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

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.