In this paper we establish $L^p-L^q$ estimates for the solution of the strictly hyperbolic first-order linear systems with bounded time-dependent coefficients. In the equation setting, Reissig and others obtained such estimates by using WKB representations of the solutions. Here the crucial point is to find minimal assumptions on the coefficients of the system so that Reissig's approach still works for systems.

In this paper, we study the existence of solitary waves associated to the two-dimensional Benjamin equation. This equation governs the evolution of waves at the interface of a two-fluid system in which surface-tension effects cannot be ignored. We classify the existence and nonexistence cases according to the sign of the transverse dispersion coefficients. Moreover, we show that the solitary waves of the 2D Benjamin equation, when they exist, converge to those of the KPI equation as the parameter preceding the nonlocal operator $H\partial^2_x$ goes to zero. We also prove the regularity of solitary waves, as well as their symmetry with respect to the transverse variable and their algebraic decay at infinity.

The first aim of this paper is to give a general variational framework for bilinear forms depending on two parameters tending to zero and to infinity respectively, allowing us to analyze the three limit problems. Secondly, we give different illustrative applications for transmission problems involving some elasticity systems, diffusion problems, and Maxwell systems where one parameter tends to infinity and/or a part of the domain squeezes to a smooth surface. These limit procedures lead to new transmission problems, like a coupling between the Lamé system and the Stokes system.

In this paper we prove the existence and multiplicity of nontrivial weak solutions for quasilinear elliptic equations of the form $-L_p u +V(x)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$, where $L_p u\doteq \epsilon^{p}\Delta_p u +\epsilon^{p}\Delta_p (u^2)u$ and $V$ is a positive continuous potential bounded away from zero satisfying some conditions and the nonlinear term $h(u)$ has a subcritical growth type. Here, we use a variational method to get the multiplicity of positive solutions involving the Lusternick-Schnirelman category of the set where $V$ achieves its minimum value.

In ordinary differential equations of singular perturbation type, the dynamics of solutions near saddle-node bifurcations of equilibria are rich. Canard solutions can arise, which, after spending time near an attracting equilibrium, stay near a repelling branch of equilibria for long intervals of time before finally returning to a neighborhood of the attracting equilibrium (or of another attracting state). As a result, canard solutions exhibit bifurcation delay. In this article, we analyze some linear and nonlinear reaction-diffusion equations of singular perturbation type, showing that solutions of these systems also exhibit bifurcation delay and are, hence, canards. Moreover, it is shown for both the linear and the nonlinear equations that the exit time may be either spatially homogeneous or spatially inhomogeneous, depending on the magnitude of the diffusivity.

We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 {\varphi} _1+\ldots +\mu _n {\varphi} _n+e(x) \; \; \mbox{for $x \in U$}, \; \; u=0 \; \; \mbox{on $\partial U$}, \] where ${\varphi} _k$ is the $k$-th eigenfunction of the Laplacian on $U$ and $e(x) \perp {\varphi} _k$, $k=1, \ldots,n$. We write the solution in the form $u(x)= \sum _{i=1}^n \xi _i {\varphi} _i+U_{\xi } (x)$, with $ U_{\xi } \perp {\varphi} _k$, $k=1, \ldots,n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\xi =(\xi _1, \ldots,\xi _n)$ fixed, but allowing $\mu =(\mu _1, \ldots,\mu _n)$ to vary. We then study the map $\xi \rightarrow \mu$, which provides existence and multiplicity results for the above problem.

In this paper, we consider a class of nonlinear Boussinesq-Stefan type systems: a Navier-Stokes equation for the velocity $u$ and the pressure $p$ with second member $F(\theta)$ where $\theta$ is the temperature field, the incompressibility condition and a scalar equation for $\theta$ having a convection term and a nonlinear diffusion operator, in which the right-hand side $\mu(\theta) |D u|^2$ is the dissipation energy. The function $F(\theta)$ is the buoyancy force which satisfies a growth assumption in dimension $2$ and is bounded in dimension $3$. We present some existence results through a fixed-point argument. We use the traditional results of Navier-Stokes equations and those of renormalized solutions. One of the difficulties is the coupling between the two equations for $u$ and $\theta$ through the dissipation energy $\mu(\theta) |D u|^2$. This prevents us from showing compactness, at least if we use the classical results of renormalized solutions for a Stefan problem with $L^1$ data.

Replace the first equation after (2.44) by $$\int_{-L}^L(\eta_t Q+u_t P) dx=\frac{1}{2} \int_{-L}^L\partial_t(\eta^2+u^2+\eta u^2-c\eta_x^2)dx. $$