Existence, regularity, and boundedness of equilibrium solutions for the self- and cross-diffusion reaction–diffusion model

Establish existence, regularity, and L2-boundedness of stationary (equilibrium) solutions (u1,u2) on Ω = [−Lx,Lx] with periodic boundary conditions for the nonlinear elliptic system −Δ μ1(u1,u2) = R1(u1,u2) and −Δ μ2(u1,u2) = R2(u1,u2), where μ1 = d1 u1 + d11 u1^3 + d12 u2^2 u1 and μ2 = d2 u2 + d22 u2^3 + d12 u1^2 u2 are the diffusion-induced chemical potentials, and R1 = η(u1 + a u2 − C u1 u2 − u1 u2^2) and R2 = η(b u2 + H u1 + C u1 u2 + u1 u2^2) are the reaction terms of the reaction–diffusion model with self- and cross-diffusion. In particular, prove that such equilibria belong to an appropriate Sobolev space Hm and satisfy uniform L2 bounds sufficient to justify the convergence assumptions used for the Fourier spectral analysis.

Background

In Section 2.2.1, the paper derives a Fourier spectral method and proves a convergence estimate for approximating equilibria, contingent on assumptions that the model admits equilibrium solutions that are bounded and sufficiently smooth (belonging to a Sobolev space Hm). The proposition’s proof relies on these regularity and boundedness properties to control aliasing and truncation errors.

Immediately after the proposition, the authors note that these properties are supported by extensive numerical observations but lack a rigorous proof. Thus, establishing existence, regularity, and boundedness of stationary solutions for the nonlinear elliptic system associated with the reaction–diffusion model (which includes linear, self-, and cross-diffusion terms) is identified as an open problem needed to fully justify the numerical convergence analysis.