We study, in this article, the well-posedness of a planetary geostrophic (PG) model of the ocean. This model is derived from the PG studied in [17] by adding a (non-linear) advection term in the horizontal momentum equation. We address in particular the question of existence and uniqueness of global (in time) weak solution to the model. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions. We prove that any weak solution to the model converges exponentially with time to a weak solution to the primitive equations of the ocean provided that the data are small.

We consider the following anisotropic problem $$-\div\big( a(x)\nabla u\big)+a(x)u=0 \quad \mbox{in ${\Omega }$,}\qquad \frac{{\partial} u} {{\partial}\nu}={\varepsilon } e^u \quad\mbox{on ${\partial\Omega }$,} $$ where ${\Omega }\subseteq \mathbb{R}^2$ is a bounded smooth domain, ${\varepsilon }$ is a small parameter and $a(x)$ is a positive smooth function. First, we establish a decomposition result for the regular part of a relative Green's function, which yields its Hölder continuous character and the smoothness of its diagonal. Next, we employ this result to derive the accumulation of bubbles at given local maximum points of $a(x)$ on the boundary, which verifies the existence of large energy solutions to the problem in [17].

We establish existence results for singular semilinear elliptic systems on bounded domains with homogeneous Dirichlet boundary conditions. The systems considered are the paradigmatic mathematical models of chemical reactions, morphogenesis (singular Gierer-Meinhardt system) and population dynamics. In these systems the operator need not be in divergence form and the systems need not be cooperative. The results have been obtained by the method of sub and supersolutions (appropriately modified) and Schauder's fixed point theorem. Some uniqueness results have been obtained extending a "concavity" argument used for a single equation. We extend some existence results to general elliptic operators and more general nonlinearities and we prove existence for systems that have not been considered in the literature.

We consider the existence of multiple nodal bubbling solutions to the equation $-{\Delta } u=2\ {\varepsilon }^2|x|^{2\alpha}\sinh u$ posed on a bounded smooth domain ${\Omega }$ in $\mathbb{R}^2$ with homogeneous Dirichlet boundary conditions. By construction, we show that there exists a solution such that $2{\varepsilon }^2|x|^{2\alpha}\sinh u_{{\varepsilon }}$ develops not only many positive and negative Dirac deltas with weights $8\pi$ and $-8\pi$ respectively, but also a Dirac delta with weight $8\pi(1+\alpha)$ at the origin, where $\alpha\not\in \mathbb{N}$. In particular, we provide explicit examples to show the existence of nodal bubbling solutions to our problem in the unit disc in $\mathbb R^2$.

This paper is concerned with the asymptotic limit of the solution of a nonlinear reaction-diffusion system when the reaction is very large in a localized region.

This paper is composed of two parts. In the first part, via a reduction dimension method, we derive a one-dimensional minimization problem involving $S^2$-valued maps for a thin T-shaped multidomain. In the second one, we analyze this limit model.

We study the sign of the solution of a linear parabolic biharmonic Cauchy problem in $\mathbb R^n$ by varying both the source and the initial datum. Eventual local positivity is proved under different assumptions and the problem of global positivity is discussed.

We propose a modification of the classical Navier-Stokes-Boussinesq system of equations, which governs buoyancy-driven flows of viscous, incompressible fluids. This modification is motivated by unresolved issues regarding the global solvability of the classical system in situations where viscous heating cannot be neglected. A simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. In the present paper, we establish the local-in-time existence and uniqueness of strong solutions for the model problem. The full system of equations and the global-in-time existence of weak solutions will be addressed in forthcoming work.