Fujita-type phenomenon of nonlinear coupled nonlocal diffusion system

Abstract

This paper studies a Cauchy problem for a nonlinear coupled nonlocal diffusion system ut=J⁎u−u+vp, vt=J⁎v−v+uq. We first deal with the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. It turns out that the results are coincident with the classical heat system. We not only generalize the results in the scalar case obtained by J. García-Melián and F. Quirós (2010) [6] to the case of a system, but also delete the monotonicity condition on the kernel function J. The key techniques are modifying the rescaled test function method with a new test function rather than the eigenfunction of the nonlocal operator to deal with blow-up of the solution, and selecting a suitable Banach space via the estimates obtained by J. Terra and N. Wolanski (2011) [13] to obtain the global solutions.