Existence of a global weak solution for a reaction–diffusion problem with membrane conditions

Abstract

Several problems, issued from physics, biology or the medical science, lead to parabolic equations set in two sub-domains separated by a membrane with selective permeability to specific molecules. The corresponding boundary conditions, describing the flow through the membrane, are compatible with mass conservation and energy dissipation and are called the Kedem-Katchalsky conditions. Additionally, in these models, written as reaction-diffusion systems, the reaction terms have a quadratic behaviour. M. Pierre and his collaborators have developed a complete L 1 theory for reaction-diffusion systems with different diffusions. Here, we adapt this theory to the membrane boundary conditions and prove the existence of weak solutions when the initial data have only L 1 regularity using the truncation method for the nonlinearities. In particular, we establish several estimates as the W 1,1 regularity of the solutions. Also, a crucial step is to adapt the fundamental L 2 (space, time) integrability lemma to our situation. G. Ciavolella and B. Perthame have equal contributions.

Effect of a Membrane on Diffusion-Driven Turing Instability

“…We questioned the eect on pattern formation of a permeable membrane at which we have dissipative conditions. This interest follows both a path started in the study of membrane problems [7] and their importance in biology. Then, we have studied Turing instability from both an analytical and a numerical point of view for a reaction-diusion membrane problem of two species u and v as in (1).…”

“…In our case with the membrane in the middle point x m , we infer that N l = N r . Concerning the membrane, the key aspect is to discretize this point as two distinct ones since the Kedem-Katchalsky conditions are constructed dening the right and left limit of the density on the membrane (see [7]). Moreover, the space step turns out to be ∆x = xm−a N l = xm−a Nr , with N l , N r ∈ N. The mesh is formed by the intervals…”

“…Since we are working with condition (7), the rst order coecient of this polynomial is positive. Consequently, we need to impose that…”

“…In [7], the reader can nd a previous analytical study on a reaction-diusion system of m ≥ 2 species with membrane conditions of the Kedem-Katchalsky type. The main result concerns the existence of a global weak solution in the case of low regularity initial data and at most quadratic non-linearities in an L 1 -setting.…”

“…Consider the linearised system with D v > 0 xed. We assume appropriate conditions on the linearisation coecients (see Equations (7) , (8)) and on λ n , η n , D φl , k φ , where φ = u, v (see Equations (12) , ( 13)). Then, for D u suciently small, the steady state is linearly unstable.…”