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On the Cauchy problem for the fast diffusion equation. (English) Zbl 1140.35496
Summary: For \(u_0\), \({1\over u_0}\in L^1_{\text{loc}}(\mathbb{R}^n)\), the author studies the existence of a kind of weak solution to the Cauchy problem
\[ \begin{aligned} u_t= \text{div}(u^{m-1} Du)&\quad\text{in }\mathbb{R}^n\times (0,T],\\ u(x,0)= u_0(x)\geq 0 &\quad\text{in }\mathbb{R}^N, \end{aligned} \] where \(m< 0\) is a constant. The uniqueness and regularity of solutions are also discussed.
MSC:
35K65 Degenerate parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
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