Existence and nonexistence of Fujita-type critical exponents for isotropic and anisotropic semi-linear parabolic systems (Q5957737)

The paper is concerned with conditions for blow-up in finite time for solutions with non-negative non-trivial initial data of the following system of two degenerate parabolic equations: NEWLINE\[NEWLINEu_t=\Delta_1 u+v^p,\;v_t=\Delta_2 v+u^q\text{ in }\mathbb{R}^N \times[0,T).NEWLINE\]NEWLINE Here \(p,q\geq 1\) and \(pq>1\) holds and \(\Delta_i\) is the Laplacian in a subspace \(R_i\cong \mathbb{R}^{N_i}\) such that \(R_1+R_2= \mathbb{R}^N\). The main result states that blow-up occurs for any of the initial data above whenever NEWLINE\[NEWLINE\max\left({p+1 \over pq-1}-{N_1\over 2}-{N-N_1\over 2q}, {q+1\over pq-1}-{N_2\over 2}-{N-N_2 \over 2p}\right)>0.NEWLINE\]