Asymptotic behavior of the solution of a nonlinear integro-differential diffusion equation. (English) Zbl 1172.35500
Differ. Equ. 44, No. 4, 538-550 (2008); translation from Differ. Uravn. 44, No. 4, 517-539 (2008).
Summary: We study the asymptotic behavior as \(t\to \infty\) of the solution of an initial-boundary value problem for the singular nonlinear integrodifferential equation
\[
\frac{\partial U}{\partial t}=\frac{\partial}{\partial x}\bigg[a\bigg( \int^t_0\bigg(\frac{\partial U}{\partial x}\bigg)^2d\tau\bigg)\frac {\partial U}{\partial x}\bigg ],
\]
where \(a(S)=(1+S)^p\), \(0<p\leq 1\). We consider problems with both homogeneous boundary conditions and a nonhomogeneous boundary condition on part of the boundary. We establish the orders of convergence of their solutions.
MSC:
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 45J05 | Integro-ordinary differential equations |
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