The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

We consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that (0, c, 0, 0) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

We consider the unique solvability of an inverse-source problem with integral transmitting condition for a time-fractional mixed type equation in rectangular domain where the unknown source term depends only on the space variable. The solution is based on a series expansion using a bi-orthogonal basis in space, corresponding to a non-self-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. The influence of the transmitting condition on the solvability of the problem is also demonstrated. Two different transmitting conditions are considered — viz. a full integral form and a special case. In order to simplify the bulky expressions appearing in the proof of our main result, we establish a new property of the recently introduced Mittag-Leffler type function in two variables.

We consider a mathematical model which describes the quasi-static evolution of a thermo-viscoelastic linear body with taking into account the effects of internal forces which generate a non linear viscous dissipative function. We derive a variational formulation of the system of equilibrium equation and energy equation. An existence result of weak solutions was obtained in an appropriate function space.

In this paper, we model laser-gas interactions and propagation in some extreme regimes. After a mathematical study of a micro-macro Maxwell-Schrödinger model [1] for short, high-frequency and intense laser-gas interactions, we propose to improve this model by adding a plasma equation in order to precisely take into account free electron effects. We examine if such a model can predict and explain complex structures such as filaments, on a physical and numerical basis. In particular, we present in this paper a first numerical observation of nonlinear focusing effects using an ab-initio gas representation and linking our results with existing nonlinear models.

We investigate the multidimensional non-isentropic Euler–Poisson (or full hydrodynamic) model for semiconductors, which contain an energy-conserved equation with non-zero thermal conductivity coefficient. We first discuss existence and uniqueness of the non-constant stationary solutions to the corresponding drift–diffusion equations. Then we establish the global existence of smooth solutions to the Cauchy problem with initial data, which are close to the stationary solutions. We find that these smooth solutions tend to the stationary solutions exponentially fast as $t\rightarrow+\infty$.