Global solutions for a nonlinear integral equation with a generalized heat kernel
Global solutions for a nonlinear integral equation with a generalized heat kernel
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DOI：
10.3934/dcdss.2014.7.767
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发表时间：
2014-02
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影响因子：
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通讯作者：
Kazuhiro Ishige;Tatsuki Kawakami;Kanako Kobayashi
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文献类型：
--
作者：
Kazuhiro Ishige;Tatsuki Kawakami;Kanako Kobayashi
研究了一类广义热核方程\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*}的全局时解的存在性和大时性，其中$\varphi\in W^{\ell,\infty}({\mathbb R}^N)$和$\ell\in\{0,1,\dots\}$。本文的论点适用于各种非线性抛物方程如分数阶半线性抛物方程、高阶半线性抛物方程和粘性Hamilton-Jacobi方程的Cauchy问题。
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.