The positive solutions are studied for the nonlinear third-order three-point boundary value problem u (t) = f(t,u(t)), a.e. t ∈ [0,1], u(0) = u (η) = u (1) = 0, where the nonlinear term f(t,u) is a Carathodory function and there exists a nonnegative function h ∈ L1[0,1] such that f(t,u) ≥ -h(t). The existence of n positive solutions is proved by considering the integrations of "height functions" and applying the Krasnosel'skii fixed point theorem on cone.
This paper is devoted to the analysis of the Cauchy problem for a system of PDEs arising in radiative hydrodynamics. This system, which comes from the so-called equilibrium diffusion regime, is a variant of the usual Euler equations, where the energy and pressure functionals are modified to take into account the effect of radiation and the energy balance containing a nonlinear diffusion term acting on the temperature. The problem is studied in the multi-dimensional framework. The authors identify the existence of a strictly convex entropy and a stability property of the system, and check that the Kawashima-Shizuta condition holds. Then, based on these structure properties, the wellposedness close to a constant state can be proved by using fine energy estimates. The asymptotic decay of the solutions are also investigated.
