Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations

Abstract

Abstract In this paper, we are interested in studying the Cauchy problem for a weakly coupled system of two semi-linear structurally damped σ k {\sigma_{k}} -evolution equations, where σ k ≥ 1 {\sigma_{k}\geq 1} for k = 1 , 2 {k=1,2} . Our first purpose involves the proof of global (in time) existence of small data energy solutions by mixing additional L m k {L^{m_{k}}} regularity for the data, where m k ∈ [ 1 , 2 ) {m_{k}\in[1,2)} . We want to point out that in some cases of choosing suitable parameters m k {m_{k}} , with k = 1 , 2 {k=1,2} , the obtained lower bound of one exponent p or q related to power nonlinearities on the right-hand sides is really smaller than the critical exponent, the so-called modified Fujita exponent. The second aim of this paper is to discuss a blow-up result for Sobolev solutions with any different fractional values of σ k ≥ 1 {\sigma_{k}\geq 1} when m 1 = m 2 {m_{1}=m_{2}} .