Abstract
We prove the existence of a unique solution of the following Neumann problem \(u_t = (\phi_m(u))_{xx}\), u > 0, in (a, b) × (0, T), u(x, 0) = u 0(x) ≥ 0 in (a, b), \((\phi_m(u))_x(a,t) = g(t)\) and \((\phi_m(u))_x(b,t)=-f(t) \forall 0 < t < T\), where \(\phi_m(u) = u^m/m\) if m < 0, \(\phi_m(u) = \log u\) if m = 0, and \(T = \sup\{t' > 0: \int_{-a}^bu_0dx > \int_0^{t'}(f+g)\,dt\}\) m≤ 0, \(0 \le f, g \in L_{loc}^{\infty}([0,\infty))\), \(0\le u_0\in L^{\infty}(a,b)\) and the case −1 < m ≤ 0, \(f, g \in L_{loc}^{\infty}([0,\infty))\), \(0 \le u_0 \in L^p(a,b)\) for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in \(\mathbb{R} \times (0,T)\) for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity.
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Hui, K.M. Existence of solutions of the very fast diffusion equation in bounded and unbounded domain. Math. Ann. 339, 395–443 (2007). https://doi.org/10.1007/s00208-007-0119-x
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DOI: https://doi.org/10.1007/s00208-007-0119-x