Global existence results for coupled parabolic systems with general sources and some extensions involving degenerate coefficients

Abstract

This work presents optimal conditions for the existence of global solutions for the coupled parabolic system: $u_t-\Delta u=h\left(t\right)f\left(v\right)$ and $v_t-\Delta v=l\left(t\right)g\left(u\right)$ in $\Omega \times (0,T)$ with the Dirichlet boundary conditions. Here, $\Omega \subset R^N$ is a bounded or unbounded domain, the initial data belong to ${\left[C_0\right(\Omega \left)\right]}^2$, and the functions $f,g,h,l\in C[0,\infty )$. We also study the coupled parabolic system with degenerate coefficients: $u_t-\mathrm{div}\left(\omega \right(x\left)\nabla u\right)=h\left(t\right)f\left(v\right)$ and $v_t-\mathrm{div}\left(\omega \right(x\left)\nabla v\right)=l\left(t\right)g\left(u\right)$ in $R^N\times (0,T)$, where ω belong to the class $A_2$ of Muckenhoupt functions and may exhibit singularities along the line $x_1=0$. This problem is related to the fractional Laplacian through the Caffarelli-Silvestre extension given in [2]. In addition, critical Fujita-type exponents are derived for both systems.