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Existence of solutions of the very fast diffusion equation in bounded and unbounded domain. (English) Zbl 1145.35075
The paper is devoted to proving the existence of a unique solution of the problem \[ \begin{aligned} u_t & = (\varphi_m(u))_{xx} \quad \text{in } (a,b)\times (0,T),\\ u(x,0) & = u_0(x) \geq 0\quad \text{in }\;(a,b),\\ (\varphi_m(u))_x(a,t) & = g(t), (\varphi_m(u))_x(b,t) = - f(t), \quad 0 < t < T, \end{aligned} \] where \(\varphi_m(u) = u^m/m\) if \(m < 0\), \(\varphi_m = \log u\) if \(m = 0\), and
\[ T = \sup \left\{t' > 0:\int_{-a}^bu_0(x)\,dx > \int_0^{t'}(f + g)\,dt \right\} \] for the case \(m \leq 0\), \(0 \leq f,g \in L_{\text{loc}}^{\infty}([0,\infty))\), \(0 \leq u_0 \in L^{\infty}\) and the case \(-1 < m \leq 0\), \(f\),\(g \in L^{\infty}_{\text{loc}}(a,b)\) for some \(p > 1 - m\). This result is extended for the case of higher dimension. The exact decay rate of the solution obtained at infinity is given. Moreover, the author gives a new proof of a result obtained by Rodriguez and Vazquez about the existence of many finite mass solutions of the above equation in \(\mathbb R \times (0,T)\) for \(-1 < m \leq 0\).

MSC:
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B25 Singular perturbations in context of PDEs
35K55 Nonlinear parabolic equations
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