Blowup estimates for a parabolic system in a three-species cooperating model
The authors study the blow-up estimates for the solutions of the following semilinear parabolic system \[ \begin{alignedat}{3} &u_{1t}- d_1 \Delta u_1= u_1(a_1- c_1u_1+ e_1u_2) && &\quad\text{in } &\Omega\times (0,T),\\ &u_{2t}- d_2 \Delta u_2= u_2 (a_2+ b_2u_1- c_2u_2+ e_2u_3) && &\quad\text{in } &\Omega\times (0,T),\\ &u_{3t}- d_3 \Delta u_3= u_3(a_3+ b_3u_2- c_3u_3) && &\quad\text{in }&\Omega\times (0,T),\\ &u_i(x,t)= 0 &\text{for }&i=1,2,3 &\quad \text{on } &\partial\Omega\times (0,T),\\ &u_i(x,0)= u_{i,0}(x) \quad &\text{for }&i=1,2,4 &\quad \text{in }&\Omega. \end{alignedat} \] The authors derive the upper and lower bound for blow-up rates in the case \(N=1\). Note that \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\). Moreover the comparison principle for a bounded and unbounded domain are given and the corresponding scalar problem is studied.
scientific article; zbMATH DE number 2082170