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Existence and large-time behavior for the exponential surface diffusion flow and its graphical PDE

Determine the well-posedness (at least local-in-time existence and uniqueness) and characterize the large-time behavior of solutions to the exponential surface diffusion flow V = Δ_Γ f(−κ) with f(r) = e^r, and to its one-dimensional graphical form on ℝ given by the fourth-order quasilinear PDE derived from V = ∂_s^2 f(−κ); in particular, prove existence (even local-in-time) for the Cauchy problem of the graphical equation and analyze its asymptotics.

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Background

Before presenting their results, the authors emphasize that for the exponential surface diffusion flow and its graphical formulation, the foundational questions of existence and large-time behavior had not been established in the literature. This contrasts with the conventional surface diffusion flow, where Koch–Lamm, Du–Yip, and others have developed existence and asymptotic theories using scale-invariant frameworks.

The authors attribute the difficulty to the lack of scale invariance caused by the nonlinear curvature dependence f(r) = er, which complicates the adaptation of known techniques and function space frameworks from the conventional case.

References

On the other hand, for an exponential surface diffusion flow expSDE and its graphical form SDE', the existence and large time behavior of solutions are widely open. To our knowledge, there are no known results on the existence of a solution to the initial value problem SDE', even in the sense of a local-in-time solution.

expSDE:

V=ΔΓf(−κ)V = \Delta_\Gamma f(-\kappa)

SDE':

ut=(1(1+ux2)1/2(f(−uxx(1+ux2)3/2))x)x in R×(0,T), u(⋅,0)=u0 in R.u_t=\left(\frac{1}{(1+u_x^2)^{1/2}}\left(f\left(-\frac{u_{xx}}{(1+u_x^2)^{3/2}}\right)\right)_x\right)_{x}\ \textrm{in}\ R\times(0,T),\ u(\cdot,0)=u_0\ {\textrm{in}\ R.}

— Large time behavior of exponential surface diffusion flows on $\mathbb{R}$ (2411.17175 - Giga et al., 26 Nov 2024) in Introduction, end of Subsection 1.1 (A general surface diffusion flow)