APDE Seminar @BCAM: Euler-flocking system with nonlocal dissipation in 1D: periodic entropy solutions and Critical non-linearity for some evolution equations with Fujita-type critical exponent

1) Title: Euler-flocking system with nonlocal dissipation in 1D: periodic entropy solutions

Abstract

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in a periodic domain in one-space dimension with linear pressure term. The main result is the global existence of periodic entropy weak solutions, for periodic initial data having finite total variation and initial density bounded away from zero.

2) Title: Critical non-linearity for some evolution equations with Fujita-type critical exponent

Abstract

We consider the Cauchy problem for a class of non-linear evolution equations in the form L u=F(u), where L is a linear partial differential operator with constant coefficients and F is a non-linear term.  

For several different choices of L, many authors have investigated the existence of global (in time) solutions to this problem when F(u)=|u|^p is a power non-linearity, looking for a critical exponent p_c>1 such that global (in time) small data solutions exist in the supercritical case p>p_c, whereas no global (in time) weak solutions exist, under suitable sign assumptions on the data, in the subcritical case 1<p<p_c.

In this talk we consider a more general non-linear term F(u)=|u|^ph(|u|) where h is a modulus of continuity; for a large class of models, we provide an integral condition on h which allows to distinguish more precisely the region of existence of a global small data solution from that in which the problem admits no global weak solutions, refining the existing results about the critical exponents for power type non-linearities.

The talk is based on the results obtained in [1] and [2]

[1] Ebert, M. R., Girardi, G. and Reissig, M. Critical regularity of nonlinearities in semilinear classical damped wave equations. Math. Ann. (2020). 

[2] Girardi, G.  Critical non-linearity for some evolution equations with Fujita-type critical exponent. Nonlinear Differ. Equ. Appl. (2025).