The evolution fractional p-Laplacian equation in $\mathbb{R}^N$. Fundamental solution and asymptotic behaviour

Juan Luis Vázquez

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arXiv · 2004.05799

The evolution fractional p-Laplacian equation in $\mathbb{R}^N$. Fundamental solution and asymptotic behaviour

Juan Luis Vázquez

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Abstract

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and show that the solutions form a family of non-expansive semigroups with regularity and other interesting properties. As main results, we construct the self-similar fundamental solution for every mass value $M,$ and prove that general finite-mass solutions converge towards that fundamental solution having the same mass in all $L^q$ spaces.A number of additional properties and estimates complete the picture.

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Juan Luis Vázquez. 2020-04-13. The evolution fractional p-Laplacian equation in $\mathbb{R}^N$. Fundamental solution and asymptotic behaviour. https://arxiv.org/abs/2004.05799

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