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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Global solutions for a nonlinear integral equation with a generalized heat kernel

Pages: 767 - 783, Volume 7, Issue 4, August 2014      doi:10.3934/dcdss.2014.7.767

 
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Kazuhiro Ishige - Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan (email)
Tatsuki Kawakami - Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan (email)
Kanako Kobayashi - Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan (email)

Abstract: We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad +\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \end{eqnarray*} where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.

Keywords:  Global solutions, generalized heat kernel, semilinear parabolic equations, weak $L^r$ space, nonlinear integral equation.
Mathematics Subject Classification:  Primary: 35A01, 35K55; Secondary: 35K91.

Received: September 2013;      Available Online: February 2014.

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