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The -elastic flow for planar closed curves with constant parametrization
Shinya Okabe, Glen Wheeler
In this paper, we consider the -gradient flow for the modified -elastic energy defined on planar closed curves. We formulate a notion of weak solution for the flow and prov…
Existence of solutions to nonlinear parabolic equations via majorant integral kernel
Kazuhiro Ishige, Tatsuki Kawakami, Shinya Okabe
We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order s…
Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations
Hans-Christoph Grunau, Nobuhito Miyake, Shinya Okabe
This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic pr…
Existence of solutions for a higher-order semilinear parabolic equation with singular initial data
Kazuhiro Ishige, Tatsuki Kawakami, Shinya Okabe
We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary condi…
A supercritical scalar field equation with a forcing term
Kazuhiro Ishige, Shinya Okabe, Tokushi Sato
This paper is concerned with the elliptic problem for a scalar field equation with a forcing term \begin{equation} \tag{P}-Δu+u=u^p+ κμ\quad \mbox{in} \quad{\bf R}^N, \quad u>0 \qu…