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Blowup for nonlocal nonlinear diffusion equations with Dirichlet condition and a source. (English) Zbl 1470.35189

Summary: This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source \(u_t(x, t) = \int_{- \infty}^{+ \infty} J((x - y) / u(y, t)) d y - u(x, t) + u^p(x, t)\), \(x \in(- L, L)\), \(t > 0\), \(u(x, t) = 0\), \(x \notin(- L, L)\), \(t \geq 0\), and \(u(x, 0) = u_0(x) \geq 0\), \(x \in(- L, L)\), which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.

MSC:

35K55 Nonlinear parabolic equations
45K05 Integro-partial differential equations
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