We investigate a two-phase free-boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function $u(x,t)$ defined in a domain ${{\mathcal D}}\subset {\mathbb R}^N\times(0,T)$ which satisfies the equation $$ \Delta u+\sum a_i\,u_{x_i}-u_t=0\quad $$ whenever $ u(x,t)\ne 0$, i.e., in the subdomains ${{\mathcal D}}_+=\{(x,t)\in {{\mathcal D}}: u(x,t)>0\}$ and ${{{\mathcal D}}}_-=\{(x,t)\in {{\mathcal D}}: u(x,t) <0\}$. Besides, we assume that both subdomains are separated by a smooth hypersurface, the free boundary, whose normal is never time-oriented and on which the following conditions are satisfied: $$ u=0 ,\quad |\nabla u^+|^2 - |\nabla u^-|^2 =2 M. $$ Here $M>0$ is a fixed constant, and the gradients are spatial side-derivatives in the usual two-phase sense. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of ${{\mathcal D}}$. The problem admits classical solutions only for good data and for small times. To overcome this problem several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. Continuing the work done for the one-phase problem we investigate conditions under which the three concepts agree and produce a unique solution for the two-phase problem.

The local controllability of $H^3$ steady-state solutions to 3-D Navier--Stokes equations with locally distributed control in $\Omega$ is established. This extends a recent result of O.Yu. Imanuvilov [8].

The Korteweg--de Vries equation occurs as a model for unidirectional propagation of small amplitude long waves in numerous physical systems. The aim of this work is to propose a well-posed mixed initial--boundary value problem when the spacial domain is of finite extent. More precisely, we establish local existence of solutions for arbitrary initial data in the Sobolev space $H^1$ and global existence for small initial data in this space. In a second step we show global strong regularizing effects.

We study properties of the solutions of two semilinear parabolic equations, the singular Zhang equation and its hydrodynamic version, $$ u_t=u_{xx}+\ln|u_x|\quad {\rm and} \,\,\,v_t = v_{xx} + {v_x}/v , $$ which exhibit strong singularities on the sets $\{u_x=0\}$ and $\{v = 0\}$ respectively. Using the concept of maximal solutions constructed by a regular monotone approximation, we prove existence and boundedness of such solutions. We show that for some classes of initial data, the solutions become instantly entirely singular: $v(t) \equiv 0$ and $u(t)\equiv -\infty$ for any arbitrarily small $t>0$. In general, we establish that for both equations the solutions cannot satisfy the initial condition in any weak sense; i.e., the corresponding semigroups are not continuous at $t=0$. Such discontinuous semigroups are exhibited by a wide class of singular nonlinear parabolic equations.

We establish an analogue of the Sobolev critical exponent for the inclusion $V^p(a)\hookrightarrow L^q(b)$, where $a$ and $b$ are weight functions, $V^p(a)$ is a weighted Sobolev space, and $L^q(b)$ is a weighted Lebesgue space. We use this result to study existence of radial solutions to the problem with weights $$(D_w)\quad\quad\begin{cases} -{\rm div}(\tilde{a}(|x|)|\nabla u|^{p-2}\nabla u)= \tilde{b}(|x|)|u|^{q-2}u \qquad \mbox{in} \quad \Omega\subset \mathbb R^N ,\\~~u=0\qquad \mbox{on}\quad \partial\Omega, \end{cases} $$ where $\Omega$ is a ball, $1 <p <q,$ and $\tilde{a}(|x|)=|x|^{1-N}a(|x|),$ $\tilde{b}(|x|)=|x|^{1-N}b(|x|)$. We are interested in the interplay between $q$ and a suitable critical exponent and its consequences for the existence and nonexistence of positive solutions of problem $(D_w).$