We consider the Cauchy problem for semilinear parabolic equations with strongly singular initial data and nonlinear terms with superlinear or sublinear growth at infinity. We show, under a certain link between the growth at infinity of the nonlinear term and the order of the maximal singularity of the initial data, existence and uniqueness theorems for local and global solutions. For this we introduce anisotropic weighted Hölder type spaces, following T. Kato in[16]. We examine the regularity up to the initial plane of these solutions.

We consider a class of semilinear diffractive-diffusion systems of the form $$ \begin{cases} -d_{A_{i}}\Delta u_i = f_{A_{i}}(u) & x\in A \\ -d_{B_{i}}\Delta \tilde u_i = f_{B_{i}}(\tilde u) &x\in B \\ u_i=\tilde u_i & \hbox{on } \partial A \\ d_{A_{i}} { \partial u_i\over \partial\eta_A} = d_{B_{i}} { \partial \tilde u_i \over \partial \eta _A}& \hbox{on } \partial A \\ \tilde u_i = g_i& \hbox{on } \partial B\backslash \partial A \end{cases} \tag{P} $$ where $A$ and $B$ are smooth bounded domains in $\Re^n$ such that there exists a smooth bounded domain $\Omega\subseteq \Re^n$ so that $A$ is a strict subdomain of $\Omega$ and $\bar A\cup B =\Omega$. We assume that $d_{A_{i}}$, $d_{B_{i}}>0$, each $g_i$ is nonnegative and smooth, and $f_A = (f_{A_{i}})$ and $f_B=(f_{B_{i}})$ are locally Lipschitz vector fields which are quasi-positive, nearly balanced, and polynomial bounded. We prove that these conditions guarantee the existence of a nonnegative solution of (P) for the case of $n=2$. In addition, for the case of $n=3$, we show that nonnegative solutions of (P) exist provided that $f_A$, $f_B$ satisfy a quadratic intermediate sum property. In particular, our results imply that, for space dimensions $n=2,3$, if (P) arises from standard balanced quadratic mass action kinetics, then nonnegative solutions of (P) are guaranteed. We apply our results to two multicomponent chemical models.

We devote this work to the long time behaviour of the solution to the incompressible Navier-Stokes equations for two viscous immiscible fluids contained in a bounded domain and subjected only to gravity forces. When there is surface tension at the interface or not, for the model linearized around the steady-state of minimal energy or for the standard nonlinear model, we investigate the following question. Do the equations reproduce the behaviour expected from experiment, namely a convergence to zero of the velocity field, and a convergence of the interface to its stable position. Our results show a wide variety of behaviours, depending on the case considered.

In this paper we study flows of viscoelastic weakly compressible fluids having a differential constitutive equation. In both cases, Jeffreys and Maxwell type constitutive equations, we establish a result of existence and uniqueness of solutions. We also show that, when the compressibility goes to zero, then the weakly compressible steady solution goes to the incompressible one.