اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExistence of ground states for aggregation-diffusion equations
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We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.
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Recently, the authors of the current paper established in [9] the existence of a ground-state solution to the following bi-harmonic equation with the constant potential or Rabinowitz potential: begin{equation} (-Delta)^{2}u+V(x)u=f(u) text{in} mathbb
{R}^{4}, end{equation} when the nonlinearity has the special form $f(t)=t(exp(t^2)-1)$ and $V(x)geq c>0$ is a constant or the Rabinowitz potential. One of the crucial elements used in [9] is the Fourier rearrangement argument. However, this argument is not applicable if $f(t)$ is not an odd function. Thus, it still remains open whether the above equation with the general critical exponential nonlinearity $f(u)$ admits a ground-state solution even when $V(x)$ is a positive constant. The first purpose of this paper is to develop a Fourier rearrangement-free approach to solve the above problem. More precisely, we will prove that there is a threshold $gamma^{*}$ such that for any $gammain (0,gamma^*)$, the above equation with the constant potential $V(x)=gamma>0$ admits a ground-state solution, while does not admit any ground-state solution for any $gammain (gamma^{*},+infty)$. The second purpose of this paper is to establish the existence of a ground-state solution to the above equation with any degenerate Rabinowitz potential $V$ vanishing on some bounded open set. Among other techniques, the proof also relies on a critical Adams inequality involving the degenerate potential which is of its own interest.
We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then t
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