This paper proposes a volumetric penalty method to simulate the boundary conditions for a non-linear hyperbolic problem. The boundary conditions are assumed to be maximally strictly dissipative on a non-characteristic boundary. This penalization appears to be quite natural since, after a natural change of variable, the penalty matrix is an orthogonal projector. We prove the convergence towards the solution of the wished hyperbolic problem and that this convergence is sharp in the sense that it does not generate any boundary layer, at any order. The proof involves an approximation by asymptotic expansion and energy estimates in anisotropic Sobolev spaces.

We are interested in finding a family of solutions of a singularly perturbed biharmonic equation, which has a concentration behavior. The proof is based on variational methods and uses a weak version of the Ambrosetti-Rabinowitz condition.

This work deals with the semilinear equation $-\Delta u+u-u^p=0$ in ${\mathbb R}^N$, $2\leq p < {N+2\over N-2}$. We consider the positive solutions, which are ${2\pi\over\varepsilon}$-periodic in $x_1$ and decreasing to $0$ in the other variables, uniformly in $x_1$. Let a periodic configuration of points be given on the $x_1$-axis, which repel each other as the period tends to infinity. If there exists a solution which has these points as peaks, we prove that the points must be asymptotically uniformly distributed on the $x_1$-axis. Then, for $\varepsilon$ small enough, we prove the uniqueness up to a translation of the positive solution with some peaks on the $x_1$-axis, for a given minimal period in $x_1$, and we estimate the difference between this solution and the groundstate solution.

We consider the long time behavior of weak and strong solutions of the $n$-dimensional viscous Boussinesq system in the half space, with $n\geq3$. The $L^r(\mathbb{R}^n_+)$-asymptotics of strong solutions and their first three derivatives, with $1\leq r\leq\infty$, are derived by combining $L^q-L^r$ estimates and properties of the fractional powers of the Stokes operator. For the $L^\infty-$asymptotics of the second order derivatives, the unboundedness of the projection operator $P: L^\infty(\mathbb{R}^n_+)\rightarrow L^\infty_\sigma(\mathbb{R}^n_+)$ is dealt with by an appropriate decomposition of the nonlinear term.

In this article, we prove results concerning the decay and first order asymptotic behavior of solutions to a system of equations that model heat transfer in a porous medium by an incompressible flow with fractional dissipation.

This work studies the initial value problem (i.v.p.) for a generalized Camassa-Holm equation with $(k+1)$-order nonlinearities (g-$k$bCH) and containing, as its members, three integrable equations: the Camassa-Holm, the Degasperis-Procesi and the Novikov equations. For $s>3/2$, using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces $H^s$ on both the circle and the line in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates.