Abstract
Let \({N \geq 3}\) and u be the solution of u t = Δ log u in \({\mathbb{R}^N \times (0, T)}\) with initial value u 0 satisfying \({B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}\) for some constants k 1 > k 2 > 0 where \({B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}\) is the Barenblatt solution for the equation and \({u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}\) for some constant k 0 > 0 if \({N \geq 4}\). We give a new different proof on the uniform convergence and \({L^1(\mathbb{R}^N)}\) convergence of the rescaled function \({\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}\), on \({\mathbb{R}^N}\) to the rescaled Barenblatt solution \({\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}\) for some k 0 > 0 as \({s \rightarrow \infty}\). When \({N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}\) in \({\mathbb{R}^N}\), and \({|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}\) for some constant k 0 > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L 1 space in \({\mathbb{R}^N}\) of the rescaled solution \({\tilde{u}(x, s)}\) to \({\tilde{B}_{k_0}(x)}\) as \({s \rightarrow \infty}\).
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Hui, K.M., Kim, S. Extinction profile of the logarithmic diffusion equation. manuscripta math. 143, 491–524 (2014). https://doi.org/10.1007/s00229-013-0634-x
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DOI: https://doi.org/10.1007/s00229-013-0634-x