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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
##### Keywords:
Neumann boundary problem; exact decay rate
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##### References:
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