Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type
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Journal of Evolution Equations
Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type Leon Bungert
and Martin Burger
Abstract. 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.
1. Introduction This work studies the fine asymptotic behavior of the abstract gradient flow ∂t u + ∂ J (u) 0, u(0) = f.
(GF)
Here, J : H → R ∪ {∞} is an absolutely p-homogeneous convex functional on a real Hilbert space H, and f ∈ H is an initial datum (see Sect. 1.2 for precise definitions). There is a variety of partial differential equations to which our theory applies and for some of which similar issues have been studied before in a specialized setting. The most prominent and most studied equations are the parabolic p-Laplacian equations for p ≥ 1 with the total variation flow as special case for p = 1 ∂t u − div |∇u| p−2 ∇u = 0, p ≥ 1. These equations can also be studied as a fourth-order gradient flow in H −1 , i.e., ∂t u − Δ div |∇u| p−2 ∇u = 0, p ≥ 1. Mathematics Subject Classification: 35K90, 35P30, 47J10, 47J35 Keywords: Gradient flow, Homogeneous functionals, Nonlinear evolution equations, Asymptotic profile, Extinction profile, Nonlinear eigenfunctions, Asymptotic behavior, Extinction time, Convergence rates.
L. Bungert and M. Burger
J. Evol. Equ.
Another class of examples are the fast diffusion equations for 1 < p < 2, the linear heat equation for p = 2, and the porous medium equation for p > 2, i.e., ∂t u − Δu p−1 = 0,
p > 1,
which, complemented with suitable boundary conditions, can also be interpreted as Hilbert space gradient flows (cf. [30] for the porous medium/fast diffusion case). Furthermore, as long as homogeneity is preserved, our general model covers nonlocal versions of the equations above, as well. Remarkably, we can also address an eigenvalue problem similar to that of the ∞-Laplacian operator [25,32] with our framework. To this end we set J (u) = ∇u ∞ for u ∈ W 1,∞ ∩ L 2 and J (u) = ∞ else, which meets all our assumptio
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