In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem.

Soit $ℰ$ un espace topologique, ${ℰ}^{\text{'}}$ un espace métrique et $\left(S\right)$ un système d’équations d’évolution admettant une solution dans ${ℰ}^{\text{'}}$ pour toute donnée initiale dans $ℰ$ et stable vis-à-vis des données initiales sur $ℰ$. On montre que l’ensemble des données initiales pour lesquelles $\left(S\right)$ admet une unique solution est un $G_{\delta }$ de $ℰ$. En particulier, si l’unicité est vraie sur un sous-ensemble dense de $ℰ$, elle l’est génériquement.

Let $\Delta u={\Sigma }_i\frac{{\partial }^2}{{\partial }_{x_i}^2}$, $Lu={\Sigma }_{i,j}{a}_{ij}\frac{{\partial }^2}{\partial x_i\partial x_j}u+{\Sigma }_i{b}_i\frac{\partial }{\partial x_i}u+cu$ be elliptic operators with Hölder continuous coefficients on a bounded domain $\Omega \subset {\mathbf{R}}^n$ of class $C^{1,1}$. There is a constant $c\>0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that$$c^{-1}{G}_{\Delta }^{\Omega }\le G_L^{\Omega }\le c{G}_{\Delta }^{\Omega }\text{on}\Omega \times \Omega ,$$where ${\gamma }_{\Delta }^{\Omega }$ (resp. $G_L^{\Omega }$) are the Green functions of $\Delta$ (resp. $L$) on $\Omega$.

We prove the uniqueness, up to shifts, of pulsating traveling fronts for reaction-diffusion equations in periodic media with Kolmogorov–Petrovskiĭ–Piskunov type nonlinearities. These results provide in particular a complete classification of all KPP pulsating fronts. Furthermore, in the more general case of monostable nonlinearities, we also derive several global stability properties and convergence to pulsating fronts for solutions of the Cauchy problem with front-like initial data. In particular,...

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized...