WEIGHTED FUNCTIONAL INEQUALITIES AND NONLINEAR DIFFUSIONS OF POROUS MEDIUM TYPE

Abstract : The main topic of this thesis is the study of the asymptotic behaviour of solutions to certain nonlinear diffusion equations, whose most important models are the porous medium equation and the fast diffusion equation. In the first chapter we analyse in detail the connections between Lp smoothing and decay properties of weighted versions of the porous medium equation and the validity of suitable functional inequalities involving the weights. In the second chapter we investigate the asymptotics of the solutions to the fractional porous medium equation with power-type weights: this is strictly linked with a similar fractional parabolic problem having as initial datum a positive finite measure, which we study separately. The third chapter is mostly devoted to the characterization of the optimal functions for a family of Caffarelli-Kohn-Nirenberg interpolation inequalities: it turns out that if the power of the weight that appears in the Lp norms is small enough, then such optimal functions are radial. As a consequence, solutions to the Euclidean fast diffusion equation with the same power weight converge towards special solutions of Barenblatt type with an optimal rate, at least for m larger than a suitable critical value. In the fourth and last chapter we consider the fast diffusion equation on hyperbolic space: the most important result we obtain, for m close to one, is the convergence of radial solutions, as t tends to the extinction time, to a separable solution in the uniform norm of the relative error.
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Matteo Muratori. WEIGHTED FUNCTIONAL INEQUALITIES AND NONLINEAR DIFFUSIONS OF POROUS MEDIUM TYPE. Analysis of PDEs [math.AP]. Politecnico di Milano; Université Paris 1 Panthéon-Sorbonne, 2015. English. ⟨tel-01289874⟩

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