We study the asymptotic behaviour of an optimal control problem for the Ukawa equation in a thick multi-structure with different types and classes of admissible boundary controls. This thick multi-structure consists of a domain (the junction's body) and a large number of $\varepsilon$-periodically situated thin cylinders. We consider two types of boundary controls, namely, the Dirichlet $H^{1/2}$-controls on the bases $\Gamma_{\varepsilon}$ of thin cylinders, and the Neumann $L^2$-controls on their 'vertical' sides. We present some ideas and results concerning of the asymptotic analysis for such problems as ${\varepsilon}\to 0$ and derive conditions under which the homogenized problem can be recovered in the explicit form. We show that the mathematical description of the homogenized optimal boundary control problem is different from the original one. These differences appear not only in the control constraints, limit cost functional, state equations, and boundary conditions, but also in the type of admissible controls for the limit problem - one of them is the Dirichlet $L^2$-control, whereas the second one is appeared as the distributed $L^2$-control.

We look for series solutions of a non linear reaction diffusion equation that is similar to the K.P.P. equation by using a variant of the decomposition method. The solution is expressed through a sum of Hermite functions, and is shown to be a classical one. The obtained algorithm can be used to approximate these solutions.

We prove local higher integrability for gradients of solutions to a large class of elliptic equations of the type $\text{div} A(x,\nabla u) = 0$ by studying the deviation, in a suitable sense, of the operator $A$ to be constant with respect to $x.$

The paper studies the existence of multiple solutions to abstract equation \begin{eqnarray*} J_p u = N_f u, \end{eqnarray*} where $J_p$ is the duality mapping on a real reflexive and smooth Banach space $X$, corresponding to the gauge function $\varphi(t) = t^{p-1}, 1 < p < +\infty $. It is assumed, that $X$ is compactly imbedded in a Lebesgue space $L^q(\Omega), p \leq q < p^*$, and continuously imbedded in $L^{p^*}(\Omega)$, $p^*$ being the Sobolev conjugate exponent, $N_f :L^q(\Omega) \rightarrow L^{q'}(\Omega)$ , ${\frac{1}{q}+ \frac{1}{q'}=1 }$, being the Nemytskii operator generated by a function $f \in {\mathcal{C}}(\bar{\Omega} \times {\mathbf{R}}, {\mathbf{R}})$, which satisfies some appropriate conditions. These assumptions allow the use of many procedures appearing essentially in Li and Zhou [9]. As applications we obtain in a unitary manner the multiplicity results already given in [9] for a Dirichlet problem with $p$-Laplacian as well as some new multiplicity results for the Neumann problem \begin{eqnarray*} - \Delta_{p} u + | {u} | ^{p-2} u & =& f(x,u) \qquad \textrm{in}\quad \Omega,\\ | {\nabla u} | ^{p-2}\frac{\partial u}{\partial n} & =& 0 \qquad \textrm{on}\quad \partial \Omega. \end{eqnarray*}

In this paper, we study a new model for dislocation dynamics with a mean curvature term. The model is a non-local Hamilton-Jacobi equation. We prove a short time existence and uniqueness result for this equation. We also prove a Lipschitz estimate in space and an estimate of the modulus of continuity in time for the solution.

In this paper we investigate questions of existence of positive solutions for nonlocal problems. We study a single equation and a system of two equations.

Solid-liquid transitions in thermal insulators and weakly conducting media are modeled through a phase-field system with memory. The evolution of the phase variable ${\varphi}$ is ruled by a balance law which takes the form of a Ginzburg-Landau equation. A thermodynamic approach is developed starting from a special form of the internal energy and a nonlinear hereditary heat conduction flow of Coleman-Gurtin type. After some approximation of the energy balance, the absolute temperature ${\theta}$ obeys a doubly nonlinear ``heat equation" where a third-order nonlinearity in ${\varphi}$ appears in place of the (customarily constant) latent-heat. The related initial and boundary value problem is then formulated in a suitable setting and its well--posedness and stability is proved.

In this paper, a problem of boundary feedback stabilization for a class of parabolic equations is considered. The problem is reformulated to convert a parabolic equation into a well known one by using an integral transformation with a kernel required to satisfy an appropriate partial differential equation (PDE). The well-posedness of the kernel PDE and the smoothness of the solution are studied. By varying a parameter ${\lambda}>0$ in the kernel PDE, the solution is exploited to explicitly construct a boundary feedback law such that the solution of the closed loop system decays exponentially at the desired rate of ${\lambda}.$ Moreover, a control which, simultaneously, stabilizes the output function and minimizes an appropriate cost functional is derived.

We study a degenerate scalar conservation law on a bounded domain with non homogeneous boundary condition: $b(v)_t + \mbox{div } \Phi(v)=f $ on $Q:= (0,T) \times \Omega$, $v(0,\cdot)=v_0$ on $\Omega$ and $v=a$ on the boundary $(0,T) \times \partial \Omega$. The function $b$ is assumed to be continuous nondecreasing and to verify the normalization condition $b(0)=0.$ Existence and uniqueness of a renormalized entropy solution is established for any $\Phi \in C({{\mathbb R}};{{\mathbb R}}^N)$, $v_0\in L^\infty(\Omega)$, $f \in L^\infty(Q)$, and boundary data $a\in L^\infty(\Sigma).$

In the present paper we prove existence results for a class of nonlinear elliptic equations whose prototype is: $$ - {\rm div} \left( \left\vert { \nabla u} \right\vert ^{{ p-2}}{ \nabla u\varphi (x)}\right) { +b(x)}\left\vert { \nabla u}\right\vert ^{ { \sigma }}{ \varphi (x)=g\varphi ,} $$ where $ \Omega $ is an open set, $ u=0 $ on $ \partial \Omega , $ where the function $ \varphi (x)=(2\pi )^{-\frac{n}{2}}$ $\exp \left( -\left\vert x\right\vert ^{2}/2\right) $ is the density of Gauss measure and $ g \! \in \! L^{r}(\log L)^{-\frac{1}{2}}(\varphi ,\Omega )$ for $ 1 < r < p^{\prime }.$