Blow up of fractional reaction-diffusion systems with and without convection terms (Q1751565)

The author studies blow up in finite time of positive mild solutions \((u_1,u_2)\) to the Cauchy problem for fractional reaction-diffusion system \[ \begin{aligned} \partial_t u_1(t,x) = ( \Delta_{\alpha_1} + b_1(x) \cdot \nabla ) u_1(t,x) + u_2^{\beta_1}(t,x), \quad t>0,\: x\in\mathbb{R}^d,\\ \partial_t u_2(t,x) = ( \Delta_{\alpha_2} + b_2(x) \cdot \nabla ) u_2(t,x) + u_1^{\beta_2}(t,x), \quad t>0,\: x\in\mathbb{R}^d,\\ u_i(0,x) = f_i(x), \quad x\in\mathbb{R}^d,\: i=1,2, \end{aligned} \] where \(d\geq 1\), \(\beta_i > 1\), \(\alpha_i\in(1,2)\), \(\Delta_{\alpha_i} = -(-\Delta)^{\alpha_i/2}\) is the fractional power of the Laplacian, and \(f_i\) are nonnegative, not identically zero, bounded continuous functions. It is shown that the blow up properties of the Cauchy problem with and without convection terms \(b_i\) are the same, provided each \(b_i\) belongs to the Kato class \(\mathcal{K}_d^{\alpha_i-1}\) on \(\mathbb{R}^d\). The author also considers the Cauchy problem in the convectionless case, i.e. for \(b_1 = b_2 = 0\), and obtains some sufficient conditions for blow up in finite time of its positive mild solutions.