Existence and nonexistence of the global solution for the semilinear parabolic system on Riemannian manifold Original Research Article

In this paper, we study the global existence and nonexistence of solutions to the following semilinear parabolic system View the MathML source{ut−Δu=vpin Mn×(0,∞),vt−Δv=uqin Mn×(0,∞),u(x,0)=u0(x)in Mn,v(x,0)=v0(x)in Mn, Turn MathJax on where MnMn (n≥3n≥3) is a non-compact complete Riemannian manifold, ΔΔ is the Laplace–Beltrami operator. We assume that both u0(x)u0(x) and v0(x)v0(x) are nonnegative, smooth and bounded functions, constants p,q≥1p,q≥1.