Two initial--boundary-value problems for the Korteweg--de~Vries equation in a half-strip with two boundary conditions and in a bounded rectangle are considered and results on local and global well-posedness of these problems are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a ``boundary potential" type for this linearized KdV equation.

In this paper, we consider the Klein-Gordon-Schrödinger system with the higher-order Yukawa coupling in $ \mathbb{R}^{1+1} $, and prove the local and global well-posedness in $L^2\times H^{1/2}$. The method to be used is adapted from the scheme originally by J. Colliander, J. Holmer, and N. Tzirakis [8] to use the available $L^2$ conservation law of $u$ and control the growth of $n$ via the estimates in the local theory.

Here, we prove Hölder continuity of the state function in a shape-optimization problem arising, for instance, in the electromagnetic shaping of a liquid metal. The equilibrium state is obtained as minimizing the total energy, which is given in the form $$ \mathcal{E}_\lambda(\Omega) = J(\Omega) + P(\Omega) + \lambda ||\Omega| - m| , $$ where $\Omega$ is the domain occupied by the liquid, $P(\Omega)$ and $|\Omega|$ being respectively the perimeter and the Lebesgue measure of $\Omega$. We prove that the state function associated with the optimal shape is $\frac{1}{2}$-H\"older continuous, the main difficulty coming from the fact that the state function is not assumed to be non-negative.

We construct solutions of a supercritical parabolic equation which blow up in finite time but possess multiple global continuations (as weak solutions).

In this paper, we establish some results to show the existence and uniqueness of classical solution of the non-autonomous second-order integrodifferential equation of the type: \begin{eqnarray*} u''(t) & = & A(t)\;[u(t)+\int_0 ^{t} k_{1}(t,s)u(s)ds]'+B(t)\;[u(t)+\int_0 ^{t} k_{2}(t,s)u(s)ds]\\ & & +f(t),\; 0\leq t \leq T, \\ u(0) & = & u_{0} \in E,\quad u'(0)=u_{1} \in E, \end{eqnarray*} on a Banach space $E$, by means of the matrix operator method.

This paper is concerned with non-linear parabolic integrodifferential equations arising in continuum mechanics, phase-field models, and elsewhere where memory terms are important. More precisely, we will consider a non-linear initial-value problem depending on a small parameter and related to a uniformly elliptic second-order differential operator $A$. The integrodifferential character of our problem is expressed by a convolution term multiplying $A$, the scalar kernel approximating a delta-type function. Consequently, the limit problem reduces to a nonlinear parabolic differential problem involving a multiple of operator $A$. Two different non-linearities, which are locally Lipschitz-continuous in suitable metrics, are considered. The basic aim of the paper consists in determining the rate of convergence of the approximating solutions to the exact one. The results proved in the linear case (cf. [10]) are used here as starting points to solve our problems.