Ishige, Kazuhiro; Kawakami, Tatsuki; Kobayashi, Kanako Global solutions for a nonlinear integral equation with a generalized heat kernel. (English) Zbl 1288.45004 Discrete Contin. Dyn. Syst., Ser. S 7, No. 4, 767-783 (2014). Summary: We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \[ u(x,t)=\int_{\mathbb R^N} G(x-y,t)\varphi(y)dy +\int_0^t \int_{\mathbb R^N} G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \] 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. Cited in 21 Documents MSC: 45G10 Other nonlinear integral equations 35K58 Semilinear parabolic equations 35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian Keywords:global solutions; generalized heat kernel; semilinear parabolic equations; weak \(L^r\) space; nonlinear integral equation; asymptotics; Cauchy problem; fractional semilinear parabolic equations; viscous Hamilton-Jacobi equations PDFBibTeX XMLCite \textit{K. Ishige} et al., Discrete Contin. Dyn. Syst., Ser. S 7, No. 4, 767--783 (2014; Zbl 1288.45004) Full Text: DOI References: [1] L. Amour, Global existence and decay for viscous Hamilton-Jacobi equations,, Nonlinear Anal., 31, 621 (1998) · Zbl 1023.35049 [2] D. G. Aronson, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30, 33 (1978) · Zbl 0407.92014 [3] P. Biler, Fractal Burgers equations,, J. Differential Equations, 148, 9 (1998) · Zbl 0911.35100 [4] P. 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