In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p−laplacian in the whole R n .
In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type (2 p < •) under a strong absorption condition:This model is interesting because it yields the formation of dead-core sets, i.e, regions where nonnegative solutions vanish identically. We shall prove sharp and improved parabolic C a regularity estimates along the set F 0 (u, W T ) = ∂ {u > 0} \ W T (the free boundary), where a = p p 1 q 1 + 1 p 1 . Some weak geometric and measure theoretical properties as non-degeneracy, positive density, porosity and finite speed of propagation are proved. As an application, we prove a Liouville-type result for entire solutions. A specific analysis for Blow-up type solutions will be done as well. The results are new even for dead-core problems driven by the heat operator.2000 Mathematics Subject Classification. 35B53, 35B65, 35J60, 35K55, 35K65. Key words and phrases. p-Laplacian type operators, dead-core problems, sharp and improved intrinsic regularity, Liouville type results.
In this work an opinion formation model with heterogeneous agents is proposed. Each agent is supposed to have different power of persuasion, and besides its own level of zealotry, that is, an individual willingness to being convinced by other agent. In addition, our model includes zealots or stubborn agents, agents that never change opinions.We derive a Bolzmann-like equation for the distribution of agents on the space of opinions, which is approximated by a transport equation with a nonlocal drift term. We study the long-time asymptotic behavior of solutions, characterizing the limit distribution of agents, which consists of the distribution of stubborn agents, plus a delta function at the mean of their opinions, weighted by they power of persuasion.Moreover, explicit bounds on the rate of convergence are given, and the time to convergence is shown to decrease when the number of stubborn agents increases. This is a remarkable fact observed in agent based simulations in different works.
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