4 papers
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…
A gradient flow for the p-elastic energy defined on closed planar curves
Shinya Okabe, Paola Pozzi, Glen Wheeler
We study the evolution of closed inextensible planar curves under a second order flow that decreases the -elastic energy. A short time existence result for is…