Global in-time behavior for the semilinear nonlocal heat exchanger system with different diffusion orders

Abstract

We consider a semilinear nonlocal heat exchanger system whose two equations may have different fractional diffusion orders. Although these orders need not coincide, both components have the same leading diffusion for large time. Its order is ${\sigma }_{\ast }:=\min \{{\sigma }_1,{\sigma }_2\}$, while its coefficient depends on which equation contains the lower order. If ${\sigma }_1\ne {\sigma }_2$, we further estimate the difference between the linear solution and the effective fractional heat flow. The additional decay changes at the threshold ${\sigma }^{\ast }=2{\sigma }_{\ast }$, where ${\sigma }^{\ast }:=\max \{{\sigma }_1,{\sigma }_2\}$. Based on these linear estimates, we prove global in-time existence for small data if $\min \{p,q\}>1+\frac{2{\sigma }_{\ast }}{n}$, derive the corresponding $L^m$ asymptotic profiles, and obtain lower bounds for the lifespan in the sub-critical and critical cases.

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