A note on the global solutions of a degenerate parabolic system
The authors deal with the Cauchy problem of the degenerate parabolic system \[ \partial_t u=\Delta u^\alpha +v^p\;x\in\mathbb{R}^n,\;t>0, \quad \partial_t v=\Delta v^\beta +u^q \tag{*} \] with \(u(x,0)= u_0(x)\geq 0\), \(v(x,0)= v_0(x)\geq 0\). They establish the following theorem about the ``very fast diffusion'' case: \(0<\alpha\), \(\beta< (N-2)_+/N\), where \((M)_+ =\text{Max} \{M,0\}\). Theorem. When \(0<\alpha\), \(\beta< (N-2)_+/N\), \(p,q>1\), there exist nontrivial global solutions of the problem (*) for small initial functions.
scientific article; zbMATH DE number 1054591