Bungert L, Burger M (2019)
Publication Language: English
Publication Type: Journal article
Publication year: 2019
URI: https://arxiv.org/abs/1906.09856
DOI: 10.1007/s00028-019-00545-1
This work is concerned with the gradient flow of absolutely p-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite (p< 2) or infinite extinction time (p≥ 2). We give upper bounds for the finite extinction time and establish sharp convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and non-local versions of PDEs like p-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting. We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here, we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions.
APA:
Bungert, L., & Burger, M. (2019). Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type. Journal of Evolution Equations. https://dx.doi.org/10.1007/s00028-019-00545-1
MLA:
Bungert, Leon, and Martin Burger. "Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type." Journal of Evolution Equations (2019).
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