We study in this article the mathematical formulation of the planetary geostrophic (PG) equations of large-scale ocean circulation, in the case where small-scale processes are parameterized by the traditional Laplacian eddy diffusion and eddy viscosity. We prove the existence and uniqueness of global in time strong solutions of these equations with either $L^\infty$ or $H^2$ initial data. Due essentially to the high nonlinearity (comparable to a squared gradient) of the equations, two problems remain open. First, the existence of more regular solutions with $L^\infty \cap H^1$ initial data is still unknown, although more regular solutions are obtained with $H^2$ initial data. Second, the existence of global attractor and its dimension estimates are open, and related to that are the time uniform boundedness of the norm in $H^2$ and higher order Sobolev spaces of the solutions.

We determine nontrivial solutions of some semilinear and quasilinear elliptic problems on $\mathbb R^n$; we make use of two different nonsmooth critical point theories which allow to treat two kinds of nonlinear problems. A comparison between the possible applications of the two theories is also made.

We investigate the existence of a local or global semi-group for a complex Ginzburg-Landau like equation in $u$ coupled with a Poisson equation in $\phi$ defined on the whole space $\mathbb R^d$. At first, we consider the Cauchy problem in the classical Sobolev spaces $L_{\mathbb C}^{p}(\mathbb R ^{d})$, and later we study it in the weighted Sobolev spaces $L_{\rho\mathbb C }^{p}(\mathbb R ^{d})$, where $p\geq 3/2$, $p\geq d$, $d$ is a positive integer and the weight $\rho$ is increasing. Using the smoothing properties of the linear part, we obtain, for initial data in $L_{\mathbb C }^{p}(\mathbb R ^{d})$, a continuous strong solution in $W_{\mathbb C }^{1,p}(\mathbb R ^{d})$ with a singularity at $t=0$ behaving like $t^{-\frac{1}{2}}$. We obtain analogous results in weighted Sobolev spaces.

Criteria for non-explosion of solutions to semilinear parabolic evolution equations on Banach spaces are given. The abstract results are applied to classical semilinear parabolic equations: heat equation with first order nonlinear perturbations, fourth order Cahn-Hilliard, FitzHugh-Nagumo system.

In this paper we prove the well-posedness of the Rotenberg's model [16] describing the growth of cell population which resembles that used in classical transport theory. In this model the density of cell population are distinguished by their degree of maturation and the maturation velocity with mitotic boundary condition. We prove the positivity, irreducibility of the generated semigroup which converges asymptotically to a projection of the rank one.

Whether Dancer's fixed point index formulas for compact maps hold true for maps which are strict-set-contractions has been an open problem. We show that this problem has a positive answer. This implies in particular that these formulas are applicable to many differential equation systems where the underlying domain is the whole~$R^n$.

Let $\Omega\subset\mathbb {R}^2$ be a smooth bounded domain and $H\in\mathbb {R}^3$. The static Landau-Lifshitz equation with external magnetic field $H$ and boundary data $\gamma\in C^{\infty}( \Omega;\mathbb {S}^2)$ is the following: $$ \Delta u+|\nabla u|^2u-(H,u)u+H=0\quad\text{in $\Omega$}, \quad u=\gamma\quad\text{on $\partial\Omega$}. $$ Here $u\in C^{\infty}(\Omega;{\mathbb {S}^2})$. We establish some connection between solutions of the Landau-Lifshitz equation and critical points of the function $\Psi$ introduced in [9] (see $\S1$ Theorem B for the definition). In particular, we show that local nondegenerate minimums of $\Psi$ have their associated solutions of the Landau-Lifshitz equation for small $|H|$ and $\gamma\equiv\,{\rm const}$.

We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers (KP-Burgers) equation in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts which makes it quite particular. After showing local existence by contraction principle for initial data $ \varphi\in H^s(\mathbb R^2) $ such that $ {\mathcal F}^{-1} (\frac{k_2}{k_1} \widehat{\varphi})\in H^r(\mathbb R^2) $, $ 0{\leqslant} r {\leqslant} s- 1 $, we try to extend the solutions for all positive times. Whereas for $ {\varepsilon}=-1 $ and $ 1{\leqslant} p < 4/3 $ this will be done without any assumption on the the initial data, we will require a smallness condition on the initial data otherwise. In a last part we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in $ L^2(\mathbb R^2) $ when $ {\varepsilon}=-1 $ and $ 1{\leqslant} p < 4/3 $.

