DCDS
Global solutions for a semilinear heat equation in the exterior domain of a compact set
Kazuhiro Ishige Michinori Ishiwata
Discrete & Continuous Dynamical Systems - A 2012, 32(3): 847-865 doi: 10.3934/dcds.2012.32.847
Let $u$ be a global in time solution of the Cauchy-Dirichlet problem for a semilinear heat equation, $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u+u^p,\quad & x\in\Omega,\,\, t>0,\\ u=0,\quad & x\in\partial\Omega,\,\,t>0,\\ u(x,0)=\phi(x)\ge 0,\quad & x\in\Omega, \end{array} \right. $$ where $\partial_t=\partial/\partial t$, $p>1+2/N$, $N\ge 3$, $\Omega$ is a smooth domain in ${\bf R}^N$, and $\phi\in L^\infty(\Omega)$. In this paper we give a sufficient condition for the solution $u$ to behave like $\|u(t)\|_{L^\infty({\bf R}^N)}=O(t^{-1/(p-1)})$ as $t\to\infty$, and give a classification of the large time behavior of the solution $u$.
keywords: global solutions. exterior domain Semilinear heat equation
PROC
Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent
Michinori Ishiwata
Conference Publications 2005, 2005(Special): 443-452 doi: 10.3934/proc.2005.2005.443
In this paper, we discuss the asymptotic behavior of some solutions for nonlinear parabolic equation in ${\Bbb R}^N$ involving critical Sobolev exponent. For the subcritical problem (with bounded domain), it is well-known that the solution which intersects the "stable set" must be a global one. But for the critical problem, it is not known whether the same conclusion holds or not. In this paper, we shall show that, in the critical case, the same conclusion actually holds true. The proof requires the concentration compactness type argument.
keywords: lack of compactness Stable set critical Sobolev exponent concentration-compactness principle.
DCDS
Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms
Michinori Ishiwata Makoto Nakamura Hidemitsu Wadade
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 4889-4903 doi: 10.3934/dcds.2015.35.4889
The Cauchy problem of Klein-Gordon equations is considered for power and exponential type nonlinear terms with singular weights. Time local and global solutions are shown to exist in the energy class. The Caffarelli-Kohn-Nirenberg inequality and the Trudinger-Moser type inequality with singular weights are applied to the problem.
keywords: Brézis-Gallouët-Wainger inequality. Caffarelli-Kohn-Nirenberg inequality weighted Trudinger-Moser inequality Klein-Gordon equation

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