We prove the existence of infinitely many radial solutions for a system of equations in $\mathbb R^3$. Numerically, that is supposed to give the profile of an asymptotic self-similar blow-up solution of the Zakharov system in dimension three. For this, we use several techniques of ordinary differential equations and especially a kind of shooting method. Moreover, we give some properties of solutions, monotonicity, estimates at infinity and integral relations.

We consider a reaction-diffusion-advection system which is a model for chemotaxis with growth. An appropriate singular limit of this system yields a free-boundary problem where the interface motion depends on the mean curvature and on some nonlocal term. We prove local-in-time existence, uniqueness and regularity for this free-boundary problem and investigate some qualitative properties (lack of monotonicity, loss of convexity). We then establish the convergence of the solution of the reaction-diffusion-advection system to the solution of the free-boundary problem.

A {\em nonresonance result} is obtained for the Dirichlet boundary value problem $$ (P)\qquad\begin{cases}-(\phi(u'))'=f(u)+e(t), \quad t\in(0,T),\ T>0, \\ u(0)=u(T)=0, \end{cases} $$ where $\phi$ is an odd, increasing homeomorphism from $\mathbb R$ onto $\mathbb R$, $f:\mathbb R \to \mathbb R$ is a continuous function satisfying some resonance condition and $e \in L^\infty [0,T]$. This result is obtained by using a time-map approach and topological degree.

We study a system of parabolic equations introduced by E.F. Keller and L.A. Segel to describe the chemotactic feature of slime molds. Concentration toward the boundary is shown for the blowup solution with the total mass less than $8\pi$. For this purpose, a variant of the concentration lemma of Chang and Yang's type, and also a parabolic version of an inequality due to Brezis and Merle, are provided.