The diffusive limit of Carleman-type models in the range of very fast diffusion equations

Francesco Salvarani (1) , Giuseppe Toscani (2)

1 Dipartimento di matematica F. Casorati
2 UNIPV - Dipartimento di Matematica “Felice Casorati” Department of Mathematics [Univ Pavia]

We improve the existing results on the limiting behavior of the Cauchy problem for a class of Carleman-like models with power-type interaction rate in the diffusive scaling with data in the spaces $L^p$, $1\le p\le \infty$. The convergence result, which has been carefully established before for exponents of the interaction rate $\alpha\le 1$, is extended here to the range of exponents $1<\alpha<4/3$. In addition, we discuss the problem of establishing a good theory in the still remaining range $\alpha\in [4/3,2)$, by introducing a modified kinetic system which admits an explicit self-similar solution. The analysis of this solution clarifies the role of the exponent $\bar\alpha =4/3$.

Domaines: Equations aux dérivées partielles [math.AP]

Francesco Salvarani, Giuseppe Toscani. The diffusive limit of Carleman-type models in the range of very fast diffusion equations. 2008. ⟨hal-00272382v2⟩