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.


35K55 Nonlinear parabolic equations
45K05 Integro-partial differential equations
Full Text: DOI


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