Minimizers $u_{\varepsilon}$ of some anisotropic Ginzburg-Landau functional $E_{\varepsilon}$ defined in (1.4) below on a smooth domain ${\Omega}\subset R^2$ with smooth boundary data of degree $d$ are shown to subconverge (as ${\varepsilon}\to 0$) locally in $C^{1+{\alpha}}$ away from finitely many points $a_1,\ldots,a_N$ to an anisotropic harmonic map $u_*$: ${\Omega}\setminus \{a_1,\ldots,a_N\}$ $\to$ $R^2$ where $\{a_1,\ldots,a_N\}\subset{\Omega} \cap a^{-1}(m)$, $N$ is related to the degree $d$, $m,a^{-1}(m)$ are defined in the following.

We consider a reaction diffusion system whit a triangular matrix of diffusion coefficients satisfying a balance law on a bounded domain with no-flux boundary condition. We demonstrate that globally bounded solutions exist for any reaction term provided a condition on the diffusion coefficients is satisfied. The proof makes use of some properties of the Neumann function for the heat equation posed in a bounded domain recently obtained in [12]. When the spatial domain is {\rm \bf R}$^N$, the proof relies on well-known properties of the fundamental solution of the heat equation.

We consider nonlocal reaction--diffusion equations in $m$--dimensional space. An existence theory is established using standard techniques. It is shown that when local monotonicity conditions are imposed, the stationary solutions that can be stable are those that are stable for an auxiliary local problem. This contrasts with what happens in the general case, where more complex solutions may be stable. An example of such a case is given. These results are obtained using comparison techniques and a generalization of previous results to the $m$--dimensional case.

We will be concerned with the multiplicity results for periodic solutions of the wave equation $(*)$ $u_{tt}-u_{xx}= f(x,t,u)$ satisfying some standard boundary and periodicity conditions. Our aim is to show that under some reasonable conditions on $f$ the above problem possesses at least three solutions in the non--equivariant case and to estimate the number of periodic solutions of $(*)$ when an action of $Z_2$ is involved.

In this article we prove the existence of an attractor for a dissipative nonlinear Schrödinger equation in the critical case for a two-dimensional thin domain. Moreover we prove that this attractor is smooth, i.e., made of smooth functions when the forcing term is smooth enough. The proofs use a splitting of the Fourier series of the solutions according to the geometry of the domain, together with anisotropic Sobolev inequalities.

We consider time dependent perturbations $B(t)$ of a non--autonomous Cauchy problem $\dot{v}(t)=A(t)v(t)$ $(CP)$ on a Banach space $X$. The existence of a mild solution $u$ of the perturbed problem is proved under Miyadera type conditions on $B(\cdot)$. In the parabolic case and $X=L^d(\Omega)$, $1 < d < \infty$, we show that $u$ is differentiable a.e. and satisfies $\dot{u}(t)=(A(t)+B(t))u(t)$ for a.e. $t$. Our approach uses perturbation results due to one of the authors, [29], and S. Monniaux and J. Prüß, [14], which are applied to the evolution semigroup induced by the evolution family related to $(CP)$. As an application we obtain solutions of a second order parabolic equation with singular lower order coefficients.

We examine a model equation describing spatial patterns in a class of physical systems where instabilities to travelling waves occur. The spatial patterns are modelled by a system of two second order ordinary differential equations, in which the cross-coupling coefficient is spatially dependent. The system has two clearly distinct types of stationary states, of which the stability depends on the cross-coupling coefficient. Under mild assumptions on the cross-coupling coefficient, we apply a variational method to prove the existence of a heteroclinic orbit between both types of states, corresponding to a domain-wall in the physical picture. This solution is found as a minimizer of a Lyapunov functional and the variational structure is exploited to obtain detailed information about the shape of the solution. In the case of a constant cross-coupling coefficient we find heteroclinic solutions connecting stationary states of the same type.