solutions. This paper provides a complete result for such a coupled degenerate parabolic system.