36 research outputs found
Energy solutions to one-dimensional singular parabolic problems with data are viscosity solutions
We study one-dimensional very singular parabolic equations with periodic
boundary conditions and initial data in , which is the energy space. We
show existence of solutions in this energy space and then we prove that they
are viscosity solutions in the sense of Giga-Giga.Comment: 15 page
The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials
This paper is concerned with the analysis of the Cauchy problem of a general
class of two-dimensional nonlinear nonlocal wave equations governing anti-plane
shear motions in nonlocal elasticity. The nonlocal nature of the problem is
reflected by a convolution integral in the space variables. The Fourier
transform of the convolution kernel is nonnegative and satisfies a certain
growth condition at infinity. For initial data in Sobolev spaces,
conditions for global existence or finite time blow-up of the solutions of the
Cauchy problem are established.Comment: 15 pages. "Section 6 The Anisotropic Case" added and minor changes.
Accepted for publication in Nonlinearit
Very Singular Diffusion Equations-Second and Fourth Order Problems
This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example
A variational view at the time-dependent minimal surface equation
We present a global variational approach to the L2-gradient flow of the area functional of cartesian
surfaces through the study of the so-called weighted energy-dissipation (WED) functional. In particular, we prove a relaxation result which allows us to show that minimizers of the WED converge in a quantitatively prescribed way to gradient-flow trajectories of the relaxed area functional. The result is then extended to general parabolic quasilinear equations arising as gradient flows of convex functionals with linear growth