12 problems

Let $(\rho_\delta,c_\delta)$ and $(\rho,c)$ be the solutions in Theorem 1.1, and suppose the convergence estimate … holds for every $\lambda>0$ and sufficiently small $\delta>0$. O…

Let $\lambda$ be the entropy parameter in the stochastic Shigesada–Kawasaki–Teramoto system, currently assumed to satisfy $\lambda>\frac{1}{2}$. Entropy conjecture. We conjecture t…

Global-existence analogy conjecture. The system $$ with nonconstant $d_1$ and $\chi_1$ should possess properties similar to those of its parabolic–elliptic simplifications, for whi…

Let $g\in H^1(\Omega)^N$ take values in $\mathcal{D}$, prescribe $\rho=g$ on $\Gamma_D\subset\partial\Omega$, and suppose that $s'(g)\in L^\infty(\Omega)$. Define the relative entr…

Global-diagram convergence conjecture. Even the global bifurcation diagram could be captured asymptotically by a convergence rate towards the singular limit.

Self- versus cross-diffusion conjecture. Opposed to standard diffusion, self-diffusion is a losing strategy whereas cross-diffusion is a winning one.

The finite-element scheme introduces a regularization term with parameter $\eps>0$ to obtain coercivity. Regularization conjecture. This regularization is only technical and should…

Let $u(t,x)$ be a solution whose support has length $\ell(t)=\sigma_+(t)-\sigma_-(t)$, with initial length $\ell(0)=\xi_+-\xi_-$. Suppose that the total mass satisfies…

Let the chemoattractant concentration be the relevant environmental concentration for a motile bacterium, and let the turning time denote the waiting time between velocity jumps. L…

Let $u(t)$ be a solution of the cross-diffusion herding system, with the other parameters fixed as in the preceding exponential-decay theorem. The source excludes $\delta=0$ from t…

Consider the cross-diffusion herding system with a single varying parameter, and let $\delta_{\textnormal{b}}$ denote a first local bifurcation point. One natural conjecture is tha…

Consider the parabolic cross-diffusion system under assumptions -, with cross-diffusion parameter $\gamma>0$ and $\xi\in(\delta,1)$. Let $\mathfrak{S}$ be the continuum of positive…